37965893 Bansal CLasses Physics Study Material for IIT JEE

BANSAL CLASSES TARGET LIT JEE 2007 XI (PQRS) CALORIMETRY & HEAT TRANSFER CONTENTS KEYCONCEPT EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY

THERMAL Definition of Heat: EXPANSION Heat is a form of energy which is transferred between a system and its surroundi ng as a result of temperature difference only. due to increase in temperature. F or temperature change At change in length Al = l0a At Area AA= A^At volume AV = V yAt 0 Thermal Expansion : Expansion 1. Type of thermal expansion Coefficient of expansion (i) Linear (ii) Superficial (iii) Volume (a) (b) 2. . a = At—>0 / 1 A/t Lim A T 0 P = Lim 1 AA At—A0 At y = At—>o v1 AV Lim At 0 For isotropic solids otj = a = a = olids p = otj + a and y = a, + a + pansion in X , Y and Z directions. rature volume increases so density 2 3 2 2 3 2 3 0 3

a (let) so P =2a and y = 3a For anisotropic s a Here , a and a are coefficient of linear ex Variation in density : With increase of tempe decreases and vice-versa. H d =(1 + yAt)

Note For solids values of y are generally small so we can write d = d (1-yAt) (u sing bimomial expansion) (0 (ii) y for liquids are in order of 10~ For water den sity increases from 0 to 4°C so y is -ve (0 to 4° C) and for 4° C to higher temperatur e y is +ve. At 4° C density is maximum. 3. Thermal Stress: Arod of length 1 is cla mped between two fixed walls with distance 1 . If temperature is changed by amou nt At then F stress A (area assumed to be constant) 0 0 : so, or A/ strain = I F/A F/ Y = A/// AAI F =YAa A t 0 0 F AaAt (!l Bansal Classes Calorimetry & Heat Transfer [3]

4. If a is not constant (i) (a varies with distance) Let a = ax+b Total expansion = Jexpansion of length dx i = |(ax + b)dxAt " x 1 (ii) ( a varies with tempearture) Let a = f (T) T2 0 dx A/ _ j"a/ dT T i Caution: If a is in °C then put Tj and T in °C. similarly if a is i n K then put Tj and T in K. 2 2 CAL ORIMETR Quantity of heat transfered and specific heat Y The amount ofheat needed to incerase the temperature of 1 gmofwaterfrom 14.5°Cto 1 5.5°CatSTP is 1 calorie dQ = mcdT Q = m [ C dT (be careful about unit of temperatu re, use units according to the given units of C) T i Heat transfer in phase change 'h Q = rnL L = latent heat of substance in cal/ gm/ °C or in Kcal/ kg/ °C L = 80 cal/ gm for ic e ice L steam = 5 4 0 C a l / g m (A) (i) (ii) Note: 1.

vibration and collision of medium particles. Steady State : In this state heat a bsorption stops and temperature gradient throughout the rod dT becomes constant i.e. — = constant. dx Before steady state : Temp of rod at any point changes If sp ecific heat of any substance is zero, it can be considered always in steady stat e. Let the two ends of rod of length 1 is maintained at temp Tj and T ( Tj > T ) dQ i ~ 2 I Thermal current D 1 = K-XH L T 2 2 T T 1 Conduction : Due to HEATTRANSFER Ohm's law for Thermal Conduction in Steady State : / Where thermal resistance R = K A Th 1 1 2. Differential form of Ohm's Law T-dT dQ dT — =KA— dT dx dT — = temperature gradient dx dx (!lBansal Classes Calorimetry & Heat Transfer [3]

(B) (Q 1. Heat transfer due to movement ofmedium particles. Radiation: Every body radiates electromagnetic radiation of all possible wavelength at all temp>0 K. Stefan's Law: Rate of heat emitted by a body at temp T K from per unit area E = GT J/sec/ m d = P = oAT watt Q Radiation power — dl If a body is placed in a surrounding of temperature T dQ Convection: 4 2 4 s valid only for black body heat from general body Emissmty or emmisive power e = ~ Iftemp ofbody falls by dT in time dt dT _ _ j4x (dT/dt=rate of cooling) dt ~ m S h e a t f r o m s ^ =cA(T -T ) 4 s 4 Newton's law of cooling Iftemp difference ofbody with surrounding is small i.e. T = T eA then, dT 4mS -a T ( T - T ) dt dT a ( T - T ) so dt rr3/ 2 s Average form of Newtons law of cooling If a body cools from T j to T in time 51 T - T _ K T, +T, -T (used generally in objective questions) 5t mS s 2 dt 4. mS (for better results use this generally in subjective) At every temperature (>0K) a body radiates energy radiations ofall wavelengths. According to Wein's displacement law if the wavelength corresponding to maximum energy is X then X T = b where b = is a constant (Wein's constant) T=temperature of body m m Wein's black body radiation T3>T2>T, ess (!l Bansal Classes

Calorimetry & Heat Transfer [3]

EXERCISE -1 Q. 1 An aluminium container of mass 100 gm contains 200 gm of ice at - 20°C. Heat is added to the system at the rate of 100 cal/s. Find the temperature of the sys tem after 4 minutes (specific heat of ice = 0.5 and L = 80 cal/gm, specific heat of A1 = 0.2 cal/gm/°C) Q. 2 A U-tubefilledwith a liquid ofvolumetric coefficient of 10 /°C lies in a vertical plane. The height of liquid column in the left vertic al limb is 100 cm. The liquid in the left vertical limb is maintained at a tempe rature = 0°C while the liquid in the right limb is maintained at a temperature = 1 00°C. Find the difference in levels in the two limbs. _5 Q.3 A thin walled metal tank of surface area 5m is filled with water tank and contai ns an immersion heater dissipating 1 kW. The tank is covered with 4 cm thick lay er of insulation whose thermal conductivity is 0.2 W/m/K. The outer face of the insulation is 25°C. Find the temperature of the tank in the steady state 2 Q.4 A glassflaskcontains some mercury at room temperature. It is found that at diffe rent temperatures the volume of air inside the flask remains the same. If the vo lume of mercury in the flask is 300 cm , thenfindvolume of the flask (given that coefficient of volume expansion of mercury and coefficient oflinear expansion o f glass are 1.8 x 10^(°C) and9x 10~ (°C) respectively) 3 _1 6 _1 Q.5 Q.6 Q.7 A clock pendulum made of invar has a period of 0.5 sec at 20°C. If the clock is us ed in a climate where average temperature is 30°C, aporoximately. How much fast or slow will the clock run in 10 sec. (a =lxlO /°C) 6 ilwar -6 A pan filled with hot food cools from 50.1 °C to 49.9 °C in 5 sec. How long will it take to cool from 40.1 °C to 39.9°C if room temperature is 30°C? A composite rod made of three rods of equal length and cross-section as shown in the fig. The thermal conductivities of the materials of the rods are K/2, 5K and K respectively. The end A and end B are at constant temperatures. All heat entering the face A goes out of the end B there being no loss of heat from the sides of the bar. Find th e effective thermal conductivity of the bar A I Q.8 Q.9 K/2 I 11 5K 2 6 1 K

1 B An iron bar (Young's modulus = 10 N/m , a = 10" /°C) 1 m long and 10~ m in area is heated from 0°C to 100°C without being allowed to bend or expand. Find the compress ive force developed inside the bar. 3 2 A solid copper cube and sphere, both of same mass & emissivity are heated to sam e initial temperature and kept under identical conditions. What is the ratio of their initial rate of fall of temperature? Q. 10 A cylindrical rod with one end in a stream chamber and other end in ice ca use melting of 0.1 gm of ice/sec. If the rod is replaced with another rod of hal f the length and double the radius of first and thermal conductivity of second r od is 1/4 that of first, find the rate of ice melting in gm/sec (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.ll Three aluminium rods of equal length form an equilateral triangle ABC. Taki ng O (mid point of rod BC) as the origin. Find the increase in Y-coordinate per unit change in temperature ofthe centre ofmass of the system. Assume the length of the each rod is 2m, and a = 4 v3 x10" /°C d 6 Q.12 Three conducting rods of same material and cross-section are shown in figur e. Temperature of A, D and C are maintained at 20°C, 90°C and 0°C. Find the ratio of l ength BD and BC if there is no heat flow in AB 20°C 90'C 0°C Q. 13 If two rods of layer L and 2 L having coefficients of linear expansion a a nd 2a respectively are connected so that total length becomes 3 L, determine the average coefficient of linear expansion of the composite rod. Q.14 A volume of 120 ml of drink (half alcohol + half water by mass) originally at a temperature of 25°C is cooled by adding 20 gm ice at 0°C. If all the ice melts, find the final t emperature of the drink, (density of drink = 0.833 gm/cc, specific heat of alcoh ol = 0.6 cal/gm/°C) Q.15 A solid receives heat by radiation over its surface at th e rate of 4 kW. The heat convection rate from the surface of solid to the surrou nding is 5.2 kW, and heat is generated at a rate of 1.7 kW over the volume of th e solid. The rate of change of the average temperature of the solid is 0.5 Cs . Find the heat capacity of the solid. o -1 Q.16 The figure shows the face and interface temperature of a composite slab con taining offour layers oftwo materials having identical thickness. Under steady s tate condition, find the value of temperature 6. 20°C 10°C E -5°C -10°C 2k 2k k = thermal conductivity Q.17 Two identical calorimeter A and B contain equal quantity of water at 20°C. A 5 gm piece of metal X of specific heat 0.2 cal g (C°) is dropped into A and a 5 gm piece of metal Y into B. The equilibrium temperature in A is 22°C and in B 23°C. Th e initial temperature of both the metals is 40°C. Find the specific heat of metal Y in cal g" (C°)~ 4 _1 1 l Q.18 Two spheres of same radius R have their densities in the ration 8 . 1 and t he ratio of their specific heats are 1 : 4. If by radiation their rates of fall of temperature are same, thenfindthe ratio of their rates of losing heat. Q.19 I n the square frame of side I of metallic rods, the corners A and C are maintaine d at Tj and T respectively. The rate of heat flow from A to Cisa. IfA and D are instead maintained Tj & T respectivleyfind,findthe total rate ofheat flow. 2 2 Q.20 A hot liquid contained in a container of negligible heat capacity loses tem perature at rate 3 K/min, just before it begins to solidify. The temperature rem ains constant for 30 min, Find the ratio of specific heat capacity of liquid to

specific latent heat of fusion is in Kr (given that rate of losing heat is const ant). 1 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q. 21 A thermostatted chamber at small height h above earth's surface maintained at 30°C has a clock fitted in it with an uncompensated pendulum. The clock design er correctly designs it for height h, but for temperature of 20°C. Ifthis chamber is taken to earth's surface, the clock in it would click correct time. Find the coefficient oflinear expansion ofmaterial of pendulum, (earth's radius is R) Q.2 2 The coefficient of volume expansion of mercury is 20 times the coefficient of linear expansion of glass. Find the volume of mercury that must be poured into a glass vessel ofvolume V so that the volume above mercury may remain constant at all temperature. Q. 23 Two 50 gm ice cubes are dropped into 250 gm ofwater ion a glass. Ifthe water was initially at a temperature of 25°C and the temperature of ice -15°C. Find the final temperature of water, (specific heat ofice = 0.5 cal/gm /°C and L = 80 cal/gm) Q.24 Water is heated from 10°C to 90°C in a residential hot wat er heater at a rate of 70 litre per minute. Natural gas with a density of 1.2 kg /m is used in the heater, which has a transfer efficiency of 32%. Find the gas c onsumption rate in cubic meters per hour, (heat combustion for natural gas is 84 00 kcal/kg) 3 Q.25 A metal rod A of 25cm lengths expands by 0.050cm. When its temperature is r aised from 0°C to 100°C. Another rod B of a different metal of length 40cm expands b y 0.040 cm for the same rise in temperature. A third rod C of 50cm length is mad e up of pieces of rods A and B placed end to end expands by 0.03 cm on heating f rom 0°C to 50°C. Find the lengths of each portion of the composite rod. Q.26 A subst ance is in the solid form at 0°C. The amount of heat added to this substance and i ts temperature are plotted in the following graph. If the relative specific heat capacity of the solid substance is 0.5, find from the graph (i) the mass of the substance; (ii) the specific latent heat of the melting process, and (iii) the specific heat of the substance in the liquid state. Q. 27 One end of copper rod ofuniform cross-section and of length 1.5 meters is in contact with melting ice and the other end with boiling water. At what point along its length should a te mperature of200°C be maintained, so that in steady state, the mass ofice melting i s equal to that of steam produced in the same interval oftime? Assume that the w hole system is insulatedfromthe surroundings. Q.28 Two solids spheres are heated to the same temperature and allowed to cool under identical conditions. Compare : (i) initial rates of fall of temperature, and (ii) initial rates of loss of he at. Assume that all the surfaces have the same emissivity and ratios of their ra dii of, specific heats and densities are respectively 1 : a, 1 : p, 1 : y. Q.29 A vessel containing 100 gm water at 0°C is suspended in the middle of a room. In 1 5 minutes the temperature of the water rises by 2°C. When an equal amount of ice i s placed in the vessel, it melts in 10 hours. Calculate the specific heat offusi on ofice. Q. 3 0 The maximum in the energy distribution spectrum of the sun is a t 4753 A and its temperature is 6050K. What will be the temperature of the star whose energy distribution shows a maximum at 9506 A. (!l Bansal Classes Calorimetry & Heat Transfer [3]

EXERCISE-II Q. 1 A copper calorimeter of mass 100 gm contains 200 gm of a mixture of ice and water. Steam at 100°C under normal pressure is passed into the calorimeter and th e temperature of the mixture is allowed to rise to 50°C. If the mass of the calori meter and its contents is now 330 gm, what was the ratio of ice and water in the beginning? Neglect heat losses. Given : Specific heat capacity of copper = 0.42 x 10 J kg K" , Specific heat capacity of water = 4.2 x 10 J kg^Kr , Specific he at of fusion of ice = 3.36 x 10 J kg Latent heat of condensation of steam = 22.5 x 1Q Jkg" 3 _1 x 3 1 5 -1 5 1 Q.2 base and two thin rods each of length l and coefficient of linear expansion a fo r the two pieces, ifthe distance between the apex and the midpoint ofthe base re main unchanged as the temperatures /, varied show that 7 2 2 l A n isoscetes triangte is form ed w ith a rod of length l and coefficient of linea r expansion OTJ for the x 2 Q.3 A solid substance of mass 10 gm at - 10°C was heated to - 2°C (still in the solid st ate). The heat required was 64 calories. Another 880 calories was required to ra ise the temperature ofthe substance (now in the liquid state) to 1°C, while 900 ca lories was required to raise the temperature from -2°C to 3°C. Calculate the specifi c heat capacities of the substances in the solid and liquid state in calories pe r kilogram per kelvin. Show that the latent heat of fusion L is related to the m elting point temperature t by L = 85400 + 200 t . m m Q.4 (a) (b) Q. 5 Q.6 Q. 7 A steel drill making 180 rpm is used to drill a hole in a block of steel. The ma ss of the steel block and the drill is 180 gm. If the entire mechanical work is used up in producing heat and the rate of raise in temperature of the block and the drill is 0.5 °C/s. Find the rate of working of the drill in watts, and the tor que required to drive the drill. Specific heat of steel = 0.1 and J = 4.2 J/cal. Use ;P = i o A brass rod of mass m = 4.25 kg and a cross sectional area 5 cm in creases its length by 0.3 mm upon heatingfrom0°C. What amount ofheat is spent for heating the rod? The coefficient of linear expansic 1 for brass is 2xl0 /K, its specific heat is 0.39 kJ/kg.K and the density of brass is 8.5 x 10 kg/m . A subm arine made of steel weighing 10 g has to take 10 g of water in order to submerge when the temperature of the sea is 10°C. How much less water it will have to take in when the sea is at 15°C? (Coefficient of cubic expansion of sea water = 2 x 10 "V°C, coefficient of linear expansion of steel = 1.2 x 10- /°C) A flow calorimeter i s used to measure the specific heat of a liquid. Heat is added at a known rate t o a stream of the liquid as it passes through the calorimeter at a known rate. T hen a measurement of the resulting temperature difference between the inflow and the outflow points of the liquid stream enables us to compute the specific heat of the liquid. A liquid of density 0.2 g/cm flows through a calorimeter at the rate of 10 cm /s. Heat is added by means of a 250-W electric heating coil, and a temperature difference of 25 °C is established in steady-state conditions between

the inflow and the outflow points. Find the specific heat of the liquid. 2 -5 3 3 9 8 5 3 3 (!lBansalClasses Calorimetry & Heat Transfer [3]

Q.8 Toluene liquid of volume 300 cm at 0°C is contained in a beaker an another quantit y of toluene of volume 110 cm at 100°C is in another beaker. (The combined volume is 410 cm ). Determine the total volume of the mixture ofthe toluene liquids whe n they are mixed together. Given the coefficient of volume expansion y = 0.001/C and all forms of heat losses can be ignored. Also find the final temperature of the mixture. Q. 9 Ice at -20°C isfilledupto height h = 10 cm in a uniform cylindr ical vessel. Water at temperature 9°C is filled in another identical vessel upto t he same height h= 10 cm. Now, water from second vessel is poured into first vess el and it is found that level of upper surface falls through Ah = 0. 5 cm when t hermal equilibrium is reached. Neglecting thermal capacity of vessels, change in density of water due to change in temperature and loss of heat due to radiation , calculate initial temperature 0 of water. Given, Density of water, p = 1 gm cm Density of ice, p. =0.9gm/cm Specific heat of water, s = 1 cal/gm °C Specific hea t of ice, s = 0.5 cal/gm°C Specific latent heat of ice, L = 80 cal/gm Q. 10 A comp osite body consists of two rectangular plates of the same dimensions but differe nt thermal conductivities K and Kg. This body is used to transfer heat between t wo objects maintained at different temperatures. The composite body can be place d such that flow of heat takes place either parallel to the interface or perpend icular to it. Calculate the effective thermal conductivities K. and Kj Of the co mposite body for the parallel and perpendicular orientations. Which orientation will have more thermal conductivity? 3 3 3 w -3 3 w ; A Q. 11 Two identical thermally insulated vessels, each containing n mole of an id eal monatomic gas, are interconnected by a rod of length I and cross-sectional a rea A. Material of the rod has thermal conductivity K and its lateral surface is thermally insulated. If, at initial moment (t = 0), temperature of gas in two v essels is T, and T (< T ), neglecting thermal capacity of the rod, calculate dif ference between temperature of gas in two vessels as a function of time. 2 } Q. 12 A highly conducting solid cylinder of radius a and length I is surrounded by a co-axial layer of a material having thermal conductivity K and negligible h eat capacity. Temperature of surrounding space (out side the layer) is T , which is higher than temperature of the cylinder. If heat capacity per unit volume of cylinder material is s and outer radius of the layer is b, calculate time requi red to increase temperature of the cylinder from T to T Assume end faces to be t hermally insulated. 0 t r Q. 13 A vertical brick duct(tube) is filled with cast iron. The lower end of the duct is maintained at a temperature T, which is greater than the melting point T of cast iron and the upper end at a temperature T which is less than the tempe rature ofthe melting point of cast iron. It is given that the conductivity of li quid cast iron is equal to k times the conductivity of solid cast iron. Determin e the fraction ofthe duct filled with molten metal. Q.14 Water is filled in a no n-conducting cylindrical vessel of uniform cross-sectional area. Height of water column is h and temperature is 0°C. Ifthe vessel is exposed to an atmosphere havi ng constant temperature of- 0°C (< 0°C) at t = 0, calculate total height h ofthe col umn at time t .Assume thermal conductivity ofice to be equal to K.Density ofwate r is p and that of ice is p.. Latent heat offusion ofice isL. m 2 0 ffi (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.15 A lagged stick of cross section area 1 cm and length 1 m is initially at a temperature of 0°C. It is then kept between 2 reservoirs of tempeature 100°C and 0°C. Specific heat capacity is 10 J/kg°C and linear mass density is 2 kg/m. Find 100°C o°c (a) temperature gradient along the rod in steady state. (b) total heat absorbed by the rod to reach steady state. Q.16 A cylindrical block of length 0.4 m an ar ea of cross-section 0.04m is placed coaxially on a thin metal disc ofmass 0.4 kg and ofthe same cross-section. The upper face of the cylinder is maintained at a constant temperature of 400K and the initial temperature of the disc is 300K. I f the thermal conductivity of the material of the cylinder is 10 watt/m-K and th e specific heat of the material of the disc in 600 J/kg-K, how long will it take for the temperature of the disc to increase to 350K? Assume, for purposes of ca lculation, the thermal conductivity of the disc to be very high and the system t o be thermally insulated except for the upper face of the cylinder. 2 2 Q.17 A copper calorimeter of negligible thermal capacity isfilledwith a liquid. The mass of the liquid equals 250 gm. A heating element of negligible thermal ca pacity is immersed in the liquid. It is found that the temperature of the calori meter and its contents risesfrom25°C to 30°C in 5 minutes when a or rent of 20.5 amp ere is passed through it at potential difference of 5 volts. The liquid is throw n off and the heater is again switched on. It is now found that the temperature ofthe calorimeter alone is constantly maintained at 32°C when the current through the heater is 7A at the potential difference 6 volts. Calculate the specific hea t capacity ofthe liquid. The temperature ofthe surroundings is 25°C. Q.18 A solid copper sphere cools at the rate of 2.8°C per minute, when its temperature is 127°C. Find the rate at which another solid copper sphere oftwice the radius lose its t emperature at 327°C, ifin both the cases, the room temperature is maintained at 27°C . Q.19 A calorimeter contains 100 cm of a liquid of density 0.88 g/cm in which a re immersed a thermometer and a small heating coil. The effective water equivale nt of calorimeter, thermometer and heater may be taken to be 13 gm. Current of 2 A is passed through the coil. The potential difference across the coil is 6.3 V and the ultimate steady state temperature is 55°C. The current is increased so th at the temperature rises slightly above 55°C, and then it is switched off. The cal orimeter and the content are found to cool at the rate of 3.6°C/min. (a) Find the specific heat of the liquid. (b) The room temperature during the experiment was 10°C. If the room temperature rises to 26°C, find the current required to keep the l iquid at 55°C. You may assume that Newton's law is obeyed and the resistance of th e heater remains constant. 3 3 Q.20 End A of a rod AB of length L = 0.5 m and of uniform cross-sectional area i s maintained at some constant temperature. The heat conductivity of the rod is k = 17 J/s-rn°K. The other end B of this rod is radiating energy into vacuum and th e wavelength with maximum energy density emitted from this end is XQ = 75000 A. If the emissivity of the end B is e = 1, determine the temperature of the end A. Assuming that except the ends, the rod is thermally insulated. Q.21 A wire of l ength 1.0 m and radius 10" m is carrying a heavy current and is assumed to radia te as a blackbody. At equilibrium temperature of wire is 900 K while that of the surroundings is 300 K. The resistivity of the material of the wire at 300 K is n x 10" O-m and its temperature coefficient of resistance is 7.8 x 10' /°C. Find t he current in the wire, [a = 5.68 x 10" w/m K ]. 3 2 8 3 8 2 4 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.22 The temperature distribution of solar radiation is more or less same as tha t of a black body whose maximum emission corresponds to the wavelength 0.483 jam . Find the rate of change of mass due to radiation. [Radius of Sun = 7.0 x 10 m] 8 Q.23 A black plane surface at a constant high temperature T , is parallel to ano ther black plane surface at constant lower temperature T . Between the plates is vacuum. In order to reduce the heatflowdue to radiation, a heat shield consisti ng oftwo thin black plates, thermally isolated from each other, it placed betwee n the warm and the cold surfaces and parallel to these. After some time stationa ry conditions are obtained. By what factor r) is the stationary heatflowreduced due to the presence of the heat shield? Neglect end effects due to thefinitesize of the surfaces. h ; Q.24 The shell of a space station is a blackened sphere in which a temperature T = 500K is maintained due to operation of appliances of the station. Find the te mperature of the shell if the station is enveloped by a thin spherical black scr een of nearly the same radius as the radius of the shell. Blackened envelop Q.25 A liquid takes 5 minutes to coolfrom80°C to 50°C. How much time will it take to coolfrom60°C to 30°C ? The temperature of surrounding is 20°C. Use exact method. Q .2 6 Find the temperature of equilibrium of a perfectly black disc exposed normally to the Sun's ray on the surface of Earth. Imagine that it has a nonconducting b acking so that it can radiate only to hemisphere of space. Assume temperature of surface of Sun = 6200 K, radius of sun = 6.9 * 10 m, distance between the Sun a nd the Earth = 1.5 x lo m. Stefan's constant = 5.7 x i0~ W/m .K . What will be t he temperature ifboth sides of the disc are radiate? s 11 s 2 4 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q. 1 Q.2 The temperature of 100 gm of water is to be raised from 24° C to 90° C by adding ste am to it. Calculate the mass of the steam required for this purpose. [JEE '96] T wo metal cubes A & B of same size are arranged as shown in figure. The extreme e nds of the combination are maintained at the indicated temperatures. The arrange ment is thermally insulated. The coefficients of thermal conductivity of A & B a re 300 W/m°C and 200 W/m°C respectively. After steady state is reached the temperatu re T of the interface will be . [JEE' 96] 2 EXERCISE - III o A B Q.3 A double pane window used for insulating a room thermally from outside consists of two glass sheets each of area 1 m and thickness 0.01 m separated by a 0.05m t hick stagnant air space. In the steady state, the room glass interface and the g lass outdoor interface are at constant temperatures of 27°C and 0°C respectively. Ca lculate the rate of heat flow through the window pane. Also find the temperature s of other interfaces. Given thermal conductivities of glass and air as 0.8 and 0.08 W nr'K- respectively. [JEE'97] 1 Q. 4 The apparatus shown in the figure consists of four glass columns connected by ho rizontal sections. The height of two central columns B & C are 49 cm each. The t wo outer columns A & D are open to the atmosphere. A & C are maintained at a tem perature of 95° C while the columns B & D are maintained at 5° C. The height of the liquid in A & D measured from the base line are 52.8 cm & 51 cm respectively. De termine the coefficient ofthermal expansion ofthe liquid, [JEE '97] A 95° C 95° Q.5 Q.6 Q.7 A spherical black body with a radius of 12 cm radiates 450 W power at 500 K . If the radius were halved and the temperature doubled, the power radiated in watt would be : (A) 225 (B) 450 (C) 900 (D) 1800 Earth receives 1400 W/m of solar pow er . If all the solar energy falling on a lens of area 0.2 m is focussed on to a block of ice of mass 280 grams, the time taken to melt the ice will be minutes. (Latent heat of fusion of ice = 3.3 x 10 J/kg) [JEE '97] 2 2 5 A solid body X of heat capacity C is kept in an atmosphere whose temperature is T = 300K. At time t = 0, the temperature of X is T = 400K. It cools according to Newton's law of cooling. At time tj its temperature is found to be 3 5 OK. At t his time t the body X is connected to a larger body Y at atmospheric temperature T , through a conducting rod of length L, cross-sectional area A and thermal co

nductivity K. The heat capacity of Y is so large that any variation in its tempe rature may be neglected. The cross-sectional area A of the connecting rod is sma ll compared to the surface area of X. Find the temperature of X at time t = 3t [ JEE' 98] A 0 p A r Q.8 A black body is at a temperature of2880 K. The energy ofradiation emitted by thi s obj ect with wavelength between 499 nm and 500 nm is U between 999 nm and 1000 nm is U and between 1499 nm and 1500nmisU . TheWienconstantb = 2.88 x 10 nmK. T hen [JEE' 98] (A) Uj = 0 (B)U = 0 (C) Uj > U (D)U >U p 2 3 6 3 2 2 1 (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.9 A bimetallic strip is formed out oftwo identical strips one ofcopper and the oth er ofbrass. The coefficient of linear expansion ofthe two metals are a and ctg. On heating, the temperature ofthe strip goes up by AT and the strip bends to for m an arc of radius of curvature R. Then R is: (A) proportional at AT (B) inverse ly proportional to AT [JEE' 99] (C) proportional to lOg - a | (D) inversely prop ortional to |a - a | c c B c Q.10 A block of ice at - 10°C is slowiy heated and converted to steam at 100°C. Whic h of the following curves represents the phenomenon qualitatively? [JEE (Scr) 20 00] (A) Heat supplied (B) Heat supplied \ (C) Heat supplied (D) Heat supplied Q. 11 The plots of intensity versus wavelength for three black bodies at tempera ture T, , T and T, respectively are as shown. Thentemperatures are such that [JE E (Scr) 2000] (A)T >T >T (B) T j > T > T (C) T > T > T (C) T. > T > T 2 1 2 3 3 2 2 3 1 2 t Q.12 Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and rig ht ends are kept at 0°C and 90°C respectively. The temperature of the junction of th e three rods will be [JEE(Scr)2001 ] o°c(A) 45°C (B) 60°C (C) 30°C (D)20°C ,S0°C "90°C Q. 13 An ideal black body at room temperature is thrown into a furnace. It is ob served that (A) initially it is the darkest body and at later times the brightes t. (B) it the darkest body at all times (C) it cannot be distinguished at all ti mes. (D) initially it is the darkest body and at later times it cannot be distin guished. [JEE(Scr)2002] Q. 14 An ice cube of mass 0.1 kg at 0°C is placed in an is olated container which is at 227°C. The specific heat S of the container varies wi th temperature T according the empirical relations = A + BT, where A= 100 cal/kg -K and B = 2 x 10~ cal/kg-K . If the final temperature of the container is 27°C, d etermine the mass of the container. (Latent heat of fusion for water = 8 x \ o c al/kg. Specific heat of water = 103 cal/kg-K) [JEE' 2001] 2 2 4 Q.15 Two rods one of aluminium of length /, having coefficient of linear expansi on a , and other steel of length l having coefficient of linear expansion a are joined end to end. The expansion in both the a 2 s

[JEE (Scr) 2003] rods is same on variation of temperature. Then the value of , h is . n +/2 ac a0 (D) None of these (A) a + a (B) a s (C) Otc r a s a - a (!l Bansal Classes Calorimetry & Heat Transfer [3]

Q.16 2 kg ice at - 20°C is mixed with 5 kg water at 20°C. Thenfinalamount ofwater in the mixture would be; Given specific heat of ice = 0.5cal/g°C, specific heat ofwa ter = 1 cal/g°C, Latent heat of fusion of ice = 80 cal/g. [JEE (Scr) 2003] (A) 6 k g (B) 5 kg (C) 4 kg (D) 2 kg Q.17 If emissivity of bodies X and Y are e and e an d absorptive power are A and Ay then [JEF (Scr) 2003] (A) e > e ; Ay > A (B) e < e ; A < A (C)e >e ;A Kj_, K| = K 1 B 3 T A R V A B ; x EXERCISE-II Q.6 Q.9 Q.ll t m m m 9.02 x 10 gm 45°C 5 \n i (T, ~T )e "3 R J 2 2 ( 4KAt N | Q.12 a s. ^log 2 (-) l0geV. 0 ~ 2 J T T Q 1 3 k(T - T ) I k(T -T ) + (T -T ) 1 Q.14 h + 0

Q.17 21000 Jkg^Kr Q.20 T = 423 K a 1 - JBL V / \ 1 \ Pi f L 12k;6t Q.15 (a) 100 °C/m, (b) 1000 J Q.18 9.72°C/min Q.21 36A 0 x Q.16 166.3 sec 9 1 Q.19 (a)0.42 cal/gm°C, (b) 1.6A Q.23 r| = 3 Q.25 10 minutes Q.I Q.4 Q.7 12 gm e Q.22 ~dt = 5.06 x 10 kg/s Q.24 T" = 500 = 600 K Q.26 T = 420 K, T = 353.6 K Q.2 60° C EXERCISE-III Q.3 Q.6 0 41.53 Watt; 26.48 °C;0.55°C 5.5 min Q.14 0.5 kg Q.19 B Q.24 A 2 x 10^ C Q.5 D log 2 ; T = 300 + 50 exp. k= Q.9 B, D Q.10 Q.16 A Q.17 Q.21 C Q. 26 A Q.22 Q.27 Q.8 D Q.15 A Q.20 D Q.25 C (!l Bansal Classes [LC tj A Q.ll B Q.12 B Q.13 D A Q.18 (a) 595 watt/m , ( b ) T * 4 2 0 K K y,= 2a s Q.23 4eaLTf+K A Q.28 B 2 0 Calorimetry & Heat Transfer [3]

BA TARGET IIT JEE 2007 XII (ALL) COHTENTS KEYCONCEPTS EXERCISE-1 EXERCISE-II EXERCISE-III ANSWER KEY

KEY 1. CAPACITANCE O F A N 0 ( CONCEPTS C = 471 e e R in a medium ISOLATED SPHERICAL CONDUCTOR : C = 47C G „ R in air This sphere is at infinite distance from all the conductors. The Capacitance C = 4T E R exists between the surface of the sphere & earth . 7 Q It consists of tw o concentric spherical shells as shown infigure.Here capacitance of region betwe en the two shells is C and that outside the shell is C . We have 471 e ab C = an d C = 471 e b b-a Depending on connection, it may have different combinations of C, and -C . t 2 n 2 Q 2 SPHERICAL CAPACITOR : 3. PARALLEL PLATE CAPACITOR : If two parallel plates each of area A & separated by a distance d are charged wi th equal & opposite charge Q, then the system is called a parallel plate capacit or & its capacitance is given by, ^ S)6 A C = — ; — .in a medium C= with air as medi um r (i) UNIFORM DI-ELECTRIC M E D I U M : This result is only valid when the electricfieldbetween plates of capacitor is c onstant, (ii) M E D I U M PARTLY A I R : C = U d-lt-i r So A When a di-electric slab of thickness t & relative permittivity e is l l l l intr oduced between the plates of an air capacitor, then the distance between P3 the plates is effectively reduced by irrespective ofthe position of BSSSSii® V ^rJ the di-electric slab . (iii) COMPOSITE M E D I U M : c= GA I I -rl r2 0 r3

4. CYLINDRICAL CAPACITOR : It consist oftwo co-axial cylinders ofradii a& b, the outer conductor is earthed . The di-electric constant ofthe mediumfilledin the space between the cylinder i s Farad e . The capacitance per unit length is C = 2ne-ne m in r y r (fe^Bansal Classes CAPACITANCE 121

CONCEPT o r VARIATION OF PARAMETERS: 6. e kA , ifeither ofk, A or d varies in the region between As capacitance ofa para llel plate capacitor isC = the plates, we choose a small dc in between the plate s and for total capacitance of system. dx -, If all dC's are in parallel C = } d C If all dC's are in series 1 e k(x)A(x) 0 T J 0 COMBINATION (i) OF CAPACITORS SERIES : : In this arrangement all the capacitors when uncharged get the same charge Q but the potential difference across each will differ (if the capacitance are unequal ). 1 — +1 1 1 —+ — + + 1 (ii) CAPACITORS I N rIMHh v, v, v, Q Q Q C| C2 C3 C 3 When one plate of each capacitor is connected to the positive terminal of the ba ttery & the other plate of each capacitor is connected to the negative terminals of the battery, then the capacitors are said to be in parallel connection. The capacitors have the same potential difference, V but the charge on each one is d ifferent (if the capacitors are unequal). eq. C CAPACITORS I N PARALLEL : I + C 2 + C 3 + +c

s 1 jC3,y 1 Q + v % 1Cj.V c,,v % ENERGY Capacitance C, charge Q & potential difference V; then energy stored is 1 U = -1 CV = — QV = 1 Q . This energy is stored in the electrostatic field set up in the di-electric - — medium between the conducting plates of the capacitor . 2 2 STORED IN A CHARGED CAPACITOR : HEAT PRODUCED IN SWITCHING IN CAPACITIVE CIRCUIT Due to charge flow always some amount of heat is produced when a switch is close d in a circuit which can be obtained by energy conservation as Heat = Work done by battery - Energy absorbed by capacitor. 9. 10 When two charged conductors of capacitance C & C at potential V & V respectively are connected by a conducting wire, the charge flows from higher potential cond uctor to lower potential conductor, until the potential of the two condensers be comes equal. The common potential (V) after sharing of charges; C,V C V q + V =n etnet charge _ C,j + q capacitance C C+C charges after sharing qj = C,'V & q = C V. In this process energy is lost in the connecting wire C C (V,-V ) as heat. T his loss of energy is U - U = ^ r ^ g s 2 } 2 2 1+ 2 2 SHARING O F CHARGES : 2 2 t 2 2 2 2 2 initial eal

250 ps, I = - 0.1 i -4000(t-250)xi (r 00t e 0.04 0.015 -o.n 6 a m p ; •t(xIO^s) Q.18 400 ^ - — P € EXERCISE # III Q.l Q.2 Q.5 (i) 0.2 x 10" 9 8 F, 1.2 x lO" J ; (ii) 4.84 x 10" J ; (iii) 1.1 x 10" 5 5 5 J 4.425 x 10~ Ampere QA = 90 Q.3 B q.4 F C K ^ /n K, (Ka-KO K, = 18 MJ pC, Q B = 150 pC, Q C = 210

pC, UJ = 4 7 . 4 MJ, U D Q= 0 CVR, Q ' 6 2^9A V48 C & Q.7 Q.10 Q.8 Ri+R2 ^ 2 e0 2 s0 Q.9 R1+R2 anda= Q.ll

XII (ALL) quesjjommm. R ifE,=E (B)C R.C, (D) f Q. 15 Four capacitors and a batteiy are connected as shown. The potential drop a cross the 7 pF capacitor is 6 V. Then the : J (A) potential difference across th e 3 pF capacitor is 10 V (B) charge on the 3 pF capacitor is 42 pC (C) e.m.f. of the battery is 3 0 V (D) potential difference across the 12 pF capacitor is 10 V. A H 3.9(.IF Jn 7F "puF 2 Q. 16 A circuit shown in the figure consists of a battery of emf 10 V and two ca pacitance C, and C of capacitances 1.0 pF and 2.0 pF respectively. The potential difference V - V is 5 V (A) charge on capacitor Cj is equal to charge on capaci tor C Ao—| |—| | — | | o B (B) Voltage across capacitor Cj is 5V. c' e q, (C) Voltage across capacitor C is 10 V (D) Energy stored in capacitor C. is two times the en ergy stored in capacitor C . B 2 2 2 Q.17 A capacitor C is charged to a potential difference V and batteiy is disconn ected. Now if the capacitor plates are brought close slowly by some di stance: ( A) some +ve work is done by external agent (B) energy of capacitor will decrease (C) energy of capacitor will increase (D) none of the above (fe Bansal Classes Question Bank on Capacitance [13]

Q.18 The capacitance of a parallel plate capacitor is C when the region between the plate has air. This region is nowfilledwith a dielectric slab of dielectric constant k. The capacitor is connected to a cell of emf E, and the slab is taken out (A) charge CE(k - 1 ) flows through the cell (B) energy E C(k - 1) is absor bed by the cell. (C) the energy stored in the capacitor is reduced by E C(k - 1 ) (D) the external agent has to do ^E C(k -1) amount ofwork to take the slab out . 2 2 2 Q.19 Two capacitors of capacitances 1 pF and 3 pF are charged to the same voltag es 5 V. They are connected in parallel with oppositely charged plates connected together. Then: (A) Final common voltage will be 5 V (B) Final common voltage wi ll be 2.5 V (C) Heat produced in the circuit will be zero. (D) Heat produced in the circuit will be 37.5 pJ Q. 20 The two plates X and Y of a parallel plate cap acitor of capacitance C are given a charge of amount Q each. X is now joined to the positive terminal and Yto the negative terminal of a cell of emfE = Q/C. (A) Charge of amount Q willflowfromthe negative terminal to the positive terminal o fthe cell inside it (B) The total charge on the plate X will be 2Q. (C) The tota l charge on the plate Y will be zero. (D) The cell will supply CE amount of ener gy. 2 Q.21 A dielectric slab is inserted between the plates of an isolated charged cap acitor. Which of the following quantities will remain the same? (A) the electric fieldin the capacitor (B) the charge on the capacitor (C) the potential differen ce between the plates (D) the stored energy in the capacitor. Q.22 The separatio n between the plates of a isolated charged parallel plate capacitor is increased . Which of the following quantities will change? (A) charge on the capacitor (B) potential difference across the capacitor (C) energy of the capacitor (D) energ y density between the plates. Q.23 Each plate ofa parallel plate capacitor has a charge q on it. The capacitor is now connected to a battery. Now, (A) the facin g surfaces of the capacitor have equal and opposite charges. (B) the two plates of the capacitor have equal and opposite charges. (C) the battery supplies equal and opposite charges to the two plates. (D) the outer surfaces ofthe plates hav e equal charges. Q. 24 Following operations can be performed on a capacitor: X connect the capacitor to a battery of emf E. Y - disconnect the battery Z - rec onnect the battery with polarity reversed. W - insert a dielectric slab in the c apacitor (A) In XYZ (perform X, then Y, then Z) the stored electric energy remai ns unchanged and no thermal energy is developed. (B) The charge appearing on the capacitor is greater after the action XWY than after the action XYW. (C) The el ectric energy stored in the capacitor is greater after the action WXY than after the action XYW. (D) The electricfieldin the capacitor after the action XW is th e same as that after WX. Q.25 A parallel plate capacitor is charged and then dis connectedfromthe source of potential difference. Ifthe plates of the condenser a re then moved farther apart by the use of insulated handle, which one of the fol lowing is true? (A) the charge on the capacitor increases (B) the charge on the capacitor decreases (C) the capacitance of the capacitor increases (D) the poten tial difference across the plate increases (fe Bansal Classes Question Bank on Capacitance [13]

Q.26 Aparallel plate capacitor is charged and then disconnected from the source steady E.M.F. The plates are then drawn apart farther. Again it is connected to the same source. Then: (A) the potential difference across the plate increases, while the plates are being drawn apart. (B) the charge from the capacitorflowsin to the source, when the capacitor is reconnected. (C) more charge is drawn to th e capacitor from the source, during the reconnection. (D) the electric intensity between the plates remains constant during the drawing apart of plates. Q.27 Wh en a parallel plates capacitor is connected to a source of constant potential di fference, (A) all the charge drawnfromthe source is stored in the capacitor. (B) all the energy drawnfromthe source is stored in the capacitor. (C) the potentia l difference across the capacitor grows very rapidly initially and this rate dec reases to zero eventually. (D) the capacity of the capacitor increases with the increase of the charge in the capacitor. Q.28 When two identical capacitors are charged individually to different potentials and connected parallel to each othe r, after disconnecting themfromthe source: (A) net charge on connected plates is less than the sum of initial individual charges. (B) net charge on connected pl ates equals the sum of initial charges. (C) the net potential difference across them is differentfromthe sum ofthe individual initial potential differences. (D) the net energy stored in the two capacitors is less than the sum ofthe initial individual energies. Q. 29 Aparallel plate capacitor of plate area A and plate s eperation d is charged to potential difference V and then the battery is disconn ected. A slab of dielectric constant K is then inserted between the plates ofthe capacitor so as tofillthe space between the plates. If Q, E and W denote respec tively, the magnitude of charge on each plate, the electricfieldbetween the plat es (after the slab is inserted) and the work done on the system, in question, in the process of inserting the slab, then e AV s KAV V AV 1 - 1 K Q. 3 0 A parall el plate capacitor is connected to a battery. The quantities charge, voltage, el ectricfieldand energy associated with the capacitor are given by Q , V , E and U respectively. A dielectric slab is introduced between plates of capacitor but b attery is still in connection. The corresponding quantities now given by Q, V, E and U related to previous ones are (A)Q>Q (B) V > V (C) E > E (D)U 6X0 ~ + 10Cr + 171^0 3 X ^ 3 + 4Cr 0 " + 26H -> 6X0 - + 8Cr + 13H 0 2 2 7 4 4 2 7 2 + 4 3+ 2 7 2 + 4 3+ 2 Q.19 Near Mount Kailash is the sacred lake, Mansorvar. In the crystal clear wate r of the lake, things at the bottom of the lake are also clearly visible. On a h ot sunny day, when the temperature at the surface is 27°C an algae at the bottom o fthe lake produces a 25 ml bubble ofpure oxygen. As the bubble rises to the top, it gets saturated with the water vapours and has a volume of 100 ml of the surf ace. The pressure at the surface is 720 mm Hg. Ifthe depth ofthe lake is 27.2 m, findthe temperature at the bottom of the lake. Vapour pressure of water at 27°C is 20 mm Hg. dj^ci = 1 gm/ml, d = 13.6 g/ml. Hg Q.20 A beam of light Ijas three X, 4144 A, 4972 A and 6216 A with a total intens ity of 3.6 x 10~ Wnr equally distributed amongst the three X. The beam falls nor mally on an area 1.0 cm of a clean metallic surface ofwork function 2.3 eV Assum e that there is no loss oflight by reflection etc. Calculate the no. of photoele ctrons emitted in 2 sec. 3 2 2 J E E Humour. A Physics teacher, a Maths teacher and a Chemistry teacher were wa lking on a sea shore. Fascinated by sea waves the physics teacher said, "I want to study the wave nature of sea waves" and went into the sea and never returned back. The maths teacher said, "I want to measure the volume of sea water" and we nt into the Sea and never returned back. The chemistry teacher concluded "Both p hysics and maths teacher are soluble in sea water under condition of 1 atm and 2 98 K. ^Bansal Classes RAkslia Bandhan Holidays Assignment [3]

ANSWER KEY soxbj§^61 VZ,£ z,l'b SITTING-I e/oos orb pneero 6ib o1ho 9ib %z,-0i x 8'i srb I r»(i-u )+ii z p= i Yn A 1a A3 E J 0 . M' b Z 8 (p) V 8 I'D d ,[X)(l-u)+l] = 0 erb 0 ld a z/b g £b xua Z6 6 8 = A '£ = %'l = Tu nt) - r /96 E A 0 = 3X 'A3 9'£I = 'A9 9'£l~ = H lib 9 d ' raro wrzz orb 6b ss'o'ao 8 b a 9b a sb a Kb q (p) vc>) 'a(q)'a00 rb o rb SITTIN G—II 9'862 x n orb V sASgi-oixzrt? z,rb 6 8I"b 6 S/Da01x8> 7 v srb r V H'6 rat>-0l x t £l'b > ? X 0 1 '0i7 '0 '0 '09 'SL ' 0 '081' % SZ 0lbJxyu^v 0 0£ T-x D V 9"b £ m ). 2 Ik A spring of mass m is pulled such that a given instant,, velocity of both of its end is v in the opposite direction. Find the kinetic energy ofthe spring. A par ticle of mass 3 kg is rotating in a circle ofradius 1 m such that the angle rota ted by its radius is given by 0 = 3 (t + sint). Find the net force acting on the particle when t = n/2. For a particle rotating in a vertical circle with unifor m speed, the maximum and minimum tension in the string are in the ratio 5 :3. If the radius of vertical circle is 2m, then find the speed of revolving body. Q.10 Two strings of length /=0.5 m each are connected to a block of mass m=2 at one end and their ends are attached to the point A and B 0.5 m apart on a tical pole which rotates with a constant angular velocity co=7 rad/sec. Find ratio T, oftension in the upper string (T,) and the lower string (T ). [Use 9.8 m/s ] 2 2

kg ver the g =

0.5 Q.ll A force F = -k(x i + y j) [where k is a positive constant] acts on a partic le moving in the x-y plane. Startingfromorigin, the particle is taken to (a, a) and then to (a/V2,o). Find the total work done by the force F on the particle. u , it does not slide on the hemisphere (i.e. leaves the surface at the top itself). O (a) For u = 2u , it lands at point P o n ground Find OP. (b) For u = u /3, Find the height from the ground at which it leaves the hemisphere. (c) Find its net acceleration at the instant it leaves th e hemisphere. Q.8 The track in Fig is straight in the horizontal section AB and is a semicircle of radius R in the vertical part BCD. A particle of mass m is gi ven a velocity of /(22gR)/5 to the left along the track. The particle moves up t he vertical section JZL and ultimately loses contact with it. How far from point B will the mass land. Q.9 A small particle of mass 1 kg slides without friction from height H=45 cm shown in figure and then loops the vertical loop of radius R from where a section of angle 6 = 60° has been removed. Find R such that after l osing contact at A and flying through the air, the particle will reach at the po int B. Also find the normal reaction between particle and path at A. Q.10 A ring of mass m slides on a smooth vertical rod. A light string is attached to the ri ng and is passing over a smooth peg distant a from the rod, and at the other end ofthe string is a mass M (> m). The ring is held on a level with the peg and re leased: Show that it first comes to rest after falling a distance: =0 2mMa M Q 0 0 A M Q.5 Q.ll Ablock ofmass m is held at rest ona smooth horizontal floor. Alight fiictio nless, small pulley isfixedat aheight of 6 mfromthe floor. Alight inextensible s tring of length 16 m, connected with Apasses over the pulley and another identic al block B is hungfromthe string. Initial height of B is 5mfromthe floor as 6m s hown in Fig. When the system is releasedfromrest, B starts to move vertically do wnwards and A slides on the floor towards right. (i) Ifat an instant string make s an angle 0 with horizontal, calculate relation between velocity u ofA and v of B Calculate v when B strikes the floor. M 2 -m2 l i l t 7 777 0, the functional form of the potential energy U (x) of the particle is [JEE (Scr.)'2002] 2 U(x) U(x) U(x)f » X U(x) » X (A) (B) (C) (D) » X Q.13 An ideal spring with spring-constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring in itially unstretched. Then the maximum extension in the spring is [JEE (Scr.)'200 2] (A) 4 Mg/k (B) 2 Mg/k (C)Mg/k (D)Mg/2k Q.14 A spherical ball of mass m is kep t at the highest point in the space between two fixed, concentric spheres Aand B (see figure). The smaller sphere A has a radius R and the space between the two spheres has a width d. The ball has a diameter veiy Sphere B slightly less than d. All surfaces are frictionless. The ball is given a gentle push (towards the right in the figure). The angle made by the radius vector ofthe ball with Sphere A the upward vertical is denoted by 9 (shown in the figure). [JEE' 2002] (a) Ex press the total normal reaction force exerted by the spheres on the ball as a fu nction of angle 9. (b) Let N and N denote the magnitudes of the normal reaction force on the ball exerted by the spheres A and B, respectively. Sketch the varia tions of N and N as functions of cos0 in the range 0 < 9 < T by T drawing two se parate graphs in your answer book, taking cos9 on the horizontal axes. Q.15 In a region of only gravitational field of mass 'M' a particle is shifted from Ato B via three different paths in the figure. The work done in different paths are W ,, W , W respectively then [JEE (Scr.)'2003] (A) W, = W = W (B) W, = w > w (C) W j > W~ > w (D) Wi < W < W Q.16 A particle ofmass m, moving in a circular path of radius R with a constant V2 V L speed v is located at point (2R, 0) at time t =

0 and a man starts moving with a velocity v, along the +ve y-axisfromorigin at t ime t=0. (0,0) Calculate the linear momentum ofthe particle w.r.t. the man as a function oftime. [JEE 2003] Q.17 A particle is placed at the origin and a force F = kx is acting on it (where k is a positive constant). If U(0)=0, the graph of U(x) versus x will be (where U is the potential energy function) a B A B 2 3 2 3 2 3 3 2 3 2 U(x) U(x) U(x) U(x) (A) (B) (C) (D) [JEE' 2004(Scr)] c = M Momentum : The total momentum of a system of particles is p = Mv Kinetic Energy: The kinetic energy of a system ofparticles c onsisits of two parts. K = K + K' 1 2 where K - — Mv , kinetic energy due to motio n of c.m. relative to the fixed origin O, c a c c c c 5. 2s,(c) ^28565 ~ 169,256 m/s (d) 44rad Q.24 0.1875 Q.25 P 2 2 2 T^. -+-> 2 4 2V2V 71R 4^5 m/s 2m g k Q.10 9 Q.14 ^2g rad/s Q.18 — m i IS Q.12 (l-V3/2)mg Q.16 6mg Q.20 2 sec 1 Q.19 - j T rad/s EXERCISE-II Q.l QF = -3ax + b, x 2 , KE 2 2b b 3V3 Q.2 2 m/s Q.3

v = v , 57ia/v 0 0 N=^^ /ra) -l g Q.5 Q.6 (i) ^,(ii)2V^g, 2a 500N/m : Q.7 40 (a) V2 r, (b) h = 2 19r , (c) g Q.8 N 1.19R Q.9 V 2 R=0.2m, ION R(vt-R)v.1/2 (2Rt-vt ) 2 3/2 Q.ll u = vsec9, v A/41 m/s Q.13 up, 10cm N Q.l Q.3 Q.8 mg max R Q.12 a = (2Rt-vt ) ' EXERCISEIII Q.14 9 =7r/2, T=mg(3sin9+3cos9-2) Q.15 4, -J^fis 25 Q.16 24 > Vo> , a=5V3 g/8, N=3mg/8 if C Q.2 Q.5 (i)36N,(ii) 11.66rad/sec,(iii) 0.1m, 0.2m D Q.6 V u = - J g | 3 2 3 y Q.7 Q.13 B Q.12 D L + 2 F=-8mgi-mgj, h=3R Q.4 A Q.9 A A Q.10 5.79 m/'s Q.ll C is 20 ohm. The ammeter reading is 0.10 Amp and voltmeter reading is 12 volt. Q Then R is equal to (A) 1 22 O (B) 140 O (C) 116 O (D)1000 Q 52 By error, a student places moving-coil vol tmeter V (nearly ideal) in series with the resistance in a circuit in order to r ead the current, as shown. The voltmeter reading will be (A) 0 (B)4V (C)6V (D) 1 2V E = 12V, R = 2Q 4FI Q.53 In a balanced wheat stone bridge, current in the galvanometer is zero. It r emains zero when; [1] battery emf is increased [2] all resistances are increased by 10 ohms \ [3 ] all resistances are madefivetimes [4] the battery and the gal vanometer are interchanged (A) only [ 1 ] is correct (B) [ 1 ], [2] and [3 ] are correct (C) [ 1 ], [3] and [4] are correct (D) [1] and [3] are correct 6Q current flows through the branch CF, then answer the f 1A following questi ons H G F E Q.21 The current through (A) branch DE is 1A (B) branch BC is 2A (C) branch BG is 4A (D) branch HG is 6 A Q.22 The emfE ofthe batteiy is (C) 18V (D) 6V (A) 24 V (B) 12 V If a zero resistance -wire is connected in parallel to bran ch CF Q.23 The current through (B) branch BC is zero (A) branch DE is zero (D) b ranch AB is 1.5 A (C)branchBGis0.5A Q.24 The emfE of the battery is (D) 10.5V (E ) 12V (C) 5.25 V (A) 9V (B) 6.6V Question No. 21 to 24 (4 questions) T 11 Inside a super conducting ring six identical resistors each of resistance R are connected as shown in figure. Q.25 The equivalent resistance(s) (A) between 1 & 3 is zero. (B) between 1 & 3 is R/2 (C) between 1 & 2, 2 & 3, 3 & 1 are all equa l. (D) between 1 & 3 is two times that between 1 & 2. Q.26 The equivalent resist ance(s) (A) between 0 & 1 is R. (B) between 0 & 1 is R/3 (C) between 0 & 1 is ze ro. (D) between 0 & 1, 0 & 2 and 0 & 3 are all equal. Q.27 Imagine a battery of emf E between the point 0 and 1, with its positive terminal connected with O. (A ) The current entering at O is equally divided into three resistances. (B) the c urrent in the other three resistances R , R , R^ is zero. (C) The resistances R^ and R^ have equal magnitudes of current while the resistance Rq, have different current. (D) Potential V = V >V,. 12 13 2 3 Question No. 25 to 27 (3 questions) The figure shows a tetrahedron, each side ofwhich has a resistance r Q.28 Choose the correct statements) related to the resistance between any two points. ( ) AB ( ) AB A R B R = = R R Question No. 28 to 30 (3 questions) (C) R is the least c d BD AC = = R R BC AD BC = = R R CD BD R = = R R CA AD BC * ^ D

= R R ( ) AB D R = R AC = R A N D CD = AD = R BD ->R Q.30 If a battery is connected between any two points ofthe tetrahedron, t hen identify the correct statement(s). (A) The potentials of the other two point s are always equal. (B) There always exists a branch through which no current fl ows. (C) The current coming out ofthe battery in each case is same. (D) None oft hese 40/\4fi Question No, 31 to 33 (3 questions) A ^ tAC The givenfigureshows a network of resistances and a battery. Q.31 Identify the correct statements) E=!2 V (A) The circuit satisfies the condition of a balanced Wheatstone bridge. (B) V - V - 0 (C) V - V = 8 (D) no currentflowsin the branch BD B D b d Q.32 Which ofthe two batteries is getting charged? (A) 8V battery (B) 12 ry (C) none Q.33 Choose the correct statement(s). (A) The current coming he 8V battery is 2A (B) The current coming out of the 12V battery is 3 A currentflowingin all the 4 0 branches is same. (D) The currentflowingin gonally opposite branches is same (D) can't be said > For random J or S, we use 1= - J • -ds f 4. In conductors drift vol. of electrons is proportional to the electric field in s ide the conductor as- v = pE where p is the mobility of electrons current densit y is given as J = — = ne v = ne(pE) = aE d RELATION IN J , E AND V D : d where a = neu is called conductivity of material and we can also write p = — -> re sistivity a of material. Thus E = p J. It is called as differential form of Ohm' s Law. 5. Dry cells, secondary cells, generator and thermo couple are the devices used for producing potential difference in an electric circuit. The potential difference between the two terminals ofa source when no energy is drawn from it is called

the " Electromotive force" or " EMF " ofthe source. The unit of potential differ ence is volt. 1 volt = 1 Amphere x 1 Ohm. il.Bansal Classes SOURCES O F POTENTIAL DIFFERENCE & ELECTROMOTIVE FORCE : Current Electricity [5]

6. ELECTRICAL RESISTANCE : The property of a substance which opposes theflowof electric current through it is termed as electrical resistance. Electrical resistance depends on the size, g eometery, temperature and internal structure ofthe conductor. LAW O F RESISTANCE : 7. The resistance R offered by a conductor depends on the following factors : R a y (cross section area of the conductor) R a L (length of the conductor) ; at a given temperature R= P ~ . Where p is the resistivity ofthe material of the conductor at the given temperature. It is also known as specific resistance of the material. 8. [ The resistance ofmost conductors and all pure metals increases with temperature, but there are a few in which resistance decreases with temperature. If R & Rbe the resistance of a conductor at 0° C and 6° C, then it is found that R = R (1 +aG). c 0 DEPENDENCE O F RESISTANCE O N TEMPERATURE : Here we assume that the dimensions ofresistance does not change with temperature if expansion coefficient ofmaterial is considerable. Then instead of resistance we use same property for resistivity as p = p (1 + a0) The materials for which resistance decreases with temperature, the temperature coefficient of resistance is negative. 0 Where a is called the temperature co-efficient of resistance. The unit of a is K " of °C reciprocal of resistivity is called conductivity and reciprocal ofresistan ce is called conductance (G). S.I. unit of G is ohm. 1 _1 9. Ohm's law is the most fundamental of all the laws in electricity. It says that t he current through the cross section or the conductor is proportional to the app lied potential difference under the given physical condition. V = R I . Ohm's la w is applicable to only metalic conductors. I - Law (Junction law or Nodal Analy sis) :This law is based on law of conservation of charge. It states that" The al gebric sum of the currents meeting at a point is zero" or total currents enterin g a junction equals total current leaving the junction. I I = I I . It is also k nown as KCL (Kirchhoffs current law). in out O H M ' S LAW : 10. KRICHHOFF'S LAW'S :

EL - Law(Loop analysis) :The algebric sum ofall the voltages in closed - v, circ uit is zero. I I I R + 2 EMF = 0 in a closed loop. The closed loop can be traver sed in any direction . While traversing a loop if higher potential point is > en tered, put a + ve sign in expression or if lower potential point is i + 4 entere d put a negative sign. -Vj -V +V -V = 0. Boxes may contain resistor or batteiy o r any other element (linear or non-linear). It is also known as KVL (Kirchhoffs voltage law). + e V 2 3 4 il.Bansal Classes Current Electricity [5]

11. A number of resistances can be connected and all the v. V, V„ complecated combinat ions can be reduced to two different types, namely series and parallel. V (i) RE SISTANCE IN SERIES : When the resistances are connected end toend then they are said to be in series, The current through each resistor is same. The effective r esistance appearing across the batter}', R = RJ + R J + R + + R and r/WV\—fyWv—A-WV3 N COMBINATION O F RESISTANCES : ••-VWV-H + Rn V = VJ + V 2 + V 3 + The voltage across a resistor is proportional to the resistance R„ i V V;V = R,+R„+. .+R R,+R-+. +R_ R 2 +V„. (ii) Aparallel circuit of resistors is one in which the same voltage is applied acros s all the components in a parallel grouping of resistors R R,, R3, , R,,. 1; RESISTANCE IN PARALLEL : CONCLUSIONS : (a) (c) Potential difference across each resistor is same. I = Ij + I + I + I 1 Effectiv e resistance (R) then ±-J_ ^ Current in different resistors is inversally proporti onal to the resistance. ,,.111; I,:l : R_ Rj R , R 2 3 2 3 (b) (d) 1 R.n A -WW-iR -WW 12. I, etc, I,l G,+G~+. + _ G G . + G2 + . . . . . . . . . + G _ 1 n I where G - — = C onductance ofa resistor. R Ij = 1 2 0 13.

If a cell of emf E an d internal resistance r be connected with a resistance R t he total resistance of the circuit is (R+r). £,r E,RE,R E,? upton I = — AB ^ 7 R+r ; E = Terminal voltage of the batten .If r 0, cell is Ideal & V -> E. AVvV V = WHERE E M F O F A CELL & ITS INTERNAL RESISTANCE : 7 GROUPING O F CELLS : (i) If n r « R t h e n I Let there be n cells each ofemf E, arranged in series,Let r be the internal resi stance of each cell, nE The total emf = n E. Current in the circuit I R+nr nE R CELLS IN SERIES : If nr » K then I E » Series combination should be used. Series combination should not be used Current Electricity il.Bansal Classes [5]

(ii) C E L L S I N PARALLEL : If m ceils each of emf E & internal resistance r be connected in parallel and if this combination be connected to an external resistance then the emf ofthe circ uit=E. Internal resistance ofthe circuit = m -^1—wU— mE 1= R+— mR+r m R •m— mE Parallel co mbination should be used. If m R « r ; 1 = If m R » r : 1 = R -» Parallel combination should not be used. upto (iii) mn=number ofidentical cells. n=number of rows m=number of cells in each rows. Th e combination ofcells is equivalent to single cell of: mr (a) emf = mE & (b) int ernal resistance = n For maximum current N = mr or R Current I = mE R+mr n C E L L S LN M U L T I P L E A R C : 12 3 m HHH>m R HHH» mr R= — = internal resistance of battery. T _ nE_mE ~ 2r~2R ' m a x W H E A T STONE N E T W O R K : When current through the galvanometer is zero (null point or balance point) — = — . When PS > QR; V < V & PS V or Q S PS = QR => products of opposite arms are equal. Potential difference between C & D at null point is zero. The null po int is not affected by resistance of G & E. It is not affected even ifthe positi ons of G & E are inter changed. I a (QR-PS). c D c D C D 14. A potentiometer is a linear conductor ofuniform cross-section with a steady curr ent set up in it. This maintains a uniform potential gradient along the length o fthe wire. Any potential difference which is less then the potential difference maintained across the potentiometer wire can be measured using this. The • • i Ii po tentiometer equation is — =— . 2 I2 E L E POTENTIOMETER : il.Bansal Classes Current Electricity [5]

15. AMMETER : It is a modified form of suspended coil galvanometer it is used to measure curre nt . A shunt (small resistance) is connected in parallel with I-Rgalvanometer to convert into ammeter. S = ; An ideal ammeter has zero resistance. where I = Max imum current that canflowthrough the galvanometer. I = Maximum current that can be measured using the given ammeter. g i J« -vwv g 16. A high resistance is put in series with galvanometer. It is used to measure pote ntial difference. V I R I = — ^ g —WW— R„+R " * + v R-»oo , Ideal voltmeter. 8 s 8 0 VOLTMETER : 17. While solving an electric circuit it is convinient to chose a reference point an d assigning its voltage as zero. Then all other potential are measured with resp ect to this point, This point is also called the common point. The energy libera ted per second in a device is called its power. The electrical power P delivered by an electrical device is given by P = VI , where V=potential difference acros s device & I = current. Ifthe current enters the higher potential point ofthe de vice then power is consumed by it (i.e. acts as load). If the current enters the lower potential point then the device supplies power (i.e. acts as source). V P ower consumed by a resistor P = I R = VI = — . 2 2 RELATIVE POTENTIAL : 18. ELECTRICAL POWER : 19. When a current is passed through a resistor energy is wested in over coming the resistances ofthe wire . This energy is converted into heat. V W = Vlt Joule; = I Rt Joule ;= — t Joule. R 2 2 HEATING EFFECT O F ELECTRIC CURRENT : 20. The heat generated (in joules) when a current ofI ampere flows through a resista nce of R ohm for T second is given by: I H = I RT Joules ; = —RT Calories. 4.2 If current is variable passing through the conductor then we use for heat produced in resistance in time t 0 tot is: =jl Rdt 2 2 H 2

JOULES LAW O F ELECTRICAL HEATING : 21. UNIT O F ELECTRICAL ENERGY CONSUMPTION : 1 unit of electrical energy = Kilowatt hour = 1 KWh = 3.6 x 10 Joules. 6 il.Bansal Classes Current Electricity [5]

EXERCISE # I Q. 1 Anetwork ofnine conductors connects six points A B, C, D, E and F as shown infigure.Thefiguredenotes resistances in ohms. Find the equivalent resistance be tween A and D. Q.2 1 In the circuit shown infigurepotential difference between point A and B is 16 V. Find the current passing through 2Q resistance. " a 2n Find the current I & vol tage V in the circuit shown. AO 4fi 9V i n 3V 4n VW-r-4 I I—I I W OB so "1 T 20V la ,60V Q. 3 Q. 4 Q.5 Q.6 Q. 7 0.443 4Q3 2Q< Find the equivalent resistance of the circuit between points A and B shown in fi gure is: (each branch is ofresistance = 10) ^ |10V |SV J 2 0 V J30V Find the cur rent through 25V cell & power supplied by T ~r 20V cell in thefigureshown. 9s If Ss t 25V Ifa cell of constant E.M.F. produces the same amount ofthe heat during the same time in two independent resistors R and R^,, when they are separately connected across the terminals of the cell, one after the another,findthe internal resista nce ofthe cell. Find the effective resistance ofthe network (seefigure)between t he points A and B. Where R is the resistance of each part. R Q.8 Q. 9 In the circuit shown infigure,all wires have equal resistance r. Find the equiva lent resistance between A and B. Find the resistor in which maximum heat will be produced. Q. 10 For what value of Rin circuit, current through 4f2 resistance is zero. Q.l l In the circuit shown infigurethe reading of ammeter is the same with both swit ches open as with both closed. Thenfindthe resistance R. (ammeter is ideal) 4y loon _ —wwh—®—f, . — w wJWt w 1 ( ison [5] il.Bansal Classes

Current Electricity W^tlv

Q.12 Ifthe switches S , S and S in thefigureare arranged such that current throu gh the battery is minimum,findthe voltage across points A and B. t 2 3 >J 6D -r-Vv 24V 6n - 9fJ w h 3£! Q.13 Thefigureshows a network ofresistor each heaving value 12H. Find the equiva lent resistance between points Aand B. Q.14 A battery of emfs = 10 Vis connected across a i m long uniform wire having resistance 1 OQ/m. Two cells ofemfgj = 2V and e = 4V having internal resistances 1Q and 5Q respectively are connected as shown in thefigure.If a galvanometer sh ows no deflection at the point P,findthe distance ofpoint P from the point a. 0 2 Q.15 A potentiometer wire AB is 100 cm long and has a total resistance of lOohm. If the galvanometer shows zero deflection at the position C, thenfindthe value ofunknown resistance R. Q.16 In thefigureshown for gives values ofRj and fL the balance point for Jockey is at 40 cmfromA When R, is shunted by a resistance of 10 O, balance shifts to 50 cm.findR, and R,. (AB = lm): -w R 3 -W2 R Q.17 A part of a circuit is shown in figure. Here reading of ammeter is 5 R -A/W WWV ampere and voltmeter is 96V & voltmeter resistance is 480 ohm. Then find the resistance R Q.18 An accumulator of emf 2 Volt and negligible internal resistan ce is connected across a uniform wire of length 10m and resistance 30Q. The appr opriate terminals ofa cell of emf 1.5 Volt and internal resistance 10 is connect ed to one end ofthe wire, and the other terminal ofthe cell is connected through a sensitive galvanometer to a slider on the wire. What length ofthe wire will b e required to produce zero deflection of the galvanometer ? How will the balanci ng change (a) when a coil ofresistance 5fi is placed in series with the accumula tor, (b) the cell of 1.5 volt is shunted with 5Q resistor ? Q.19 The resistance ofthe galvanometer G in the circuit is 25f2. The meter deflects Ri R-, full scal e for a current of 10 mA. The meter behaves as an ammeter of -v-AVrvWv- 'vVvVthr ee different ranges. The range is 0-10 A ifthe terminals O and P are taken; rang e is 0 - 1 A between O and Q; range is 0 - 0.1A between O 10A 1A 0.1 A R and R. Calculate the resistance Rj, R2 and R . List of recommended questions from I.E. Irodov, 3,147, 3.149, 3.150,3.154,3.155,3.169, 3.175, 3.176, 3.179,3.186, 3.189, 3.190, 3.194,3.196, 3.207 3 il.Bansal Classes

Current Electricity [5]

Q. 1 Atriangle is constructed using the wires AB, BC & CAof same material and of resistance a, 2a & 3a respectively. Another wire of resistance a/3 from A can m ake a sliding contact with wire BC. Find the maximum resistance ofthe network be tween points A and the point of sliding wire with BC. Q.2(a) The current density across a cylindrical conductor of radius R varies according to the equation , w here r is the distancefromthe axis. Thus the current density is a maximum J at t he axis r = 0 and decreases linearly to zero at the surface r = R. Calculate the current in terms of J and the conductor's cross sectional areaisA=7iR Suppose t hat instead the current density is a maximum J at the surface and decreases line arly to zero at the axis so that J = J —. Calculate the current. 0 0 2 0 0 EXERCISE # II (b) Q.3 Q4 What will be the change in the resistance of a circuit consisting of five identi cal conductors iftwo similar conductors are added as shown by the dashed line in figure. The current I through a rod of a certain metallic oxide is given by 1 = 0.2 V , where V is the potential difference across it. The rod is connected in series with a resistance to a 6V battery ofnegligible internal resistance. What value should the series resistance have so that: the current in the circuit is 0 .44 the power dissipated in the rod is twice that dissipated in the resistance. 5/2 © 00 Q.5 Q.6 (I) 01) Q.7 Q. 8 Apiece ofresistive wire is made up into two squares with a common side of length 10 cm. A currant enters the rectangular system at one ofthe corners and leaves at the diagonally opposite corners. Show that the current in the common side is l/5th of the entering current. What length of wire connected between input and o utput terminals wouid have an equivalent effect. A network of resistance is cons tructed with R, & R^ as shown inthefigure.The potential at the points 1,2,3,.., N are Vj, V , V ,.., V respectively each having a potential k tune smaller than previous one Find: Rj R p and p in terms of k current that passes through the re sistance R2 nearest to the V in terms V , k &R . 2 3 R 2 0 0 A hemisphere network ofradius a is made by using a conducting wire of resistance per unit length r. Find the equivalent resistance across OP. Three equal resist ance each of R ohm are connected as shown infigure.A battery of2 volts of intern al resistance 0.1 ohm is connected across the circuit. Calculate Rfor which the heat generated in the circuit is maximum. c 3 r XL. R / 2V

il.Bansal Classes Current Electricity [5]

Q.9 A person decides to use his bath tub water to generate electric power to run a 4 0 watt bulb. The bath tube is located at a height of 10 m from the ground & it h olds 200 litres ofwater. If we install a water driven wheel generator on the gro und, at what rate should the water drain from the bath tube to light bulb? How l ong can we keep the bulb on, ifthe bath tub was full initially. The efficiency o f generator is 90%. (g= lOm/s" ) 2 |36V Q.10 C O m en: In the circuit shown infigure,calculate the following: Potential difference between points a and b when switch S is open. Current through S in the circuit w hen S is closed. 3Q-" •6Q Q.ll The circuit shown infigureis made of a homogeneous wire ofuniform cross-sec tion. ABCD is a square. Find the ratio ofthe amounts of heat liberated per unit time in wire A-B and C-D. T Q.12 Arod oflength L and cross-section area Alies along the x-axis between x = 0 and x = L. The material obeys Ohm's law and its resistivity varies along the ro d according to p (x) = p e . The end ofthe rod at x = 0 is at a potential V and it is zero at x = L. (a) Find the total resistance of the rod and the current in the wire. (b) Find the electric potential in the rod as a function ofx. 0 _xL 0 Q.13 In the figure. PQ is a wire of uniform cross-section and of resistance Rq. Ais an ideal ammeter and the cells are ofnegligible resistance. Thejockey J canf reelyslide over the wire PQ making contact on it at S. If the length ofthe wire PS is f= l/n* ofPQ, find the reading on the ammeter. Find the value of'f for max imum and minimum reading on the ammeter. Q.14 An ideal cell having a steady emfo f2 volt is connected across the potentiometer wire oflength 10 m. The potentiome ter wire is ofmagnesium and having resistance of 11.5 Q/m. An another cell gives a null point at 6.9 m. Ifa resistance of 5£2 is put in series with potentiometer wire,findthe new position ofthe null point. Q.15 Find the equivalent resistance of the following group of resistances between A and B. Each resistance of the ci rcuit is R (a) -w-*A v Vr—, v»—— -oB -Vyx 2 Q.16 An enquiring physics student connects a cell to a circuit and measures the current drawn from the cell to Ij. When he joins a second identical cell is seri es with the first, the current becomes I . When the cells are connected are in p arallel, the current through the circuit is I,. Show that relation between the c urrent is 31 1 = 2 I (I +1 ) iv iv iv iv 3 2 t 2 3 n Q.17 Find the potential difference V - V for the circuit shown in the figure. A B il.Bansal Classes Current Electricity

[5]

Q.18 A resistance R of thermal coefficient of resistivity = a is connected in pa rallel with a resistance = 3R, having thermal coefficient of resistivity = 2a. F ind the value of a . 40 -AV-2Q. w 2/3 f2 -W- 4nw Q.19 Find the current through — O resistance in thefigureshown. 2Q eff I Q.20 A galvanometer having 50 divisions provided with a variable shunt s is used to measure the current when connected in series with a resistance of 90 Q and a battery of internal resistance 10 Q. It is observed that when the shunt resista nce are 10Q, 500, respectively the deflection are respectively 9 & 30 divisions. What is the resistance ofthe galvanometer? Further ifthe full scale deflection ofthe galvanometer movement is 300 mA find the emf ofthe cell. Q.21 In the prima ry circuit of potentiometer the rheostat can be variedfrom0 to 100. Initially it is at minimum iov ion resistance (zero). ^-HpvWv vwv (a) Find the length AP oft he wire such that the galvanometer shows zero 9n deflection. 12m (b) Now the rhe ostat is put at maximum resistance (100) and the switch S is closed. New balanci ng length is found to 8m. Find the internal resistance r 4.5V ofthe 4.5 V cell. 2n Q.22 A galvanometer (coil resistance 99 D) is converted into a ammeter using a shunt of 1Q and connected as shown in thefigure(i). The ammeter reads 3 A The same galvanometer is converted into a voltmeter by connecting a resistance of 10 1 O in series. This voltmeter is connected as shown infigure(ii).Its reading is found to be 4/5 of the full scale reading. Find 12V r 12 V r |H' VWv—| H'—VWV—I intern al resistance r ofthe cell (a) 2n (b) range ofthe ammeter and voltmeter -AAAA —W/v I full scale deflection current ofthe galvanometer 2n (c) (ii) G) 1£1 10 V V. il.Bansal Classes Current Electricity [5]

EXERCISE # III Q. 1 An electrical circuit is shown in the figure. Calculate the potential diffe rence across the resistance of400 ohm, as will be measured by the voltmeter V of resistance 400 ohm, either by applying Kirchhoffs rules or otherwise. [JEE'96, 6] 4000 -VvVv100Q 100Q 200£1 rwv-WAVi—vwv-h -Wr 100Q MV O Q.2(i) A steady current flows in a metallic conductor ofnonuniform cross-section . The quantity /quantities constant along the length ofthe conductor is / are: [ JEE' 97,1 +2+5] (A) current, electricfieldand drift speed (B) drift speed only ( C) current and drift speed (D) current only (ii) The dimension of electricity co nductivity is . (iii) Find the emf (E) & internal resistance (r) ofa single batt ery which is equivalent to a parallel combination oftwo batteries ofemfs V, &V & internal resistances r. & r respectively with their similar polarity connected to each other ^Wr-Wr-rW, Q.3 In the circuit shown in thefigure,the current throu gh: (A) the 3fi resistor is 0.50 A (B) the 3Q resistor is 0,25 A ^yL sq| 40(C) 4 Q resistor is 0.50 A (D) the 4Q resistor is 0.25 A 20 2£1 2SI M/W^wM-VM [JEE'98,2 ] 2 2 Q.4 In the circuit shown, P # R, the reading ofthe galvanometer is same with switch S open or closed. Then L-VWV L-—(g> (A)I = I (B) I = I (C)I = I (D)I = I [JEE'99,2 ] r 0 p G Q G Q r r-Wv p Q Wr— •K I; Q. 5 The effective resistance between the points P and Q of the electrical circuit sh own in thefigureis (A)2Rr/ (R+r) (B) 8R(R+r)/(3R+r) (C)2r + 4R (D) 5 R/2 + 2r [J EE 2002 (Scr), 3] 2 3 2R —WW-^-WA:2R t2R 2R -Wv -VWv—f-AMA 2R VM2R

Q.6 A100 W bulb Bj, and two 60 W bulbs B and B , are connected to a 250 V source, as shown in the figure. Now W W and W are the output powers ofthe bulbs B,,B and B respectively. Then (A) W > W = W (B) W, > W > W (C)Wj < w = w (D) Wj P >P >P (B)P >P >P >Pj (C)P >P >P >Pj (D)P >P >P >P Q.9 In the given circu it, no current is passing through the galvanometer. If the cross-sectional diame ter of AB is doubled then for null point of galvanometer the value ofAC would [J EE' 2003 (Scr)] (A)x (B)x/2 (C)2x (D) None Q.10 How a battery is to be connected so that shown rheostat will behave like a potential divider? Also indicate the points about which output can betaken. [JEE '2003] Q.ll Six equal resistances are connected between points P, Q and R as sho wn in thefigure.Then the net resistance will be maximum between (A) P and Q (B) Q and R (C) P and R (D) any two points [JEE' 2004 (Scr)] AWv—*B r c Q Q.12 In an RC circuit while charging, the graph of In I versus time is as shown by the dotted line in the adjoining diagram where I is the current. When the val ue ofthe resistance is doubled, which of the solid curves best represents the va riation of In I versus time? [JEE' 2004 (Scr)] (A)P (B)Q (C)R (D)S " M 1 -R -s "Q •p il.BansalClasses Current Electricity [5]

Q.13 For the post office box arrangement to determine the value ofunknown resist ance, the unknown resistance should be connected between [JEE' 2004 (Scr)] (A) B and C (B)CandD (C) A and D (D)B andC 1 1 »ooo*oogTi 'jaTotoo o o fESuSsjEEOQi 3 Q. 14 Draw the circuit for experimental verification of Ohm's law using a source ofvariable D.C. voltage, a main resistance of 100 O, two galvanometers and two resistances ofvalues 10 Q and 10* O respectively. Clearly show the positions oft he voltmeter and the ammeter. [JEE' 2004] 6 Q.15 In thefigureshown the current through 2Q resistor is (A) 2 A (B) OA (C) 4 A (D) 6 A , 10V f50 —VWv— 10Q 2fJ Wr 20V [JEE' 2005 (Scr)] Q.16 An uncharged capacitor of capacitance 4pF, a battery of emf 12 volt and a r esistor of 2.5 MO are connected in series. The time after which v = 3v is (take /n2 = 0.693) (A) 6.93 sec. (B) 13.86 sec. (C) 20.52 sec, (D) none of these [JEE' 2005 (Scr)] c R Q.17 A galvanometer has resistance 100Q and it requires current lOOpAforfull sca le deflection. Aresistor 0. ID is connected to make it an ammeter. The smallest current required in the circuit to produce the full scale deflection is (A) 1000 .1mA (B) 1.1mA (C) 10.1mA (D) 100.1mA [JEE' 2005 (Scr)] Q.18 An unknown resistan ce X is to be determined using resistances R,, R or R,. Their corresponding null points are A, B and C. Find which of the above will give the most accurate read ing and why? [JEE 2005] 2 —1| VWv -sC or R 3 A B R=R, or R 2 Q.19 Consider a cylindrical element as shown in thefigure.Current , flowing the through element is I and resistivity ofmaterial ofthe 4 cylinder is p. Choose th e correct option out * the following. (A) Power loss in second half is four time s the power loss infirsthalf. (B) Voltage drop infirsthalfis twice ofvoltage dro p in second half. (C) Current density in both halves are equal. (D) Electricfiel din both halves is equal.

r B $2r 1/2 1/2 [JEE 2006] il.Bansal Classes Current Electricity [5]

ANSWER KEY Q I in 12A-20W 4Q 9n 20 ohm Q.2 Q.5 Q.9 Q.13 Q.17 Q.l Q.4 22 I = 2.5 A V = 3.5 Volts Q.4 ^n 3r Q.7 8/7R Q.8 — Q.6 V i 2 Q.10 lQ Q.ll 600n Q. 12 I V 10 Q.16 y n , 5 n Q.14 46.67 cm Q.15 4 ohm Q.18 7.5 m, 8.75m, 6.25m Q.19 Rj = 0.0278 n , R2 = 0.25 n , R = 2.5 n 3.5 A R EXERCISE # I Q.3 R R _3 Q.3 R! 5 (3/1 l)a Q.2 (a)J A/3;(b)2J A/3 Q.5 7/5 times the length of any si de of the square (i)10.52n;(u)0.3125n (2 + 7i)ar Q.6 (i) (k - l ) ' ( kk- l )(ii ) ((k-l)/k )v Q7 R 8 Q.8 0.3n Q.9 4/9 kg/sec., 450 sec Q.10 (i) V = - 12 V, (ii) 3 amp from b to a Q.ll II + 6V2 < ^ V Af - ^ Q.12 R PoL . - I ; i = ;v = V" (e" 1 - e -1 ) A Po e - l , £ Q.13 r + R ( f - f ) ' m a f = 0 , l ; I f = l / 2 Q.14 7.2m Q.15 (a) 5/7R, (b) 9R/14 22 Q eff ^ Q.19 1A Q.20 233.3n; 144V Q.21 (a) 6m, (b) i n Q.17 - — V Q.22 (a) 1.01 W, (b) 0-5A 0-10V, (c) 0.05 A 2 0 0 2 2 0 k w 3 ab n 0 e EXERCISE # II 3 L v 0 2 f o r I X m in 1 8 a = a EXERCISE # III Vir +V r! l 2 2 2 r + r

Q. 1 20/3 V Q-4 Q.2 (i) D; (ii) M L~ T A ; (iii) Q.5 Q.6 D _1 3 3 2 ( Y ) rr t + r 2 r l 2 Q.3 D Q.7 (a) No, (b) A J 0-y) ^ B •VWV— 12 O D (c)8n Q.8 A Q9 Q.10 Battery should be connected across Aand B. Out put can be taken across the terminals Aand C or B and C Q.ll A Q.12meterB Q.13 C Volt 10' n\ r t ® - ^ Q.14 t 2 Q.15 B Q.16 B Q.17 D Q.19 A [5] Q.18 This is true for r = r ; So R, given most accurate value il.BansalClasses Current Electricity

XII (ALL) ELECTROMAGNETIC INDUCTION ALTERNATING CURRENT CONTENTS & KEY CONCEPTS EXERCISE-I EXERCISE-II EXERCJSE-llI ANSWER KEY

When a conductor is moved across a magnetic field, an electromotive force (emf) is produced in the conductor. If the conductors forms part of a closed circuit t hen the emf produced caused an electric current to flow round the circuit. Hence an emf (and thus a current) is induced in the conductor as a result of its move ment across the magnetic field. This is known as "ELECTROMAGNETIC INDUCTION." 1. MAGNETIC FLUX : KEY CONCEPTS (]) = B . A ^BA cos 9 weber for uniform B . (j) = j B . d A for non uniform B . 2. (i) (ii) An induced emf is setup whenever the magnetic flux linking that circuit changes. The magnitude of the induced emf in any circuit is proportional to the rate of change of the magnetic flux linking the circuit, s a — . dt The direction of an in duced emf is always such as to oppose the cause producing it. LAW O F EMI: LENZ'S LAWS : FARADAY'S LAWS O F ELECTROMAGNETIC INDUCTION : 3. 4. e = - — . The neaative sign indicated that the induced emf opposes the change of t he flux. dt 5. E = BLV sin 0 voltwhere B = flux densi ty in wb/m ; L = length of the conductor (m); V=velocity of the conductor (m/s); 9 = angle between direction of motion of conductor & B . 2 E M F INDUCED IN A STRAIGHT CONDUCTOR IN UNIFORM MAGNETIC FIELD : 6. COIL ROTATION IN MAGNETIC FIELD SUCH THAT A X I S O F ROTATION I s PERPENDICULAR T O Instantaneous induced emf. N = number of turns in the coil ; B = magnetic induct ion ; E = maximum induced emf. 0 THE MAGNETIC FIELD : A = area of one turn; ©= uniform angular velocity ofthe coil; E = NABco sin cot = EQ sin cot, where 7. When a current flowing through a coil is changed the flux linking with its own w

inding changes & due to the change in linking flux with the coil an emf is induc ed which is known as self induced emf & this phenomenon is known as self inducti on. This induced emf opposes the causes ofInduction. The property ofthe coil or the circuit due to which it opposes any change ofthe current coil or the circuit is known as SELF - INDUCTANCE . It's unit is Henry. Coefficient of Self inducta nce L = — or 4> = Li s SELF INDUCTION & SELF INDUCTANCE : fe Bansal Classes Electromagnetic Induction [10]

L depends only on; (i) (ii) shape of the loop & medium i = current in the circui t. II axis of solenoid J 0 vj 1H Q)re 10. R = 0 ; E = 0. Therefore (j) = constant. Thus in a superconducting loop flux nev er changes, (or it opposes 100%) total SUPER CONDUCTION LOOP IN MAGNETIC FIELD : 11.

(i) ENERGY STORED IN A N INDUCTOR : (ii) W = -2 LI . Energy of interation of two loops U = l,(j) = I ^ , = M I j I , wher e M is mutual inductance . 2 2 2 fe Bansal Classes Electromagnetic Induction [10]

12. GROWTH O F A Rt/L CURRENT IN A N L - R CIRCUIT : I = — (1 - e~ ) . [ If initial current = 0 ] R L R = time constant of the circuit. E (i) (ii) 13. L behaves as open circuit at t = 0 [If /' = 0 ] L behaves as short circuit at t = oo always. L Curve (1) > — Large R L Curve (2) — Small R 0 DECAY O F CURRENT : Initial current through the inductor = I ; Current at any instant i = I e~ M 0 Rt/L ^Bansal Classes Electromagnetic Induction [4]

EXJER CISE—I Q.l Q.2 The horizontal component ofthe earth's magneticfieldat a place is 3 x 10^T and t he dip is tan '(4/3). A metal rod of length 0.25 m placed in the north-south pos ition is moved at a constant speed of lOcm/s towards the east. Find the e.rnf. i nduced in the rod. A wire forming one cycle of sine curve is moved in x-y plane with velocity V = V i + V j. There exist a magneticfieldB = - B k Find the motio nal emf develop across the ends PQ of wire. x y 0 Q.3 A conducting circular loop is placed in a uniform magneticfieldof0.02 T, with it s plane perpendicular to thefield.If the radius ofthe loop starts shrinking at a constant rate of 1.0 mm/s, thenfindthe emfinduced in the loop, at the instant w hen the radius is 4 cm. Find the dimension of the quantity 7-7-7 , where symbols have usual meaimng. RCV A rectangular loop with a sliding connector of length I = 1.0 m is situated in a uniform magnetic field B = 2T perpendicular to the pla ne of loop. Resistance of connector is r = 2 f l Two resistances of 6 0 and 3Q a re connected as shown infigure.Find the external force required to keep the conn ector moving with a constant velocity v = 2m/s. ©B —» 3Q:? Q. 4 Q.5 >6Cl Q. 6 Two concentric and coplanar circular coils have radii a and b(»a)as shown in figur e. Resistance of the inner coil is R. Current in the outer coil is increased fro m 0 to i, thenfindthe total charge circulating the inner coil. A horizontal wire isfreeto slide on the vertical rails of a conductingframeas shown infigure.The wire has a mass m and length I and the resistance ofthe circuit is R. If a unifo rm magneticfieldB is directed perpendicular to the frame, thenfindthe terminal s peed ofthe wire as it falls under the force ofgravity. Q, 7 *B x -yww xR X Q.8 Q.9 A metal rod of resistance 200 isfixedalong a diameter of a conducting ring of ra dius 0.1 m and lies on x-y plane. There is a magnetic field B = (50T)k- The ring rotates with an angular velocity 0 = 20 rad/ sec about its axis. An external re sistance of 10Q is connected across the centre of the ring andrim.Find the curre nt through external resistance. 6Q r-VW\A 2 mH In the given current,findthe rati o of i, to i where i, is the initial (at t = 0) current and i i s steady state ( at t = 0 ) current through the battery. 0

2 2 10 Q 10 In the circuit shown, initially the switch is in position 1 for a long time . Then the switch is shifted to position 2 for a long time. Find the total heat produced in R,. R. -WVVfe Bansal Classes H HVWVR; Electromagnetic Induction [10]

Q.ll Two resistors of 1OQ and 20Q and an ideal inductor of 1 OH are connected to a 2V battery as shown. The key K is shorted at time t = 0. Find the initial (t = 0) and final (t —» oo) currents through battery. L = 10H I—W-j I V V • —W R= ion H>J 2on Q.12 There exists a uniform cylindrically symmetric magneticfielddirected along the axis ofa cylinder but varying with time as B = kt. Ifan electron is released fromrest in thisfieldat a distance of' r'fromthe axis of cylinder, its accelerat ion, just after it is released would be (e and m are the electronic charge and m ass respectively) Q.13 An emf of 15 volt is applied in a circuit containing 5 H inductance and 10 Q resistance. Find the ratio of the currents at time t = oo an d t = 1 second. Q. 14 A uniform magnetic field of 0.08 T is directed into the pl ane of the page and perpendicular to it as shown in thefigure.A wire loop in the plane of the page has constant area 0.010 m . The magnitude ofmagneticfielddecr ease at a constant rate of 3.0x10 Ts . Find the magnitude and direction ofthe in duced emf in the loop. 2 4 -1 X x x x Q.15 In the circuit shown infigureswitch S is closed at time t = 0. Find the cha rge which passes through the battery in one time constant. r ^ Li M Rn Q.16 Two coils, 1 & 2, have a mutual inductance=M and resistances R each. A curr ent flows in coil 1, which varies with time as: Ij = kt , where K is a constant and't' is time. Find the total charge that has flown through coil 2, between t = 0 and t = T. 2 Q.17 In a L-R decay circuit, the initial current at t = 0 is I. Find the total c harge that hasflownthrough the resistor till the energy in the inductor has redu ced to one-fourth its initial value. Q.18 A charged ring of mass m = 50 gm, char ge 2 coulomb and radius R=2m is placed on a smooth horizontal surface. Amagnetic fieldvarying with time at a rate of(0.21) Tesla/sec is applied on to theringin a direction normal to the surface of ring. Find the angular speed attained in a t ime t = 10 sec. x Q. 19 A capacitor C with a charge Q is connected across an inductor through a sw itch S. If at t = 0, the switch is closed, thenfindthe instantaneous charge q on the upper plate of capacitor. 0 Q^ ^ 0 c Q.20 A uniform but time varying magneticfieldB = K t - C ; ( 0 < t < C/K), where

K and C are constants and t is time, is applied perpendicular to the plane ofth e circular loop of radius' a' and resistance R. Find the total charge that will pass around the loop. Q.21 A coil ofresistance 3000 and inductance 1.0 henry is connected across an alternating voltage of frequency 3 00/271: Hz. Calculate the phase difference between the voltage and current in the circuit. Q.22 Find the value of an inductance which should be connected in series with a capacitor of 5 pF, a resistance of 10Q and an ac source of 50 Hz so that the power factor of t he circuit is unity. fe Bansal Classes Electromagnetic Induction [10]

Q.23 In an L-R series A. C circuit the potential difference across an inductance and resistance j oined in series are respectively 12 V and 16V. Find the total potential difference across the circuit. Q.24 When 100 volt D.C. is applied acro ss a coil, a current of one ampere flows through it, when 100 V ac of 5 0 Hz is applied to the same coil, only 0.5 amp flows. Calculate the resistance and induc tance of the coil. Q.25 A 50W, 100V lamp is to be connected to an ac mains of200 V, 50Hz. What capacitance is essential to be put in seirs with the lamp. List of recommended questions from I.E. Irodov. 3.288 to 3.299, 3.301 to 3.309, 3.311, 3.313, 3.315, 3.316, 3.326 to 3.329, 3.331, 3.333 to 3.335, 4.98, 4.99, 4.100, 4 .134, 4.135, 4.121, 4.124, 4.125, 4.126, 4.136, 4.137, 4.141, 4.144 fe Bansal Classes Electromagnetic Induction [10]

EXERCISE—II Q. 1 Two straight conducting rails form a right angle where their ends are joine d. A conducting bar contact with the rails starts at vertex at the time t = 0 & moves symmetrically with a constant velocity of 5.2 m/s to the right as shown in figure. A 0.35 T magnetic field points out ofthe page. Calculate: (i) The flux through the triangle by the rails & bar at t = 3.0 s. (ii) The emf around the tr iangle at that time. (iii) In what manner does the emf around the triangle vary with time. Q. 2 5.2m/s Two long parallel rails, a distance I apart and each having a resistance X per u nit length are joined at one end by a resistance R. A perfectly conducting rod M N of mass m is free to slide along the rails without friction. There is a unifor m magnetic field of induction B normal to the plane of the paper and directed in to the paper. A variable force F is applied to the rod MN such that, as the rod moves, a constant current i flows through R. Find the velocity ofthe rod and the applied force F as function of the distance x of the rod from R Q.3 A wireisben t into 3 circular segments ofradiusr = 10 cm as shown in figure. Each segment is a quadrant of a circle, ab lying in the xy plane, be lying in the yz plane & ca lying in the zx plane. (i) if a magnetic field B points in the positive x direc tion, what is the magnitude of the emf developed in the wire, when B increases a t the rate of 3 mT/s ? (ii) what is the direction ofthe current in the segment b e. Q. 4 Consider the possibility of a new design for an electric train. The engi ne is driven by the force due to the vertical component ofthe earths magneticfie ldon a conducting axle. Current is passed down one coil, into a conducting wheel through the axle, through another conducting wheel & then back to the source vi a the other rail. (i) what current is needed to provide a modest 10 - KN force ? Take the vertical component of the earth'sfieldbe 10 pT & the length of axle to be 3.0 m. (ii) how much power would be lost for each Q, of resistivity in the r ails ? (iii) is such a train unrealistic ? Q.5 A square wire loop with 2 m sides in perpendicular to a uniform magnetic field, o• o © ©o•(s © a©o« © © o o ©o with halfthe e loop in the field. The loop contains a 20 V battery with « © © 'i negligible interna l resistance. If the magnitude of the field varies with time S according to B = 0.042 - 0.871, withB in tesla&tin sec. V ' / (i) What is the total emf in the ci rcuit ? \ /\ (ii) What is the direction of the current through the battery ? Q.6 A rectangular loop of dimensions I & w and resistance R moves with constant vel ocity V to the right as shown in the figure. It continues to move with same spee d through a region containing a uniform magnetic field B directed into the plane ofthe paper & extending a distance 3 W. Sketch the flux, induced emf & external force acting on the loop as a function ofthe distance. ! : 0 0 0 v fe Bansal Classes Electromagnetic Induction [10]

Q.7 Q.8 A rectangular loop with current I has dimension as shown in figure. Find the mag netic flux $ through the infinite region to the right of line PQ. A square loop of side 'a' & resistance R moves with a uniform velocity v away from a long wire that carries current I as shown in the figure. The loop is moved awayfromthe wi re with side AB always parallel to the wire. Initially, distance between the sid e AB of the loop & wire is 'a'. Find the work done when the loop is moved throug h distance 'a' from the initial position. Two long parallel conducting horizonta l rails are connected by a conducting wire at one end. A uniform magneticfieldB exists in the region of space. A light uniform ring of diameter d which is pract ically equal to separation between the rails, is placed over the rails as shown in thefigure.Ifresistance of ring is X per unit length, calculate the force requ ired to pull theringwith uniform velocity v. B ic c a 'I'D a »| C 0 D * A Q.9 Q.10 Q.ll \ x x x x x x x x x x y. x x Q.12 Available magneticfieldcreates a constant emf E in a conductor ABCDA. The r esistances of portion ABC, CDA and AMC are R R and R respectively. What current will be shown by meter M? The magnetic field is concentrated near the axis ofthe circular conductor. p 2 3 Q .13 In the circuit shown in the figure the switched S and S are closed at time t = 0. After time t = (0.1) In 2 sec, switch S is opened. Find the current in t he circuit at time t = (0.2) In 2 sec. t 2 2 #100V 40£1« S 2 IH j Q.14 (i) (ii) (iii) (iv) Find the values of / and i immediately after the switch S is closed. long time l ater, with S closed. immediately after S is open. long time after S is opened. C ion i 30Q

i^ioov fe Bansal Classes Electromagnetic Induction [10]

Q.15 Consider the circuit shown infigure.The oscillating source ofemf deliver a sinusoidal emfof amplitude e andfrequencyco to the inductor L and two capacitors Cj and C . Find the maximum instantaneous current in each capacitor. max 2 R i(t) Q.16 Suppose the emfofthe battery, the circuit shown varies with timet so the current WvV ~ is given by /'(t) = 3 + 5t, where i is in amperes & t is in s econds. Take R = 4Q, L = 6H &findan expression for the battery emf as function o f time. Q.17 A current of 4 A flows in a coil when connected to a 12 Vdc source. Ifthe same coil is connected to a 12V, 50 rad/s ac source a current of 2.4 A fl ows in the circuit. Determine the inductance ofthe coil. Also find the power dev eloped in the circuit if a 2500 pF capacitor is connected in series with the coi l. Q.18 An LCR series circuit with 1000 resistance is connected to an ac source of2 00 V and angular frequency 300 rad/s. When only the capacitance is removed, the current lags behind the voltage by 60°. When only the inductance is removed, the c urrent leads the voltage by 60°. Calculate the current and the power dissipated in the LCR circuit. Q.19 A box P and a coil Q are connected in series with an ac s ource ofvariable frequency. The emf of source at 10 V. Box P contains a capacita nce of 1 pF in series with a resistance of 32Q coil Q has a self-inductance 4.9 mH and a resistance of 68Q series. The frequency is adjusted so that the maximum current flows in P and Q. Find the impedan-^ of P and Q at this frequency. Also find the voltage across P and Q respectively. Q.20 A series LCR circuit contain ing a resistance of 120Q has angular resonancefrequency4 x 10 rad s' . At resona nce the voltages across resistance and inductance are 60 V and 40 V respectively . Find the values of L and C. At whatfrequencythe current in the circuit lags th e voltage by 45°? 5 1 fe Bansal Classes Electromagnetic Induction [10]

EXERCISE—III Q. 1 Arectangular frame ABCD made of auniform metal wire has a straight connecti on B between E & F made ofthe same wire as shown in thefigure.AEFD is a square x x x x of side 1 m & EB = FC = 0.5 m. The entire circuit is placed in a steadily X X X X X y increasing uniform magneticfielddirected into the place ofthe paper & normal X IK X X X to it. The rate of change of the magneticfieldis 1 T/s, the resistance per unit D length ofthe wire is 1 O/m. Find the current in segments AE, BE & EF. [JEE'93, 5] Q.2 An inductance L, resistance R, battery B and switch S are j—©—i H 3 H connected in series. Voltmeters V and V are connected across L and R respectively. When switch is closed: (A) The initial reading in V will be grea ter than in V . (B) The initial reading in V will be lesser than V . (C) The ini tial readings in V and V will be the same. s (D) The reading in V will be decrea sing as time increases. [JEE'93, 2] Q.3 Two parallel vertical metallic rails AB & CD are separated by 1 m. They are connected at the two ends by resistance R & R as shown in the figure. A horizontally metallic bar L of mass 0.2 kg slides wi thoutfriction,vertically down the rails under the action of gravity. There is a uniform horizontal magnetic field of 0.6T perpendicular to the plane of the rail s, it is observed that when the terminal velocity is attained, the power dissipa ted inRj & R, are 0.76 W & 1.2 W respectively. Find the terminal velocity of bar L & value R, & R [JEE '94, 6] v L R L R L R L R L } 2 n Q. 4 Two different coils have self inductance 8mH and 2mH. The current in one coil is increased at a constant rate. The current in the second coild is also increased at the same constant. At a certain instant of time, the power given to the two coils is the same. At that time the current, the induced voltage and the energy stored in thefirstcoil are I Vj and respectively. Corresponding values for the s econd coil at the same instant are I , v and W, respectively. Then: [JEE' 94,2] ]_ Ij 1 2 „ Ii Yl. (B) (D) V, 4 IT4 p 2 2 W ( A ) Q.5

(a) (b) A metal rod OA of mass m & length r is kept rotating with a constant angular spe ed co in a vertical plane about a horizontal axis at the end O. Thefreeend Ais a rranged to slide withoutfrictionalong afixedconducting circular ring in the same plane as that ofrotation. Auniform & constant magnetic induction § is applied per pendicular & into the plane ofrotation as shown in figure. An inductor L and an external resistance R are connected through a switch S between the point O & a p oint C on the ring to form an electrical circuit. Neglect the resistance ofthe r ing and the rod. Initially, the switch is open. What is the induced emf across t he terminals of the switch ? (i) Obtain an expression for the current as a funct ion of time after switch S is closed. (ii) Obtain the time dependence ofthe torq ue required to maintain the constant angular speed, given that the rod OA was al ong the positive X-axis at t = 0. [JEE '95,10] Electromagnetic Induction fe Bansal Classes [10]

Q.6 Q.7 A solenoid has an inductance of 10 Henry & a resistance of 2 D. It is connected to a 10 volt battery. How long will it take for the magnetic energy to reach 1 / 4 of its maximum value ? [JEE'96, 3] Select the correct alternative. X X X A thi n semicircular conducting ring of radius R is falling with its plane vertical in X :•X B x • a horizontal magnetic induction B. At the position MNQ the speed of the ring is x /\ A x v & the potential difference developed across the ring is: M B VTCR (A)zero & M is at higher potential (B) (D) 2 RB V & Q is at higher potentia l (C) k RB V & Q is at higher potential Fill inthe blank. A metallic block carry ing current I is subjected to a uniform magnetic induction which results B j. Th e moving charges experience a force F given by in the lowering of the potential of the face [JEE '96, 2] [assume the speed of the carrier to be v] .«' N >: x V V * , VI A X x x xY x \ X Q.8 [JEE'96,2] A pair ofparallel horizontal conducting rails ofnegligible resistance shorted at one end is fixed on a table. The distance between the rails is L. A conducting massless rod of resistance R can slide on the rails frictionlessly. The rod is t ied to a massless string which passes over a pulleyfixedto the edge of the table . Amass m, tied to the other end of the string hangs vertically. A constant magn eticfieldB exists perpendicular to the table. If the system is releasedfromrest, calculate: the terminal velocity achieved by the rod. 0) the acceleration ofthe mass at the instant when the velocity ofthe rod is halfthe terminal velocity. ( ") [JEE '97, 5] Q.10 Acurrent/ = 3.36 (1 +2t) x 10" A increases at a steady rate in a long straight wire. A small circular loop of radius 10~ m is in the plane of the wire & is placed at a distance of 1 m from the wire. The resistance of th e loop is 8.4 x 10" D. Find the magnitude & the direction of the induced current in the loop. [REE '98, 5] Q.ll Select the correct alternative(s). [ JEE '98, 3 x 2 = 6,4x2=8] The SI unit of inductance, the Henry, can be written as : (i) (A) weber/ampere (B) volt-second/ampere (C) joule/(ampere) (D) ohm-second of side I is placed inside a large square wire of © A small square loop ofwirecentres coinc ide. The mutual inductanceloop ofsystem isside L(L » I ) .toThe loop are co-planar & their of the proportional : (D)K (A) ( B )i Q9 2 3 2 2 2 (iii) fe Bansal Classes A metal rod moves at a constant velocity in a direction perpendicular to its len gth. A constant, uniform magneticfieldexists in space in a direction perpendicul ar to the rod as well as its velocity. Select the correct statement(s)fromthe fo llowing (A) the entire rod is at the same electric potential (B) there is an ele

ctricfieldin the rod (C) the electric potential is highest at the centre of the rod & decreases towards its ends (D) the electric potential is lowest at the cen tre of the rod & increases towards its ends. Electromagnetic Induction [10]

(iv) An inductor of inductance 2.0mH,is connected across a charged capacitor of capac itance 5.0pF,and the resulting LC circuit is set oscillating at its naturalfrequ ency.Let Q denote the instantaneous charge on the capacitor, and I the current i n the circuit. It is found that the maximum value of Q is 200 pC. (a) when Q= 10 0 pC, what is the value of | dl / dt| ? (b) when Q=200 pC ,what is the value of I ? (c) Find the maximum value of I. (d) when I is equal to one half its maximum value, what is the value of | Q| Q.12 Two identical circular loops of metal wir e are lying on a table without touching each other. Loop-A carries a current whi ch increases with time. In response, the loop-B [JEE' 99] (A) remains stationary (B) is attracted by the loop-A (C) is repelled by the loop-A (D) rotates about its CM, with CM fixed Q.13 A coil of inductance 8.4 mH and resistance 6Q is conn ected to a 12V battery. The current in the coil is 1.0 A at approximately the ti me (A) 500 s (B) 20 s (C)35 ms (D) 1 ms [ JEE'99 ] Q.14 A circular loop of radiu s R, carrying current I, lies in x-y plane with its centre at origin. The total magnetic flux through x-y plane is (A) directly prop ortional to I (B) directly proportional to R (C) directly proportional to R (D) zero [JEE' 99] 2 Q.15 A magneticfieldB = (B y / a) k is into the plane ofpaper in the +z directio n. B and a are positive constants. A square loop EFGH of side a, mass m and resi stance R in x-y plane, starts falling under the influence of gravity. Note the d irections ofx and y axes in the figure. Find (a) the induced current in the loop and indicate its direction, (b) the total Lorentz force acting on the loop and indicate its direction, (c) an expression for the speed ofthe loop, v(t) and its terminal value. 0 0 E, ® F ® I ®GMH 8 — 0 0 0 1 1 [JEE '99] Q.16 Two circular coils can be arranged in any ofthe three situations shown in t hefigure.Their mutual inductance will be (A) maximum in situation (a) 0 (B) maximum in situation (b) ^ —-^Q (C) maximum in situation (c) •W-J2 r/8 r/4 ™\sr/8" T 9TIS Q.23 A Q.25 A,C Q.29 D Q.26 B Q.30 D Q.24 1 = Q.27 D Q.31 C (H ni (Dcoscot)7ta (Ld) 0 0 p27iR Q.28 C Q.32 (A) P; (B) P, Q, S; (C) Q,S ; (D) Q, R, S

XII (ALL) ELECTROMAGNETIC INDUCTION & ALTERNATING CURRENT QUESTION BANK ON

QUESTION FOR SHORT ANSWER Q.l Q.2 Q.3 Q.4 Q.5 Are induced emfs and currents different in any way from emfs and currents provid ed by a battery connected to a conducting loop? Can a charged particle at rest b e set in motion by the action of a magnetic field? If not, why not? If so, how? Consider both static and time-varying fields. In Faraday's law of induction, doe s the induced emf depend on the resistance ofthe circuit? If so, how? Figure sho ws a copper ring that is hung from a ceiling by two threads. Describe in detail how you might most effectively use a bar magnet to get this ring to swing back a nd forth. Two conducting loops face each other a distance d apart, as shown in f igure. An observer sights along their common axis from left to right. A clockwis e current i is suddenly established in the larger loop by a battery not shown, ( a) What is the direction of the induced current in the smaller loop? (b) What is the direction ofthe force (if any) that acts on the smaller loop? A circular lo op moves with constant velocity through regions where uniform magneticfieldsofth e same magnitude are directed into or out ofthe plane of the page, as indicated infigure.At which ofthe seven indicated positions will the induced current be (a ) clockwise, (b) counterclockwise, and (c) zero? Q.6 IX X X Q.7 Q.8 Q. 9 Can an induced current ever establish a magneticfieldB that is in the same direc tion as the magnetic field inducing the current? Justify your answer. A plane cl osed loop is placed in a uniform magneticfield.In what ways can the loop be move d without inducing an emf? Consider motions both oftranslation and rotation. Fig ure (a) shows a top view of the electron orbit in a betatron. Electrons are acce lerated in a circular orbit in the xy plane and then withdrawn to strike the tar get T. The magneticfieldB is along the z axis (the positive z axis is out ofthe page). The magneticfieldB along this axis varies sinusoidally as shown in figure (b). Recall that the magneticfieldmust (i) guide the electrons in their circula r path and (ii) generate the electricfieldthat accelerates the electrons. Which quarter cycle(s) infigureare suitable (a) according to (i), (b) according to (ii ), and (c) for operation of the betatron? z 2 5/-\6 '4 7\ (a) | | Bansal Classes (b) Question Bank on EMI [2]

Q.10 (i) A piece of metal and a piece of non-metallic stone are droppedfromthe s ame height near the surface of the earth. Which one will reach the ground earlie r? (ii) (iii) (iv) (v) A metallic loop is placed in a nonuniform magneticfield,w ill an emf be induced in the loop ? A wire loop is held with its plane horizonta l. Amagnet with its north pole downward is allowed to fall through itfromsome he ight. Will the magnet fall with constant acceleration? What will happen ifthe po les are reversed? A magnet is dropped down into long vertically copper tube. Sho w that, even neglecting air resistance the magnet will reach a constant terminal velocity. A magnet is droppedfromthe ceiling along the axis of a copper loop ly ingflaton thefloor.Ifthe falling magnet is photographed with a time sequence cam era, what differences, if any will be noted if, (i) the loop is at room temperat ure© the loop is packed in dry ice ? Q.ll A copper ring is suspended in a vertical plane by a thread. A steel bar is passed through the ring in the horizontal direction which is perpendicular to th e plane of the loop. Then a magnet is similarly passed through the loop. Will th e motion ofthe magnet and the bar affect the position of the ring? Q.12 If the m agneticfieldoutside a copper box is suddenly changed, what happens to the magnet icfieldinside the box ? Such low-resistivity metals are used to form enclosures which shield objects inside them against varying magnetic fields. Q.13 Metallic (nonferromagnetic) and nonmetallic particles in a solid waste may be separated a s follows. The waste is allowed to slide down an incline over permanent magnets. The metallic particles slow down as compared to the nonmetallic ones and hence are separated. Discuss the role of eddy currents in the process. Q.14 Ajet plane isflyingdue north. A potential difference is produced between he wing tips ofth e plane. Will a passenger sitting inside the plane also expect some emfbetween t he wing tips? Will a tiny bulb connected to the wing tips glow? Q.15 Is the indu ctance per unit length for a solenoid near its centre; (a) the same as(b) less t han or (c) greater than the inductance per unit length near its ends ? Q.16 Two solenoids A & B have the same diameter & length & contain only one layer of wind ings, with adj acent turns touching, insulation thickness being negligible. Sole noid A contains many turns offinewire & solenoid B contains fewer turns ofheavie r wire, (i) which solenoid has the larger inductance? (i) which solenoid has the larger inductive time constant ? (material is same) B Q.17 If thefluxN4> through each turn of a coil is the same, the inductance of th e coil may be computed passing from L = j . How might one compute L for a coil f or which this assumption is not valid . (fe Bansal Classes Question Bank on EMI [3]

Q.18 If a current in a source of emfis in the direction of the emf, the energy o fthe source decreases, if a current is in a direction opposite to the emf (as in charging a battery), the energy of the source increases Do these statements app ly to the inductor. Q.19 Does the time required for the current in particular LR circuit to build up to any given fraction of its equilibrium value depend on th e value of the applied emf. Q.20 A steady current is set up in a coil with a ver y large inductive time constant. When the current is interupted with a switch a heavy arc tends to appear at the switch blades. Explain? [Note: interrupting cur rents in highly inductive circuits can be dangerous] Q.21 What is the advantage of placing the two electric wires carrying ac close together? Q. 22 In an LR ser ies circuit the selfinduced emfis a maximum at the instant the switch is closed. How can this be since there is no current in the inductance at this instant. Q. 23 Explain what is meant by the statement "A motor acts as a motor and generato r at the same time." Can the same be said for a generator? Q. 24 In a toroid, is the energy density larger near the inner radius or near the outer radius ? Q.25 Two circular loop s are placed with their centres separated by afixeddistance. How would you orient the loops to have (a) the largest mutual inductance (b) the smallest mutual inductance ? Q.26 If the resistance R in the left hand circuit offigureis increased, what is the direction ofthe induced current in the right h and circuit ? (fe Bansal Classes Question Bank on EMI [14]

ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question. An electron is moving in a circular orbit on a. At the centre ofthe orbit is kept a The e.m.f induced in the smaller loop due ero, since charge on electron in constant a er er

of radius R with an angular accelerati conducting loop of radius r, (r « R ) . to the motion of the electron is (A) z (B) Po a 4R (D) none of these (C) M-o

y^Q-rl a conducting loop of radius R is present in a uniform magnetic field B perpendic ular the plane ofthe ring. Ifradius R varies as a function oftime't', as R = R + 1. The e.m.f induced in the loop is (A) 2TC(R + t)B clockwise ( B ) 7 T ( R + T ) B clockwise (C) 27T(R + t)B anticlockwise (D) zero A wire loop is placed in a region oftime varying magneticfieldwhich is oriented orthogonally to the plane o f the loop as shown in thefigure.The graph shows the magneticfieldvariation as t he function oftime. Assume the positive emf is the one which drives a current in the clockwise direction and seen by the observer in the direction of B. Which o f the following graphs best represents the induced emf as a function oftime. 0 d 4TTR Q 0 J J T 4 Q.5 (C) (D) ST A square wire loop of 10.0 cm side lies at right angles to a uniform magneticfieldof 20T. A10 V light bulb is in a series with the loop as shown in t hefig.The magneticfieldis decreasing steadily to zero over a time interval At. T he bulb will shine with full brightness if At is equal to (A) 20 ms (B) 0.02 ms (C) 2 ms (D) 0.2 ms A long straight wire is parallel to one edge as infig.If the current in the long wire is varies intime as I = T -fx, what will be the induce d emf in the loop?" p bl d + a ^ d+a (A) —— In ( B ) ^ f e . 711 / 2ttc V e T 0 w (A) (B) t, t. (D) TIT In d+a 7ix v a A rectangular loop with a sliding connector oflength 10 c m is situated in uniform magneticfieldperpendicular to plane of loop. The magnet ic induction is 0.1 tesla and resistance of connector (R) is 1 ohm. The sides AB and CD have resistances 2 ohm and 3 ohm respectively. Find the current in the c onnector during its motion with constant velocity one metre/sec. p 0 bl (B) 220

1 (D) 440 A (fe Bansal Classes Question Bank on EMI [5]

Q/f The magnetic flux through a stationary loop with resistance R varies during inte rval of time T as = at (T -1). The heat generated during this time neglectin g the inductance of loop will be a^T aT a ¥ a ¥ (C) 3R (A) 3R (B) 3R CD) R Q/8" The dimensions of permeability offreespace can be given by (D) [MLA^ ] (A) [MLT A" ] (B) [MLA" ] (C) [ML" T A ] Q.9 A wire as a parabola y = a x is located in a uni form magnetic field of inductance B, the vector B being perpendicular to the pla ne xy. At the moment t = 0 a connector starts translation wisefromthe parabola a pex with a constant acceleration co tofindthe emf of electromagnetic induction i n the loop this formed as a function of y 2 2 2 2 2 3 2 2 1 2 (AW =2 By c S = (BK in By (D) s ( ) in a ' 2 A thin circular ring of area 10~ m is held perpendicular t o a uniform magneticfieldofinduction 0.1 T. A small cut is made in the ring and the galvanometer is connected across the ends such that the total resistance of the circuit is 0.1 Q. The ring is squeezed to area 0.5 x 10 m in time 0.1 sec. T he average induced current in the circuit is (A) insufficient data (B) 0.05 A (C )0.5A (D)5A A closed planar wire loop ofarea A and arbitrary shape is placed in a uniform magneticfieldofmagnitude B, with its plane perpendicular to magneticfi eld.The resistance of the wire loop is R. The loop is now turned upside down by 180° so that its plane again becomes perpendicular to the magneticfield.The total charge that must have flowed through the wire ring in the process is (A) < AB/R (B)=AB/R (C) = 2AB/R (D)None D fy r A square coil ABCD is placed in x-y plane wi th its centre at origin. A long straight wire, passing through origin, carries a current in negative z-direction. ® Current in this wire increases with time. The induced current in the coil is : (A) clockwise (B) anticlockwise (C) zero (D) al ternating By 2 2 -2 2 J ^ 3 s j * r1I 5 (fe Bansal Classes A vertical bar magnet is droppedfromposition on the axis ofafixedmetallic rh s c ii coil as shown infig-1. Infig- II the magnet isfixedand horizontal coil is dr opped. The acceleration ofthe magnet and coil are al and cl, respectively —i1. N L then C ]1 J) fixed fixed N (B) aj > g, a < g (A) a, > g , a > (C) a, < g a < g fig-H (D) a. < fig-I Two identical coaxial circular loops carry a current i each circulating inthe same direction. If the loops approach each other (A) the curr ent in each will decrease (B) the current in each will increase (C) the current in each will remain the same (D) the current in one will increase and in other w ill decrease A long straight conductor is placed along axis of a circular coil o f radius R. Ifthe current, B as shown infigure,starts decreasing with time, the current induced in loop would be \ (A) clockwise (ACB) (B) anticlockwise (ABC) S e ^ (C) can not be decided (D) there will be no induced current. 2

2 2 A Question Bank on EMI [6]

Q . 16 In a long hollow vertical metal pipe a magnet is dropped. During its fall , the acceleration of magnet: (A) will decrease linearly (B) will decrease upto a value which is less than g. (C) will decrease to zero and will attain a termin al speed (D) may increase or decrease s^tyrff In the arrangement shown in givenf igurecurrentfromAto B is increasing inmagnitude. Induced current in the loop wil l (A) have clockwise direction (B) have anticlockwise direction (C) be zero (D) oscillate between clockwise and anticlockwise An electric current ij canfloweith erdirectionthroughloop(l) and induced current i, in loop (2). Positive i, is whe n current isfrom'a' to 'b' in loop (1) ' loop (l) and positive i is when the cur rent is from 'c' to'd' in loop (2) In an experiment, the graph of i against time *t* is as shown below • v 2 2 l o o p (2) Which one(s) of the following graphs could have caused i to behave as give above . 2 (A) (C) (B)o (D) ^Qr1"9 A bar magnet is releasedfromrest along the axis of a very long, vertical copper tube after some time, the magnet (A) will stop the tube (B) will move wit h almost constant speed (C) will move with acceleration g (D) will oscillate J^> 0 Figure shows a bar magnet and a long straight wire W, carrying current into th e plane of paper. Point P is the point of intersection of axis of magnet and the line N of shortest distance between magnet and the wire. If P is the midpoint o f the magnet, then which of the following statements is correct ? W (A) magnet e xperiences a torque in clockwise direction (B) magnet experiences a torque in an ticlockwise direction (C) magnet experiences a force, normal to the line of shor test distance (D) magnet experiences a force along the line of shortest distance X QrlT A square coil ABCD is lying in xy plane with its centre at origin. A lng st raight wire passing through origin carries a current i = 2t in negative z-direct ion. The induced current in the coil is (A) clockwise (B) anticlockwise (C) alte rnating (D) zero »x ^. Bansal Classes Question Bank on EMI [130]

(YXI A negative charge is given to a nonconducting loop and the loop is rotated in the IIIIII fj plane of paper about its centre as shown infigure.The magneticf ieldproduced by the ring affects a small magnet placed above the ring in the sam e plane: s (A) the magnet does not rotate (B) the magnet rotates clockwise as se enfrombelow. (C) the magnet rotates anticlockwise as seen from below (D) no effe ct on magnet is there. Two infinitely long conducting parallel rails are connect ed through a capacitor C ® ® B as shown in thefigure.A conductor of length I is move d with constant speed J7® v .'Which of the following graph truly depicts the varia tion of current L® through the conductor with time ? p % 1 0 N (A) Current T I(t) (B) Current T I(t) t (time) t (time) (C) Current t I(t) (D) t (time) Current t I(t) 1= 0 t (time) Two identical conductors P and Q are placed on twofiictionlessrails R and S in a uniform magneticfielddirected into the plane. If P is moved in the X direction shown infigurewith a constant speed then Rrod Q V (A) will be attracted towards P X (B) will be repelled away from P (C) will remain stationary (D) maybe repell ed or attracted towards P s X X X X X X X •B x X

Qs25 Thefigureshows an isosceles triangle wireframewith apex angle equal to n/2. The frame starts entering into the region ofuniform magneticfieldB with constan t velocity v at t= 0. The longest side of the frame is perpendicular to the dire ction ofvelocity. If i is the instantaneous current through theframethen choose the alternati ve showing the correct variation ofi with time. (A) (B) (C) (D) i" A thin wire of length 2m is perpendicular to the xy plane. It is moved with velo city v = (2i + 3 j + k) m / s through a region ofmagnetic induction B = (I + 2 j ) Wb / m • Then potential difference induced between the ends ofthe wire : (A) 2 v olts (D) none ofthese (B) 4 volts (C)0 volts 2 ^. Bansal Classes Question Bank on EMI [131]

y Q J Z ^ A long metal bar of 30 cm length is aligned along a north south line a nd moves eastward at a speed of 10 ms" . A uniform magnetic field of 4.0 T point s vertically downwards. If the south end of the bar has a potential of 0 V, the induced potential at the north end ofthe bar is (A) + 12 V (B)-12V (C) 0 V (D) c annot be determined since there is not closed circuit 8 A square metal loop of s ide 10 cm and resistance 1 Q is moved with a constant velocity partly inside a m agnetic field of 2 Wbrrr , directed into the paper, as shown in thefigure.This l oop is connected to a network offiveresistors each of X X X value 3 f l If a ste ady current of 1 mAflows inthe loop, then the speed of the loop is (A) 0.5 cms" (B) 1 cms (C)2cms" (D) 4 cms" Q.29 Two conducting rings P and Q ofradii r and 2r rotate uniformly in opposite directions with centre of mass velocities 2v and v respectively on a conducting surface S. There is auniform magneticfieldofmagnit ude B perpendicular to the plane ofthe rings. The potential difference between B ,2r/ o the highest points ofthe two rings is iinriimniiinnninuiii P S Q (A) zer o (B) 4 Bvr (C) 8 Bvr (D) 16Bvr ^ J ^ f ) Two coils, X and Y, are linked such th at emfE is induced in Y when the current in X is changing at the rate 1 2 1 -1 1 1 dIN dt . If a current I is now made toflowthrough Y, thefluxlinked with X will be 0 t c , (C)(EI)I (D)ioi IJ E A conductor AB of length I oriented along x-axis moves in X Y plane with velocity v = v (i - j). A magneticfieldB = B (i + j) exists in the region. The induced emf is (A)V2B /v (B)2B /V (C)B /V (D)zero A conducting rod m oves with constant velocity u perpendicular to the long, straight wire carrying a current I as shown compute that the emf generated between the ends of the rod. p ol/ p ol/ 2p ol/ p ol/ VY (A) (B) 2izr (C) (D) 4Tv 7ir %r A conducting rod of length I moves with velocity u a direction parallel to a long wire carrying a s teady current I. The axis of the rod is maintained perpendicular to the wire wit h near end a distance r away as shown in thefig.Find the emf induced in the rod. 0 (A) EI 1 (B) E \ 6 0 0 0 0 0

0 0 0 0 0 0 0 0 v / (A) (C K In T+l \ r J i ^ T+l J (B r + / 7t Vol" , K ( D ) — In T+l (fe Bansal Classes Question Bank on EMI [9]

A square loop of side a and resistance R is moved in the region of uniform magne tic field B(loop remaining completely insidefield) ,with a velocity v through a distance x. The work done is : 4B ^ vx 2B ^ 2 VX B£2vx (D) none (C) R (B) R (A) R A metallic rod oflength L and mass M is moving under the action oftwo unequal fo rces F and F (directed opposite to each other) acting at its ends along its leng th. Ignore gravity and any external magneticfield.If specific charge ofelectrons is (e/m), then the potential difference between the ends ofthe rod is steady st ate must be (A)|F -F |mL/eM (B) (F -F )mL/eM (C) [mL/eM]/n [F,/F ] (D)None Two p arallel rigid wires arefixedat a distance' d' apart, with each wire in a vertica l position. The top ends of the two wires are connected through an ideal inducto r of inductance L. Astraight connector of mass M can slidefreelyup and down, mai ntaining electrical contact with the two wires, in a horizontal position. Aunifo rm magneticfieldexists perpendicular to the plane of the wires. If the connector is releasedfromrest, the graph ofits downward velocity with time is: 2 2 2 } 2 1 2 1 2 2 (D) (C) (B) 31 A rod closing the circuit shown infiguremoves along a U shaped wi re at a constant speed v under the action of the force F. The circuit is in a un iform magnetic field perpendicular to the plane. Calculate F ifthe rate ofheat g eneration in the circuti is Q. (A) (A) F = Qv ( B ) F =v ? (- Q F = ^ Q (D) F = a/Qv Two parallel long straight con ductors lie on a smooth surface. Two other parallel conductors rest on J^ them a t right angles so as to form a square of side a initially. Auniform magneticfiel dB exists at right angles to the plane containing the conductors. They start mov ing out with a constant velocity v. If r is the resistance per unit length ofthe wire the current in the circuit will be Bv Br (B)— (C)Bvr (D) Bv (A) — r v Q (B) Current flows from Q » R > O >? > Q (C) Cur rent flows from Q > P > 0 and from Q -—>R > O (D) No current flows Current growth in two L-R circuits (b) and (c) as shown in figure (a). Let L L , R and Rj be th e \ corresponding values in two circuits. Then (D)L R (B)R =R (C)L >L —WT—v v— L R2 L, Ri Tc) p 2 t 1 2 1 2 1 2 1 2 2 (a) (b) (c) A circuit consisting of a constant e.m.f.'E', a self induction'L and a resistance'R'is closed at t = 0. The relation between the current I in the ci rcuit and time t is as shown by curve 'a' in the fig. When one or more of parame ters E, R & L are changed, the curve 'b' is obtained .The steady state current i s same in both the cases. Thenit is possible that : (A) E & R are kept constant & L is increased (B) E & R are kept constant & L is decreased (C) E & R are both halved and L is kept constant (D) E & L are kept constant and R is decreased A circuit element is placed in a closed box. At time t=0, constant current v(voits ) generator supplying a current of 1 amp, is connected across the box. Potential difference across the box varies according to graph shown in figure. The elemen t in the box is : (A) resistance of 2H (B) battery of emf 6Y (C) inductance of 2 H (D) capacitance of 0. 5F A constant current i is maintained in a solenoid. Whi ch ofthe following quantities will increase if an iron rod is inserted in the so lenoid along its axis? (A) magneticfieldat the centre. (B) magnetic flux linked with the solenoid (C) self-inductance of the solenoid (D) rate of Joule heating. The symbols L, C, R represent inductance, capacitance and resistance respective ly. Dimension of frequency are given by the combination 1 (A) 1 / RC (B) R / L ( C) (D) C / L An LR circuit with a battery is connected at t = 0. Which ofthe fol lowing quantities is not zero just after the circuit (A) current in the circuit (B) magneticfieldenergy in the inductor (C) power delivered by the battery (D) e mf induced in the inductor ^P^Z/ The switches in figures (a) and (b) are L R r^H W- - W n closed at t = 0 (A) The charge on C just after t = 0 is EC. (B) The cha rge on C long after t = 0 is EC. -A. (C) The current in L just after t = 0 is E/ R. (D) The current in L long after t = 0 is E/R. (b) s (fe Bansal Classes Question Bank on EMI [21]

Q.28 At a moment (t = 0) when charge on capacitor C, is zero, the switch is clos ed. If I be the current through inductor at that instant, for t > 0, (A) maximum current through inductor equals I /2. C, (B) maximum current through inductor e quals Cjlp 0 0 c I+ c 2 (C) maximum charge on C, = c,+c 2 (D) maximum charge on Cj = I C, ^ q QA9 For L - R circuit, the time constant is equal to (A) twice the ratio of the energy sto red in the magnetic field to the rate ofthe dissipation of energy in the resista nce. (B) the ratio ofthe energy stored in the magnetic field to the rate of diss ipation of energy in the resistance. (C) half ofthe ratio of the energy stored i n the magneticfieldto the rate of dissipation of energy in the resistance. (D) s quare of the ratio of the energy stored in the magneticfieldto the rate of dissi pation energy in the resistance. 0 An inductor L, a resistance R and two identical bulbs and B are connected to a b attery through a switch S as shown in the figure. The R —vwv [®[B, I L-; resistance of coil having inductance L is also R. Which of the following statement gives th e correct description of the happenings when the switch S is closed? (A) The bul b B„ lights up earlier than B, and finally both the bulbs shine equally bright. (B ) B, light up earlier and finally both the bulbs acquire equal brightness. (C) B„ lights up earlier and finally B, shines brighter than B,. (D) Bj and B^ light up together with equal brightness all the time. 2 h n Which of the following quantities can be written in SI units in Kgm A~ S"~ ? (A) Resistance (B) Inductance (C) Capacitance (D) Magnetic flux 2 2 3 In figure, the switch S is closed so that a current flows in the iron-core induc tor r^flftHnRT-i R which has inductance L and the resistance R. When the switch is opened, a H iB spark is obtained in it at the contacts. The spark is due to ( A) a slow flux change in L (B) a sudden increase in the emf ofthe battery B (C) a rapid flux change in L (D) a rapidfluxchange in R J* In figure, a lamp P is in series with an iron-core inductor L. When the switch S r O - » is closed, the brightness of the lamp rises relatively slowly to its full brightness than it would do without the inductor. This is due to B (A) the low resistance of P (B) the induced-emf in L (C) the low resistance of L (D) the hig h voltage of the battery B I (feBansal Classes Question Bank on EMI [22]

Q M Two coil Aand B have coefficient of mutual inductance M = 2H. The magnetic f lux passing through coil V A changes by 4 Weber in 10 seconds due to the change in current in B. Then (A) change in current in B in this time interval is 0.5 A (B) the change in current inB in this time interval is 2A (C) the change in curr ent in B in this time interval is 8A (D) a change in current of 1A in coil A wil l produce a change in flux passing through B by 4 Weber. X 5 Which ofthe following is true for an ideal transformer (A) Total magnetic flux l inked with primary coil equals flux linked with secondary coil (B) flux per turn in primary is equal to flux per turn in secondary (C) induced emf in secondary coil equals induced emf in primary (D) power associated with primary coil at any moment equals power associated with secondary coil Q.36 A circuit has three elements, a resistance of 11W, a coil of inductive resi stance 120W and a capacitive reactance of 120W in series and connected to an A.C . source of 110 V, 60 Hz. Which of the three elements have minimum potential dif ference? (A) Resistance (B) Capacitance (C) Inductor (D) All will have equal pot ential difference (X3-7 The reactance of a circuit is zero. It is possible that the circuit contains : (A) an inductor and a capacitor (B) an inductor but no ca pacitor (C) a capacitor but no inductor (D) neigher an inductor nor a capacitor. 3 8 In a series R-L-C circuit, thefrequencyof the source is half of the resonan cefrequency.The nature of the circuit will be (A) capacitive (B) inductive (C) p urely resistive (D) data insufficient 9 An a. c. source of voltage V and of freq uency 5 0 Hz is connected to an inductor of 2H and negligible resistance. A curr ent of r.m. s. value 7 flows in the coil. When the frequency of the voltage is c hanged to 400 Hz keeping the magnitude ofV the same, the current is now (A) 87 i n phase with V (B) 47 and leading by 90° from V (C) 7/4 and lagging by 90° from V (D ) 7/8 and lagging by 90° from V (fe Bansal Classes Question Bank on EMI [23]

ANSWER KEY ONLY ONE OPTION IS CORRECT Q.l Q.8 Q.15 Q.22 Q.29 Q.36 Q.43 Q.50 Q.57 Q.64 Q.71 Q.78 Q.85 Q.92 Q.99 B A D B C A A B C A A D B D B Q.l Q.5 Q.9 Q.13 Q.17 Q.21 Q.25 Q.29 Q.33 Q.37 Q.2 Q.9 Q.16 Q.23 Q.30 Q.37 Q 44 Q.51 Q.58 Q.65 Q.72 Q.79 Q.86 Q.93 Q.100 C A C C B B C A A B A D D B D Q.3 Q.10 Q.17 Q.24 Q.31 Q.38 Q.45 Q.52 Q.59 Q.66 Q.73 Q.80 Q.87 Q.94 Q.101 Q.2 Q .6 Q.10 Q.14 Q.18 Q.22 Q.26 Q.30 Q.34 Q.38 C B A A D A A C A A C C B C D C A,C,D B,C B B,D A,C D A B A Q.4 Q.ll Q.18 Q.25 Q.32 Q.39 Q.46 Q.53 Q.60 Q.67 Q.74 Q.81 Q.88 Q.95 Q.102 A C D D B C A B A D B D B D C Q.3 Q.7 Q.ll Q.15 Q.19 Q.23 Q.27 Q.31 Q.35 Q.39 Q5 Q.12 Q.19 Q.26 Q.33 Q.40 Q.47 Q.54 Q.61 Q.68 Q.75 Q.82 Q.89 Q.96 Q.l 03 C A C A C D B,D A B,D D B C B A D A D A B D D A A C C Q.6 Q.13 Q.20 Q.27 Q.34 Q.41 Q.48 Q.55 Q.62 Q.69 Q.76 Q.83 Q.90 Q.97 Q.l 04 Q.4 Q.8 Q.12 Q.16 Q.20 Q.24 Q.28 Q.32 Q.36 B C D A D B B C B B A D C D B B A B,D D D A,B,C D C A Q.7 Q.14 Q.21 Q.28 Q.35 Q.42 Q.49 Q.56 Q.63 Q.70 Q.77 Q.84 Q.91 Q.98 Q.105 A A D C A D D A C C A A C c c ONE OR MORE THAN ONE OPTION MAY BE CORRECT B B B A,B A,B,C,D B,D A,B,C A B A,D (fe Bansal Classes Question Bank on EMI [24]

TARGET IIT JEE 2007 XII (ALL) ELECTROSTATICS CONTENTS KEYCONCEPTS EXERCISE-I EXERCISE-II EXER CISE-III ANSWER KEY

1. ELECTRIC Charge of a material body is that possesion (acquired or natural) due to which i t strongly interacts with other material body. It can be postive or negative. S. I. unit is coulomb. Charge is quantized, conserved, and additive. COULOMB'S 0 CHARGE KEY CONCEPTS 2. LAW: F = — - — ^ r . In vector form F = — — — r ? where 1 s = permittivity of free space = 8.85 x 10 N m~ c or F/m and e = Relative permit tivity of the medium = Spec. Inductive Capacity = Dielectric Const. s = 1 for ai r (vacuum) = oo for metals e e = Absolute permittivity of the medium N O T E : T he Law is applicable only for static and point charges. Only applicable to stati c charges as moving charges may result magnetic R, 0 0 t 4ne i 0 Behaves as a point charge situated at the centre for these points E- = (xi) r < R where p = volume charge density Uniformly charged spherical shell (conduct ing or non-donducting) or uniformly charged solid conducting sphere. E = ^ ^ ; r > R out 2 in pr — ; 47I£qR 3£Q Qr Behaves as a point charge situated at the centre for these points E J= 0 ; r < R (xii) uniformly charged cylinder with a charge density p is -(radius of cylinde r = R) for r < R pr pR E = 2 e ; for r > R o E= 2 e r ~ (xiii) Uniformly charged cylinderical shell with surface charge density a is forrR E= pr 2 m 0 0 m e o r 6. ELECTRIC (i) (ii) (iii) (iv) (v) 7.

The line of force in an electricfieldis a hypothetical line, tangent to which at any point on it represents' the direction of electricfieldat the given point. P roperties of ( E L F ) : Electric lines offerees never intersects. ELF originate sfrompositive charge or oo and terminate on a negative charge of infinity. Prefe rence oftermination is towards a negative charge. If an ELF is originated, it mu st require termination either at a negetive charge or at oo. Quantity of ELF ori ginated or terminated from a charge or on a charge is proportional to the magnit ude of charge. ELECTROSTATIC LINES OF FORCE (ELF) (i) (ii) Position where net force (or net torque) on a charge(or electric dipole) = 0 STA BLE EQUILIBRIUM : If charge is displaced by a small distance the charge comes (o r tries to come back) to the equilibrium. UNSTABLE EQUILIBRIUM : If charge is di splaced by a small distance the charge does not return to the equilibrium positi on. EQUILIBRIUM tl^Bansal Classes ELECTROSTATICS [5]

8. ELECTRIC "Work done by external agent to bring a unit positive charge(without accelaratio n) from infinity to a point in an electricfieldis called electric potential at t hat point" . if w is the work done to bring a charge q (very small) from infinit y to a point then potential at that point is V = (W°° ) ; S.I. unit is volt (= 1 J/C ) w r POTENTIAL (Scalar Quantity) 9. POTENTIAL A B q r ext DIFFERENCE e (W ) V^ =V - V = ^ V ^ = p.d. between point A& B . W = w.d. by external source t o transfer a point charge q from B to A (Without••acceleration). • * BA 10. ELECTRIC FIELD & ELECTRIC E = - grad V = - V V {read as gradient of V} grad = iox + k — ; — oy oz Used whenEF varies in three dimensional coordinate system. Forfindingpotential difference be tween two points in electricfield,we use V - V = ~ j E • dt if£ j varying with dista nce A if E is constant & here d is the distance between points A and B. A B s POINTENIAL ? d *d *d 11. POTENTIAL DUE TO 0 (i) (iii) (iv) (v)

a point charge V = ^ 47is r (ii) . many charges V = ———+——+——— + 47tE r 47te r 47rs r 0 1 0 2 0 3 continuous charge distribution V = —i— f ^ spherical shell (conducting or non conduc ting) or solid conducting sphere non conducting uniformly charged solid sphere : V o u t 4TTS r j 0 S r ' ( ] > V i n 2 4TTER 0 ' ( r " R ) 12. EQUIPOTENTIAL In an electricfield the locus of points of equal potential is called an equipote ntial surface. An equipotential surface and the electricfieldmeet at right angle s. The region where E = 0, Potential of the whole region must remain constant as no work is done in displacement of charge in it. It is called as equipotential region like conducting bodies. SURFACE AND EQUIPOTENTIAL REGION ^Bansal Classes ELECTROSTATICS [4] #

13. MUTUAL "The work to be done to integrate the charge system." q,q For 2 particle system U = 47is— — r 2 mutual 0 POTENTIAL ENERGY OR INTERACTION ENERGY For 3 particle system U 14. (a) m u t u a l = q,q q2qs , q3qi 47is r 47i8 r 47te r 2 0 12 0 23 0 31 15. (b) (c) For n particles there will be \ ^ terms . Total energy of a system = U + U P.E. of charge q in potential field U = qV. Interaction energy of a system of two cha rges U=q V = q V . ELECTRIC DIPOLE. O is mid point of line AB (centre ofthe dipo le) - equitorial line on the axis (except points on line AB) -q +q « E= E P 27is [ r — (a /4)] 27ts r ( i f r < < a ) p = qa = Dipole moment, r = distance of the poi nt from the centre of dipole p p on the equitorial ; E= ~ 47is [r +(a /4)] 47cs r At a general point P(r, 0) in polar co-ordinate system is 2kp sin 0 Radial ele ctricfieldE„= n ( se]f mutual 1 2 2 1 0 2 2 2 0 3 0 2 2 3/2 0 3 Tangentral electric field E = T net 2 kpcos0 2 2 Net electric field at P is E = ^ E + E = ^ Vl + 3sin 0 kpsinO Potential at point P is V = N O T E : If 0 is measured from axis of dipole. Then sin0 and cos0 wil l be interchanged p (d) (e) (f) (g) (i) 16. Dipole V = ^ p=qa electric dipole moment . If 0 is angle between p and 4718 r 47is r reache

s vector of the point. Electric Dipole in uniform electric field : torque x=pxE ; F = 0 . Work done in rotation of dipole is w = PE (cos 0 - cos 0 ) P.E. of an electric dipole in electric field U = - p.E. / ^ dEc Force on a dipole when plac ed in a non uniform electricfieldis F= d (-PE)i = P.-—i. dx ' dx PG Q _ p.r 2 0 1 2 v ELECTRIC FLUX (ii) For uniform electric field; (j) = E.A = EA cos 0 where 0 = angle between § & area vector ( A ). Flux is contributed only due to the component of electricfieldwhic h is perpendicular to the plane. If E is not uniform throughout the area A, then = j" E.d A J tl^Bansal Classes ELECTROSTATICS [5]

17. GA USS'S LAW —." cj) = J>EdA = — o o (j) does not depend on the E s q p - -> (Applicable only to closed surface)" Net flux emerging out of a closed surface i s q q = net charge enclosed by the closed surface . (i) (ii) shape and size of the c losed surface The charges located Outside the closed surface. 18. 19. 20. Flux of charge q having through the circle of radius R is q/e q ( j ) = — — x O = r. (l-cos9) o Energy stored p.u. volume in an electricfield= ^ — 0 z e 2 2 CONCEPT OF SOLID ANGLE : Solid angle of coneof half angle 9 is Q=2rt(l-cos0) Electric pressure due to its own charge on a surface having charged density a is P = c . 2s Electric pressur e on a charged surface with charged density a due to external electric field is P E L E =aE 2 ele 0 t IMPORTANT POINTS TO BE REMEMBERED (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) Electric field is always perpendicular to a conducting surface (or any equipoten tial surface). No tangential component on such surfaces. Charge density at sharp points on a conductor is greater. When a conductor is charged, the charge resid es only on the surface. For a conductor of any shape E (just outside) = — o p.d. b etween two points in an electric field does not depend on the path joining them. Potential at a point due to positive charge is positive & due to negative charg e is negative. Positive charge flows from higher to lower (i. e. in the directio n of electricfield)and negative charge from lower to higher (i.e. opposite to th e electric field) potential. When p||E the dipole is in stable equilibrium p||(E) the dipole is in unstable equilibrium When a charged isolated conducting sphe re is connected to an unchaged small conducting sphere then potential (and charg e) remains almost same on the larger sphere while smaller is charged . Self pote ntial energy of a charged shell = KO . 2R 3k0 Self potential energy of an insula ting uniformly charged sphere = — — . 5R A spherically symmetric charge {i.e p depen ds only on r} behaves as if its charge is concentrated at its centre (for outsid

e points). Dielectric strength of material: The minimum electricfieldrequired to ionise the medium or the maximum electricfieldwhich the medium can bear without breaking down. 8 2 2 tl^Bansal Classes ELECTROSTATICS [5]

EXERCISE # I Q. 1 A negative point charge 2q and a positive charge q are fixed at a distance I apart. Where should a positive test charge Q be placed on the line connecting the charge for it to be in equilibrium? What is the nature ofthe equilibrium wit h respect to longitudinal motions? Q.2 Two particles A and B each carrying a cha rge Q are held fixed with a separation d between then A particle C having mass m ans charge q is kept at the midpoint ofline AB. Ifit is displaced through a sma ll distance x (x « d) perpendicular to AB, (a) thenfindthe time period ofthe oscil lations of C. (b) If in the above question C is displaced along AB,findthe time period ofthe oscillations of C. Q.3 Draw E - r graph for 0 < r < b, if two point charges a & b are located r distance apart, when (i) both are + ve (ii) both ar e - ve _ (iii) a is + ve and b is - ve (iv) a is - ve and b is + ve Q.4 A charge + 10 C is located at the origin in free space & another charge Q at (2, 0, 0). If the X-component ofthe electricfieldat (3,1,1) is zero, calculate the value of Q. Is the Y-component zero at (3,1, 1)? 9 Q.5 Six charges are placed at the vertices of a regular hexagon as shown in the figure. +Q -Q Find the electricfieldon the line passing through O and perpendicular to the pla ne -Q< +Q of thefigureas a function of distance xfrompoint O. (assume x » a) + + T hefigureshows three infinite non-conducting + plates of charge perpendicular to the plane of A + B _ • • + the paper with charge per unit area + a, + 2o + + and - a . Find the ratio ofthe net electricfieldat + 2.5m that point Ato that at point B . +C7 +2ct 5m 5m Q.7 A thin circular wire of radius r has a charge Q. If a point charge q is placed at the centre ofthe ring, then find the increase in tension in the wire. Q.8 In thefigureshown S is a large nonconducting sheet ofuniform ch arge density a. A rod R oflength / and mass 'm' is parallel to the sheet and hin ged at its mid point. The linear charge densities on the upper and lower half of the rod are shown in the figure. Find the angular acceleration of the rod just a fter it is released. Q.9 A simple pendulum of length / and bob mass m is hanging in front of a large nonconducting sheet having surface charge density a. If sud denly a charge +q is given to the bob & it is releasedfromthe position shown inf igure.Find the maximum angle through which the string is deflectedfromvertical. A particle of mass m and charge - q moves along a diameter of a uniformly charge d sphere of radius R and carrying a total charge + Q. Find the frequency of S .H .M. ofthe particle if the amplitude does not exceed R. Q.ll A charge + Q is unif ormly distributed over a thin ring with radius R. A negative point charge - Q an d mass m starts from rest at a point far awayfromthe centre ofthe ring and moves towards the centre. Find the velocity ofthis particle at the moment it passes t hrough the centre of the ring. J ^Bansal Classes ELECTROSTATICS [9]

Q.12 A spherical balloon ofradius R charged uniformly on its surface with surfac e density o. Find work done against electric forces in expanding it upto radius 2R. Q.13 A point charge + q & mass 100 gm experiences a force of 100 N at a poin t at a distance 20 cm from a long infinite uniformly charged wire. Ifit is relea sedfindits speed when it is at a distance 40 cm from wire Q.14 Consider the conf iguration of a system offour charges each of value +q. Find the work done by ext ernal agent in changing the +q configuration ofthe systemfromfigure(i) tofig(ii) . +qr fig(ii) fig (i) Q.15 There are 27 drops of a conducting fluid. Each has ra dius r and they are charged to a potential V . They are then combined to form a bigger drop. Find its potential. Q.16 Two identical particles of mass m carry ch arge Q each. Initially one is at rest on a smooth horizontal plane and the other is projected along the plane directly towards the first from a large distance w ith an initial speed V. Find the closest distance of approach. Q.17 A particle o f mass m and negative charge q is thrown in a gravity free space with speed u fr om the point A on the large non conducting charged sheet with surface charge den sity a, as shown infigure.Find the maximum distancefromAon sheet where the parti cle can strike. Q.18 Consider two concentric conducting spheres of radii a & b ( b > a). Inside sphere has a positive charge q What charge should be given to the outer sphere so that potential of the inner sphere becomes zero? How does the p otential varies between the two spheres & outside ? Q.19 Three charges 0.1 coulo mb each are placed on the corners of an equilateral triangle of side 1 m. If the energy is supplied to this system at the rate of 1 kW, how much time would be r equired to move one of the charges onto the midpoint ofthe line joining the othe r two? Q.20 Two thin conducting shells ofradii R and 3R are shown infigure.The o uter shell carries a charge +Q and the inner shell is neutral. The inner shell i s earthed with the help of switch S. Find the charge attained by the inner shell . 0 r Q.21 Consider three identical metal spheres A B and C. Spheres A carries charge + 6q and sphere B carries charge - 3q. Sphere C carries no charge. Spheres A and B are touched together and then separated. Sphere C is then touched to sphere A and separated from it. Finally the sphere C is touched to sphere B and separated fromit. Find thefinalcharge on the sphere C. y p '(0,y) Q.22 A dipole is placed at origin of coordinate system as shown infigure,find the electricfieldat point P (0, y). \p Q.23 Two point dipoles p k and -P k are located at (0,0,0) and (lm, 0,2m) respectively. Find the resultant electricfielddue to the two dipoles at t he point (lm, 0,0). Q. 24 The length of each side ofa cubical clo sed surface is /. If charge q is situated on one of the vertices of the cube, thenfindthe flux passing through shaded face of the cube. Q.25 A point charge Q is located on th e axis of a disc of radius R at a distance a from the plane of the disc. If one fourth (l/4th) of the flux from the charge passes through the disc, then find th e relation between a & R. *Q C] R) from the centre of the sphere. Find the minimum velocity v so that it can penetrate R/2 distance of the sphere. Neglect any resistance ot her than electric interaction. Charge on the small mass remains constant through out the motion. Q.8 An electrometer consists of vertical metal bar at the top of which is attached a thin rod which gets deflectedfromthe bar under the action o f an electric charge (fig.). The reading are taken on a quadrant graduated in de grees. The length ofthe rod is I and its mass is m. What will be the charge when the rod of such an electrometer is deflected through an angle a . Make the foll owing assumptions: Uu\uumuuuuuuvuft\\< (a) the charge on the electrometer is equ ally distributed between the bar & the rod (b) the charges are concentrated at p oint A on the rod & at point B on the bar. Q.l 2 0 1 81 nd £ EXERCISE # II ^Bansal Classes ELECTROSTATICS [9]

Q.9 A cavity ofradius r is present inside a solid dielectric sphere of radius R havi ng a volume charge density of p. The distance between the centres of the sphere and the cavity is a. An electron e is kept inside the cavity at an angle 6 = 45° a s shown. How long will it take to touch the sphere again? t 2 Q.10 Two identical balls of charges q & q initially have equal velocity of the s ame magnitude and direction. After a uniform electricfieldis applied for some ti me, the direction ofthe velocity ofthefirstball changes by 60° and the magnitude i s reduced by half*. The direction of the velocity of the second ball changes the re by 90°. In what proportion will the velocity of the second ball changes ? Q. 11 Electrically charged drops of mercury fallfromaltitude h into a spherical metal vessel of radius R in the upper part ofwhich there is a small opening. The mass of each drop is m & charge is Q. What is the number 'n' of last drop that can s till enter the sphere. Given that the (n + 1)* drop just fails to enter the sphe re. Q.12 Small identical balls with equal charges are fixed at vertices of regul ar 2004 - gon with side a. At a certain instant, one ofthe balls is released & a sufficiently long time interval later, the ball adjacent to the first released ball is freed The kinetic energies of the released balls are found to differ by K at a sufficiently long distancefromthe polygon. Determine the charge q of each part. Ex Q.13 The electricfieldin a region is given by E = —j— -i. Find the charge contained inside a cubical volume bounded by the surfaces x = 0, x = a, y = 0, y = a, z = 0 and z = a. Take E = 5 * 10 N/C, /=2cm and a = 1 cm. 0 0 3 Q.14 2 small metallic balls ofradii R, & R are kept in vacuum at a large distanc e compared to the radii. Find the ratio between the charges on the 2 balls at wh ich electrostatic energy ofthe system is minimum. What is the potential differen ce between the 2 balls? Total charge ofballs is constant. 2 Q.15 Figure shows a section through two long thin concentric cylinders of radii a & b with a < b . The cylinders have equal and opposite charges per unit length X. Find the electricfieldat a distance rfromthe axis for (a) r < a (b) a < r < b (c) r > b Q.16 A solid non conducting sphere of radius R has a non-uniform cha rge distribution of volume charge density, p = p —, where p is a constant and r is the distancefromthe centre ofthe sphere. Show that: R (a) the total charge on t he sphere is Q = I p R and T 0 0 0 3 (b) the electricfieldinside the sphere has a magnitude given by, E = R 4 . Q.17 A nonconducting ring ofmass m and radius R is charged as shown. The charged density i.e. charge per unit length is X. It is then placed on a rough noncondu cting horizontal surface plane. At time t=0, auniform electricfieldE = E0i is sw itched on and the ring start rolling without sliding. Determine thefrictionforce (magnitude mvmmmuum and direction) acting on the ring, when it starts moving. 1/3 Q .15 - , + , + , - , + , - Q.16 Q.17 C Q.18 D Q.19 V' = v3t, .V Q.20 AB,C,D 0 4 4 2 0 0 0 Q P Q.6 (i) C, (ii) (a) H = ~ , (b) U = mg 2^h +a -h equilibrium at h = - = , V3 o Un 2mga V3mga 1 00V Q.20 Potential difference between centre & the surface of sphere of radius R and uniform volume charge densitv p within it will be: ( C ) 0 2 1/3 273 Q.21 If the electric potential ofthe inner metal sphere is 10 volt & that of the outer shell is 5 volt, then the potential at the centre will be: £ (A) 10 volt (B ) 5 volt (C) 15 volt (D) 0 ^ pR 2 pR 2 () C 0 ( )pR D k 2 Q.22 Three concentric metallic spherical shell A, B and C or radii a, b and c (a < b < c) have surface charge densities - cr, + cr, and - a respectively. The po tential of shell A is: (A)(o/e )[a + b - c ] (B)(a/e )[a-b + c] J # ( a / e ) [ b - a - c ] (D)none Q.23 A charged particle having some mass is resting in equil ibrium at a height H above the centre ofa uniformly charged non-conducting horiz ontal ring ofradius R. The force ofgravity acts downwards. The equilibrium of th e particle will be stable i R R R (A) for all values of H (B) only if H> ^ (C) o nly if H < ^ (D) only if H = 0 0 0 Q.24 An infinite number of concentric rings cany a charge Q each alternately pos itive and negative. Their radii are 1,2,4,8 meters in geometric progression as s hown in the figure. The potential at the centre ofthe rings will be Q Q (A) zero (B) ^ (C, ^ --..4 /I )\ Q.2J When a negative charge is released and moves in electric field, it moves to ward a position of (A) lower electric potential and lower potential energy (B) l ower electric potential and higher potential energy (C) higher electric potentia l and lower potential energy (D) higher electric potential and higher potential energy (f| Bansal Classes

Question Bank on Electrostatics [15]

£ Q.26 A hollow metal sphere of radius 5 cm is charged such that the potential on its surface is 10 V. The potential at the centre of the sphere is (A)0V fe (B)10 V (C) same as at point 5 cm away from the surface out side sphere. (D) same as a point 25 cm away from the surface. Q.27 A solid sphere of radius R is charged u niformly. At what distance from its surface is the electrostatic potential half of the potential at the centre? i - (A)R (B)R/2 (C)R/3 (D)2R Q.28 An infinite no nconducting sheet of charge has a surface charge density of 10~ C/m . The separa tion ^ between two equipotential surfaces near the sheet whose potential differ by 5 V is (A) 0.88 cm (B) 0.88 mm (C) 0.88 m (D)5xl0" m Q.29 Four equal charges +q are placed at four corners of a square with its centre at origin and lying in yz plane. The electrostatic potential energy of a fifth charge +q' varies on xaxis as: u u /\ (A) (D) 7 2 7 2 Q.30 Two identical thin rings, each of radius R meter are coaxially placed at di stance R meter apart. If Qj and Q coulomb are respectively the charges uniformly spread on the two rings, the work done in moving a charge qfromthe centre of on e ring to that of the other is (A) zero (B) < f a ^ J j 2 - \ ) / ( j 2 A n e R ) (C) qV2(Q +Q )/47i£ R (D) qr(Qj-Q )(V2+l)/(V2.47rs R) Q.31 Two positively charge d particles X and Y are initially far awayfromeach other and at rest. X begins t o move towards Y with some initial velocity. The total momentum and energy of th e system are p and E. (A) If Y is fixed, both p and E are conserved. (B) If Y is fixed, E is conserved, but not p. (C) If both are free to move, p is conserved but not E. (D) If both are free, E is conserved, but not p. Q.33 Two particles X and Y, ofequal mass and with unequal positive charges, arefreeto move and are i nitially far awayfromeach other. With Y at rest, X begins to move towards it wit h initial velocity u. After a long time, finally (A) X will stop, Y will move wi th velocity u. (B) X and Y will both move with velocities u/2 each. (C) X will s top, Y will move with velocity < u. (D) both will move with velocities < u/2. Q. 34 A circular ring of radius R with uniform positive charge density X per unit l ength is located in the y-z plane with its centre at the origin O. Aparticle of mass m and positive charge q is projected from the Xq_ point P(R V3 , 0 , 0 ) on the positive x-axis directly towards O, with an initial kinetic energy *tb . (A ) The particle crosses O and goes to infinity. (B) The particle returns to P. • (C ) The particle will just reach O. (D) The particle crosses O and goes to -RV3. 0 1 2 0 2 0 d 0 (f| Bansal Classes Question Bank on Electrostatics [15]

Q.35 A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge + q. If it strikes the surface of sphere wit h speed u,findthe minimum speed u so that it can penetrate through the sphere. ( Neglect all resistance forces orfiictionacting on bullet except electrostatic fo rces) + + , ++ + \ m + I t+ + + V3q (C) yj 87i£ mR (D) yj 47ts mR (B) ^47is mR Q.36 In space of horizontal EF (E = (mg)/q) exist as shown in figure and a ////////// mass m attached at the end of a light rod. If mass m is releasedfromthe position shown in figure find the ang ular velocity of the rod when it X passes through the bottom most position (A) ^27rs mR 0 0 0 0 ( A ) v (B) - f (D) Q.37 Two identical particles of mass m carry a charge Q each. Initially one is a t rest on a smooth horizontal plane and the other is projected along the plane d irectly towardsfirstparticle from a large distance with speed v. The closed dist ance of approach be VOr 1 3Q 1 2Q 1 4Q (D) 4tc8O m v (A) 4TCS m v (B) 47is mv (C ) 4TIS MV Q.38 The diagram shows a small bead of mass m carrying charge q. The b ead can freely move on the smooth fixed ring placed on a smooth horizontal plane . In the same plane a charge +Q has also beenfixedas shown. The potential atthe point /* P due to +Q is V. The velocity with which the bead should projected fro m the point P so that it can complete a circle should be greater than I qV 3qV 6 qV (D)none (A) V m (B) m (C) m Q.39 Electricfieldgiven by the vector E = xi + yj is present in the XY plane. (0,L) A small ring carrying charge +Q, which canfre elyslide on a smooth non conducting rod, is projetced along the rod from the poi nt (0, L) such \ that it can reach the other end ofthe rod. What minimum velocit y should be given to the ring?(Assume zero gravity) (A) (QL /m) (B) 2(QL /m) (C) 4(QL /m) (D)(QL /2m) 2 0 0 2 0

2 2 2 1/2 2 1/2 2 1/2 2 1/2 Q.40 A unit positive point charge of mass m is projected with a velocity V insid e the tunnel as shown. The tunnel has been made inside a uniformly charged non c onducting sphere. The minimum velocity with which the point charge should be pro jected such it can it reach the opposite end of the tunnel, is equal to (A) [pR /4ms ] (B) [pR /24ms ] (C) [pR /6me ] (D) zero because the initial and thefinalp oints are at same potential. > 2 0 1/2 2 0 1/2 2 0 1/2 (f| Bansal Classes Question Bank on Electrostatics [15]

Q.41 A conducting sphere of radius a has charge Q on it. It is enclosed by a neu tral conducting concentric spherical shell having inner radius 2a and outer radi us 3a. Find electrostatic energy of system. 5kQi JikQl 12 a 12 a 2a 1 Q.42 A par ticle of mass 1 kg & charge. — pC is projected towards a non conducting fixed sphe rical shell having the same charge ] uniformly distributed on its surface. Find the minimum initial velocity of projection required if the particle just grazes the shell. ( A ) O ( D ) n o n e z^LZX. I l^T'j —' A "f from » (A) J 7 m/s [2 (B) 2 J— m/s [2 (C) — m/s 2 (D) none of these Y Q. 4 3 The diagram shows three infinitely long uniform line charges placed on 3X the X, Y and Z axis. The work done in moving a unit positive charge from(l, 1, l)to (0,1,1) is equal to (A)(Un2)/27is (B) (X In 2) /ne 2X (C)(3Xln2)/2ne (D)Non e £ Q. 44 A charged particle of charge Q is heldfixedand another charged particle ofmass m and charge q (ofthe same sign) is released from a distance r. The impul se of the force exerted by the external agent on the fixed charge by the time di stance between Q and q becomes 2r is Qq I Qqm Qqm Qqm 0 0 0 ( D ) P ^ 0 r Q.45 In a uniform electricfield,the potential is 10V at the origin of coordinate s, and 8 V at each of the points /T (1,0,0),(0,1,0) and (0,0,1). The potential a t the point (1,1,1) will be (A) 0 (B) 4 V (C) 8 V (D)10V Q.46 In a regular polyg on of n sides, each corner is at a distance r from the centre. Identical charges are placed at (n - 1) corners. At the centre, the intensity is E and the potent ial is V. The ratio V/E has magnitude. (A)rn (B)r(n-l) (C)(n-l)/r (D)r(n-l)/n Q.

47 The equation of an equipotential line in an electricfieldis y=2x, then the el ectricfieldstrength vector at (1,2) maybe i (A) 4i + 3j (B) 4i + 8j (C) 8 i + 4 j (D) - 8 i + 4 j Q.48 The electric field in a region is given by : E = (4axy Vz )i + (2ax Vz )j + (ax / Vz )k, where a is a positive constant. The equation of an equipotential surface will be of the form (A) z = constant / [x y ] (B) z = c onstant / [xy ] (C) z - constant / [x y ] (D) None Q.49 A charge 3 coulomb exper iences a force 3000 N when placed in a uniform electricfield.The potential diffe rence between two points separated by a distance of 1 cm along thefieldlines is ^ (A) 10 V (B) 90 V (C) 1000V (D) 9000V Q. 5 0 Two point charges of+Q each have been placed at the positions (-a /2,0,0) and (a / 2,0,0). The locus of the point s where - Q charge can be placed such the that total electrostatic potential ene rgy of the system can become equal to zero, is represented by which of the follo wing equations? (A) Z + (Y-a) = 2a (B) Z + (Y-a) = 27a /4 (C) Z + Y = 15a /4 (D) None 1 2 2 3 2 2 4 2 2 2 2 2 2 2 2 2 (f| Bansal Classes Question Bank on Electrostatics [15]

Q.51 Figure shows equi-potential surfaces for a two charges system. At which of the labeled points point will an I electron have the highest potential energy? ( B) Point B (A) Point A (C) Point C (D) Point D Q.52 Auniform electricfieldhaving strength £ is existing in x-y plane as shown infigure.Find the p.d. between origi n O & A(d, d, 0) (A) Ed (cos0 + sin0) (B) -Ed (sin6 - cos0) (C) 4 l Ed (D) none of these £ $ Q.53 In a certain region of space, the potential is given by: V = k[2x - y + z ] . The electric field at the point (1,1,1) has magnitude = (A) k-/6 (B)2kV6 (C)2k V3 (D) 4kV3 Q.54 Find the force experienced by the semicircular rod charged with a charge q, placed as shown in figure. Radius of the wire is R and the line of charge with linear charge density A, is passing through its centre and perpendic ular to the plane ofwire. 1 Aq Aq A,q Aq (B) 7L S R 2tt S R 2 2 2 2 0 2 0 Q.55 Uniform electric field of magnitude 100 V/m in space is directed along the line y = 3 + x. Find the potential difference between point A (3,1) & B (1,3) (A ) 100 V (B)200V2V (C)200V (D)0 Q.56 A wheel having mass m has charges +q and -q on diametrically opposite points. It remains in equilibrium on a rough inclined plane in the presence of uniform vertical electricfieldE = mg mg tan 0 mg (D)non e (A) (B) 2q (C) 2q Q.57 An equilateral triangle wire frame of side L having 3 p oint charges at its vertices is kept in x-y plane as shown. Component of electri cfielddue to the configuration in z direction at (0,0, L) is [origin is centroid of triangle] 9 kg 9V3kq (B)zero (D) None (C) 8L 8L Q.58 A, B, C, D, P and Q are points in a uniform electricfield.The potentials B C a these points are V (A) = 2 volt. V (P) = V (B) = V (D) = 5 volt. f y* P V (C) = 8 volt. The electric fie ld at P is Q D A (A) 10 Vm" along PQ (B) 15^2 V n r along PA i — • (C) 5 V n r along PC (D) 5 V m along PA 0.2 m £ 2 2 1 1 1 _1 (f|BansalClasses Question Bank on Electrostatics [15]

Q. 5 9 A and B are two points on the axis and the perpendicular bisector respect ively of an electric dipole. A and B are far awayfromthe dipole and at equal dis tancefromit. The field at A and B are E and E . (A)E,A _: FB (B)E 2E (D) | E | = — | E |, and E is perpendicular to E^ y (C) E = - 2 E A B y A = B a R B a 0 Q.60 Figure shows the electric field lines around an electric dipole. Which ofth e arrows best represents the electricfieldat point P ? (B)\ £ (A) | Q.61 A dipole consists of two particles one with charge +lpC and mass 1kg and the other with c harge -1 pC and mass 2kg separated by a distance of 3m. For small oscillations a bout its equilibrium position, the angularfrequency,when placed in a uniform ele ctricfieldof 20kV/m is (A) 0.1 rad/s (B) 1.1 rad/s - (C) 10 rad/s (D)2.5rad/s Q. 62 The dipole moment of a system of charge +q distributed uniformly on an arc of radius R subtending an angle 7t/2 at its centre where another charge -q is plac ed is: •f qR 2qR 2V2qR V2qR (A) 71 (B) 71 (D) 71 71 Q.63 An electric dipole is kep t on the axis of a uniformly charged ring at distance R/ V2 from the centre ofth e ring. The direction ofthe dipole moment is along the axis. The dipole moment i s P, charge of the ring is Q and radius of the ring is R. The force on the dipol e is nearly 4kPQ 2kPQ i ( A ) 4kPQ2 ^ (B)^rT ( )W3V 3 R c ( D ) z e r o ( C ) / ( P ) / Q.64 Alarge sheet carries uniform surface charge density a. Arod oflength 21 has a linear charge density X on one half and -X on the second half. The rod is hin ged at mid point O and makes an angle 6 with the normal to the sheet. The torque experienced by the rod is y (A)0 (C) — s i n e GXI 2 (B) ^7~sine 2s 0 aXl 2 (D) 2 T Q.65 Two short electric dipoles are placed as shown. The energy of elect ric interaction between these dipoles will be 2kPjP cose - 2kPjP sin 6 •2kP]P cosB - 4kPjP cos 6 (A) (B) (C) (D)

2 2 2 2 aXl Q.66 Point P lies on the axis of a dipole. Ifthe dipole is rotated by 90° anticloc k wise, the electricfieldvector E P u (A)at90°will rotate by (B) 180° clock wise (C) 90° anti clock wise (D) none clock wise (f| Bansal Classes Question Bank on Electrostatics [15]

Q.67 4 charges are placed each at a distance 'a' from origin. The dipole moment of configuration is (A) 2qaj (B) 3qaj (C)2aq[i + j] (D)none " y 2q -2q Q.68 Both question (a) and (b) refer to the system of charges as shown in the fi gure. A spherical shell with an inner radius 'a' and an outer radius 'b' is made of conducting material. A point charge +Q is placed at the centre of the spheri cal shell and a total charge - q is placed on the shell. (a) Charge - q is distr ibuted on the surfaces as (A) - Q on the inner surface, - q on outer surface (B) - Q on the inner surface, - q + Q on the outer surface (C) +Q on the inner surf ace, -q - Q on the outer surface (D) The charge - q is spread uniformly between the inner and outer surface. Assume that the electrostatic potential is zero at an infinite distancefromthe spherical shell. The electrostatic potential at a di stance R (a < R < b)fromthe centre of the shell is (A) o (where K = 1 47IS 0 (b) (B)^ ) 0 Q.69 In a as E = E rallel to (B)s E a 3 0 0 3 E

region of space, the electric field is in the x direction and is given x i. Consider an imaginary cubical volume of edge a, with its edges pa the axes of coordinates. The charge inside this volume is: 1„ 1 (A) zero (C) — o (D) s E a a 3 7 0 0 2

Q. 7 0 Electricfluxthrough a surface of area 100 m lying in the xy plane is (in V-m) if E = i + V2 j + V3k (A) 100 (B) 141.4 (C) 173.2 (D)200 2 Q. 71 An infinite, uniformly charged sheet with surface charge density a cuts th rough a spherical Gaussian surface of radius R at a distance xfromits center, as shown in the figure. The electric flux O through the Gaussian surface is HIV a 7tR u (A) " 7 — o 2 E ztiiin. R- x x ) a 27t( - (B) ~ o 2 2 £ / V _ x /\ N ' o o Q. 72 Two spherical, nonconducting, and very thin shells ofuniformly distr ibuted positive charge Q and radius d are located a distance 1 Odfromeach other. A positive point charge q is placed inside one ofthe shells at a distance d/2 f rom the center, on the line connecting the centers of the two shells, as shown i n the figure. What is the net force on the charge q? b s (C) —

TT(R-X) 2 CT (D) * TT(R -X 2 ) A 2 (A) 36l7T8 d 0 qQ 2 t o t h e l e f t (B) 36l7ts d totheright 0 2 qQ ' 362qQ < > 36l7TE d C 0 2 t o t h e l e f t 360qQ CD) 2 totheright 3 6 l 7 r g ( ) d « iod w (f| Bansal Classes Question Bank on Electrostatics [15]

Q.73 A positive charge q is placed in a spherical cavity made in apositively cha rged sphere. The centres of sphere and cavity are displaced by a small distance / . Force on charge q is: (A) in the direction parallel to vector J (B) in radia l direction (C) in a direction which depends on the magnitude of charge density in sphere (D) direction can not be determined. Q.74 There are four concentric sh ells A, B, C and D ofradii a, 2a, 3a and 4a respectively. Shells B and D are giv en charges +q and -q respectively. Shell C is now earthed. The potential differe nce V - V is: Kq Kq Kq Kq (B)' 3a — (D) 6a ^ y A c Q.75 A metal ball of radius R is placed concentrically inside a hollow metal sph ere of inner radius 2R and outer radius 3R. The ball is given a charge +2Q and t he hollow sphere a total charge - Q. The electrostatic potential energy ofthis s ystem is: 7Q 5Q 5Q (D) None (A) 247is R (B) 167t8 R (C) 87is R Question No. 76 t o 80 % fcfcjtft wuLuffx' b e soVv€, (C) T = 1/2 7r r (2KA,q/m) (D) T = l/27tr (m/KTtAq) where K = l/4n e i/2 2 2 3 1/2 172 0 (f| Bansal Classes Question Bank on Electrostatics [15]

Q.30 Charge Q is distributed non-uniformly over a ring of radius R, P is a point on the axis of ring at a distance •Sr from its centre. Which ofthe following is a wrong statement. KQ (A) Potential at P is -—2R V3KQ (B) Magnitude of electric fie ld at P may be greater than y~ 8R V3KQ (C) Magnitude of electric field at P must be equal to j 8R V3KQ (D) Magnitude of electric field at P cannot be less than — 8R Q.31 An electric dipole moment p = (2.0i + 3.0j) pC. m is placed in a uniform el ectric field E = (3.0i + 2.Ok) x 10 NC" . (A) The torque that E exerts on p is ( 0.6i - 0.4 j - 0.9k) Nm. (B) The potential energy of the dipole is -0.6 J. (C) T he potential energy ofthe dipole is 0.6 J. (D) Ifthe dipole is rotated in the el ectric field, the maximum potential energy of the dipole is 1.3 J. 5 1 Q.32 Which ofthe following is true for thefigureshowing electric lines of force? (E is electricalfield,V is potential) (A)E >E (B)E >E (C)V >V (D)V >V a b b a A B b a Q.33 If we use permittivity s, resistance R, gravitational constant G and voltag e V as fundamental physical quantities, then (A) [angular displacement] = s R G°V (B) [Velocity] = e - ' R ' W (C) [dipole moment] = s ^ V V (D) [force] = e ' R ^ V Q.34 Units ofelectric flux are 0 0 0 1 2 < c ) v o l , m ( D ) V o l t m 3 Q.35 Which ofthe following statements are correct? (A) Electricfieldcalculated b y Gauss law is thefielddue to only those charges which are enclosed inside the G aussian surface. (B) Gauss law is applicable only when there is a symmetrical di stribution of charge. (C) Electric flux through a closed surface will depends on ly on charges enclosed within that surface only. (D) None of these Q.36 Mark the correct options: (A) Gauss's law is valid only for uniform charge distributions . (B) Gauss's law is valid only for charges placed in vacuum. (C) The electricfi eldcalculated by Gauss's law is thefielddue to all the charges. (D) The flux of the electricfieldthrough a closed surface due to all the charges is equal to the flux due to the charges enclosed by the surface. (f| Bansal Classes Question Bank on Electrostatics [15]

Q.37 A thin-walled, spherical conducting shell S ofradius R is given charge Q. T he same amount of charge is also placed at its centre C. Which of the following statements are correct? Q (A) On the outer surface of S, the charge density is 2 7tR • (B) The electric field is zero at all points inside S. (C) At a point just o utside S, the electric field is double the field at a point just inside S. (D) A t any point inside S, the electricfieldis inversely proportional to the square o fits distancefromC. 2 Q.38 A hollow closed conductor of irregular shape is given some charge. Which of the following statements are correct? (A) The entire charge will appear on its o uter surface. (B) All points on the conductor will have the same potential. (C) All points on its surface will have the same charge density. (D) All points near its surface and outside it will have the same electric intensity. Q.39 Charges Q, and Q lies inside and outside respectively of a closed surface S. Let E be th efieldat any point on S and < be the flux of E over S. ( > (A) If Q j changes, b oth E and , tL Hy / ! (C)~ A -JL (D) mMk Q.17 Which ofthe following reading is most accurate (A) 4.00 cm (B) 0.004 mm (C) 40.00 cm (D) 4.00 m Q. 18 The least count of a stop watch is 1/5 sec. The time of 20 oscillations of a pendulum is measured to be 25 sec. The minimum percentage error in the measur ement oftime will be (A) 0.1% (B) 0.8% (C) 1.8% (D)8% Q.19 A vernier callipers h aving 1 main scale division = 0.1 cm is designed to have a least count of 0.02 c m. If n be the number of divisions on vernier scale and m be the length ofvernie r scale, then (A) n= 10, m=0.5 cm (B) n=9, m=0.4 cm (C)n=10,m=0.8 cm (D) n=10, m =0.2 cm Q.20 Solve with due regard to significant digits >.91x0,3842 (i) V6.5-6. 32 (ii) 0.080 Q.21 Abody travels uniformly a distance of (13.8 ± 0.2)m in time (4. 0 ± 0.3) sec. Calculate its velocity. Q.22 The main scale of a vernier calipers re ads in millimeter and its vernier is divided into 10 divisions which coincide wi th 9 divisions ofthe main scale. When the two jaws ofthe instrument touch each o ther the seventh division of the vernier scale coincide with a scale division an d the zero of the vernier lies to the right of the zero of main scale. Furthermo

re, when a cylinder is tightly placed along its length between the two jaws, the zero of the vernier scale lies slightly to the left of 3.2 cm and the fourth ve rnier division coincides with a scale division. Calculate the measured length of the cylinder. Q.23 A short circuit occurs in a telephone cable having a resista nce of 0.45 Onr . The circuit is tested with a Wheatstone bridge. The two resist ors in the ratio arms ofthe Wheatstone bridge network have values of 100Q and 11 10Q respectively. Abalance condition is found when the variable resistor has a v alue of4000. Calculate the distance down the cable, where the short has occurred . 1 Q.24 5.74 gm of a substance occupies a volume of 1.2 cm . Calculate its density with due regard for significant figures. 3 n > V2 1 . ° } viscosity. Ifthe cross s ection at C is one halfthat at D and if D is at a distance h, below the level of liquid in A to what height h (in r terms of h, )will liquidrisein pipe E ? 8m Q .8 For the system shown in thefigure,the cylinder on the left at L has a mass of 600kg and a cross sectional area of800 cm . The piston on the right, at S, has c ross sectional area 25cm and negligible weight. 600kg If the apparatus isfilledw ith oil.(p = 0.75 gm/cm ) Find the force F required to hold the system in equili brium. 2 3 Q.l EXERCISE # I I 2 \ «r~"i 2 2 3 Q.9 (a) (b) (c)

A siphon has a uniform circular base of diameter ~j= cm with its crest V7t 1.8m A1.8 m above water level as infigure.Find velocity of flow discharge rate ofthe flow in m /sec. absolute pressure at the crest level A. [Use P = 10 N/m & g = 10 m/s ] 3 0 5 2 2 8 3.6m ^Bansal Classes Fluid Mechanics [12]

Q.10 A large tank isfilledwith two liquids of specific gravities 2a and c. Two h oles are h,4 made on the wall ofthe tank as shown. Find the ratio ofthe distance sfromO ofthe points on the ground where the jets fromholes A& B strike. h/2 Ml Q .ll The horizontal pipe shown in the diagram has a cross sectional area of 40cm at the wider position and 10cm at the narrow poriton. A liquid of 7 JT» V _ • 10cm 4 0cm specific gravity 1.6 isflowingin the pipe with volumeflowrate equal to 5 x 1 0 m /s. Find the difference in the heights h between the mercury hJ mercury colu mn in the manometer tube. (p = 13.6 * 10 kg/m ) Q.12(a)A spherical tank of 1.2m radius is half filled with oil of relative density 0.8 . Ifthe tank is given a h orizontal acceleration of 10 m/s . Calculate the inclination ofthe oil surface t o horizontal and maximum pressure on the tank. (b) The volume of an air bubble i s doubled as it rises from the bottom of a lake to its surface. If the atmospher ic pressure is H m ofmercury & the density of mercury is n times that of lake wa ter. Find the depth of the lake. Q.13 A test tube of thin walls has some lead sh ots in it at its bottom and the system floats vertically in water, sinking by a length 1= 10cm. A liquid of density less than that of water, is poured into the tube till the levels inside and outside the tube are even. If the tube now sinks to a length / =40cm, the specific gravity ofthe liquid is . Kerosene W 2 2 2 2 -3 3 mg 3 3 A 2 o s=0.8 Q.14 For the arrangement shown in thefigurethe value ofh ifthe pressure differen ce between the vessel Aand B is 3 kN/m is 2 Q.15 An open cubical tank completelyfilledwith water is kept on a horizontal sur face. Its acceleration is then slowly increased to 2m/s as shown in the Fig. The side of the tank is lm. Find the mass ofwater that would spill out ofthe tank. 2 1m •2m/s 2 lm Q.16 In air an object weighs 15N, when immersed completely in water the same object weighs 12N. When immersed in another liquid completely, it weighs 13N. Fi nd (a) the specific gravity of the obj ect and (b) the specific gravity ofthe ot her liquid. Q.17 Compute the work which must be performed to slowly pump the wat er put of a hemispherical reservoir of radius R - 0.6 m. Q.18 Block Ainfigurehan gs by a cord from spring balance D and is submerged in a liquid C contained in a

beaker B. The mass of the beaker is 1 kg & the mass of theliquidis 1.5kg. The b alance Dreads 2.5 kg & balance E reads 7.5kg.. The volume ofblock Ais 0.003 m . (i) What is the density ofblock & the liquid, (ii) What will each balance read i fblock is pulled out of the liquid. 3 Q.19 A solid cube, with faces either vertical or horizontal, isfloatingin a liqu id of density 6 g/cc. It has two third ofits volume submerged. If enough water i s addedfromthe top so as to completely cover the cube, whatfractionof its volume will remain immersed inthe liquid? foBansalClasses Fluid Mechanics [10]

Q.20 A ball is given velocity v (greater than the terminal velocity v ) in downw ard direction inside a highly viscous liquid placed inside a large container. Th e height ofliquid in the container is H. The ball attains the terminal velocity just before striking at the bottom of the container. Draw graph between velocity ofthe ball and distance moved by the ball before getting terminal velocity. 0 T 2 2 XT Q.21 Two arms of a U-tube have unequal diameters d, = 1.0 mm and d = 1.0 cm. Ifw ater (surface tension 7 x 10~ N/m) is poured into the tube held in the vertical position,findthe difference of level ofwater in the U-tube. Assume the angle ofc ontact to be zero. Q.22 A spherical ball of radius 1 x 10" m and density 10 kg/m falls freely under gravity through a distance h before entering a tank of water . If after entering the water the velocity of the ball does not change, find h. The viscosity of water is 9.8 x 10 N-s/m . 4 4 3 -6 2 Q. 23 Calculate the rate of flow of glycerine of density 1.25 x 10 kg/m through the conical section of a pipe if the radii of its ends are 0. lm & 0.04m and the pressure drop across its length is 10N/m . 3 3 2 Q. 24 The tank infigdischarges water at constant rate for all water levels above the air inlet R. The height above datum to which water would rise in the manome ter tubes M and N respectively are & Q.25 A uniform cylindrical block of length / density d, and area of cross section A floats in a liquid of density d contain ed in a vessel (d >d ). The bottom of the cylinder just rests on a spring of con stant k. The other end ofthe spring is fixed to the bottom ofthe vessel. The wei ght that may be placed on top ofthe cylinder such that the cylinder is just subm erged in the liquid is 2 2 1 Open io atmosphere 40cm 20cm 11 Q.26 Find the speed ofrotation of 1 m diameter tank, initially full ofwater such that water surface makes an angle of 45° with the horizontal at a radius of 30 cm . What is the slope ofthe surface at the wall ofthe tank. Q. 27 A vertical unifo rm U tube open at both ends contains mercury. Water is poured in one limb until the level of mercury is depressed 2cm in that limb. What is the length ofwater c olumn when this happens. Q. 28 Apiece of copper having an internal cavity weigh 264gm in air and 22 lgm in water. Find the volume of cavity. Density of copper i s 8.8 gm/cc. Q.29 A vessel contains oil density = 0. 8gm/cm . A homogeneous sphe re floats with halfits volume immersed in mercury and the other half in oil. The density ofthe material ofthe sphere in gm/cm is Q.30 An expansible balloon fill ed with air floats on the surface of a lake with 2/3 of its volume submerged. Ho w deep must it be sunk in the water so that it is just in equilibrium neither si nking further nor rising ? It is assumed that the temperature of the water is co nstant & that the height of the water barometer is 9 meters. 3 3 fo Bansal Classes Fluid Mechanics [10]

EXERCISE # II Q.l Q.2 (a) (b) (c) Q.3 A solid block ofvolume V=10~ m and density d=800kg/m is tied to one end of a str ing, the other end ofwhich istiedto the bottom ofthe vessel. The vessel contains 2 immiscible liquids of densities p.^l 000kg/ m and p =15 00kg/m . The solid bl ock is immersed with 2/5 th of its volume in the liquid of higher density & 3/5t h in the liquid of lower density. The vessel is placed in an elevator which is m oving up with an acceleration of a=g/2. Find the tension in the string. [g= 1 Om /s ] An open rectangular tank 5m x 4m * 3 m high containing water upto a 3m heig ht of 2m is accelerated horizontally along the longer side. "Wirier 2m Determine the maximum acceleration that can be given without spilling the water. 5m Calcu late the percentage ofwater split over, ifthis acceleration is increased by 20%. If initially, the tank is closed at the top and is accelerated horizontally by 9m/s ,findthe gauge pressure at the bottom of thefrontand rear walls ofthe tank. A level controller is shown inthefigure.It consists ofa thin circular plug of d iameter 10cm and a cylindricalfloatof diameter 20cm tied together with a lightri gidrod Float of length 10cm. The plugfitsin snugly in a drain hole at the bottom of the tank which opens into atmosphere. As waterfillsup and the level reaches height h, the 10cm plug opens. Findh. Determine the level ofwater in the tank wh en the plug closes -plug again. Thefloathas a mass 3kg and the plug may be assum ed as massless. 3 3 3 3 2 3 2 2 Q.4 A closed tube in the form of an equilateral triangle of side / contains equal vo lumes of three liquids which do not mix and is placed vertically with its lowest side horizontal. Find x in the figure if the densities of the liquids are in A. P. A ship sailingfromsea into ariversinks X mm and on discharging the cargorises Y mm. On proceeding again into sea the shiprisesby Z mm. Assuming ship sides to be vertical at water line,findthe specific gravity of sea water. A conical vesse l without a bottom stands on a table. Aliquid is poured with the vessel & as soo n as the level reaches h, the pressure of the liquid raises the vessel. The radi us of the base ofthe vessel is R and half angle of the cone is a and the weight of the vessel is W. What is the density ofthe liquid ? As the arrangement shown in thefigis released the rod of mass M moves down into the water. Friction is ne gligible and the string is inextensible. Find the acceleration of the system w.r .t. the distance moved by each mass. Find the time required to completely immers e the rod into water if ™ _ P - P water M p density of rod water density of water : Q.5 Q.6 Q.7 (a) (b) ////J/// w M JL Q. 8 The interface of two liquids of densities p and 2p respectively lies at the poin t A ^ in a U tube at rest. The height of liquid column above A is 8 a/3 where AB -a. The cross sectional area of the tube is S. With what angular velocity the tu be must be whirled about a vertical axis at a distance 'a' such that the interfa ce of the liquids shifts towards B by 2a/3. ^Bansal Classes

Fluid Mech an ics [9]

Q.9 A closed cylindrical tank 2m high & 1 m in diameter contains 1.5 m of water. Whe n the angular velocity is constant at 20.0 rad/s, how much of the bottom of the tank is uncovered? (The cylinder is rotated about vertical axis of symmetry pass ing through its length.) o o Q.10 A cylinder of height H isfilledwith water to a height h (h < H), & is place d on a horizontal floor. Two small holes are punched at time t = 0 on the vertic al line along the length of the cylinder, one at a height hj from the bottom & t he other a depth hj below the level ofwater in the cylinder. Find bo the relatio n between hj & hj such that the instantaneous waterjets emerging from the cylind er from the two holes will hit the ground at the same point. Q.ll A cylindrical tank with a height of h = lm isfilledwith water up to its rim. What time is requ ired to empty the tank through an orifice in its bottom? The cross sectional are a of the orifice is (l/400)th ofthe tank. Find the time required for the same am ount ofwater to flow out of the tank ifthe water level in the tank is maintained constant at a height of h = 1 m from the orifice. Q.12 A Conical funnel whose h eight H=20cm isfilledwith water. The radius of the upper opening R - 12 cm. The lower opening through which the water begins to flow out ofthe funnel has the ra dius r=0.3cm. (a) In what time is the water level in the funnel lowered by 5 cm ? (b) When will the funnel be emptied ? 20cm . Q.13 A water clock used in ancien t Greek is designed as a closed vessel with a small orifice O. The time is deter mined according to the level ofthe water in the vessel. What should be the shape of the vessel be for the time scale to be uniform. Find mathematical equation g overning curve AOB. Q.14 For the arrangement shown in thefigure.Find the time interval 80cm after which the water jet ceases to cross the wall. Area of the tank = 0.5 m . A rea of the orifice = 1 cm . 2 2 3 2 0.81 m Q.15 A cylindrical tank having cross-sectional area A = 0.5 m isfilledwith two l iquids of density p, = 900 kgm~ , to a height h=60cm as shown in the figure. A s mall hole having area a = 5 cm is made in right vertical wall at a height y=20cm from the bottom. Calculate h F*± C velocity of efflux. O © horizontal force F to ke ep the cylinder in static equilibrium, ifit is placed on a smooth horizontal pla ne, value of (iii) minimum and maximumm = 0.01.F to keep the cylinder at rest. T he coefficient of friction between cylinder and the plane is velocity of the top most layer ofthe liquid column and also the velocity ofthe boundary separating the (iv) two liquids. base S 4000 Q.16 A cylindrical wooden float whosewood area 0,8=gf/cm cm & the altitude H = 50 cm drifts on the water surface. Specific weig ht of d= . (a) What work must be performed to take the float out of the water ? (b) Compute the work to be performed to submerge completely the float into the w ater. Q.17 A10cm side cube weighing 5N is immersed in a liquid ofrelative densit y 0.8 contained in a rectangular tank of cross sectional area 15cmx 15cm. If the tank contained liquid to a height of 8cm before the immersion, determine the le vels of the bottom of the cube and the liquid surface. 2 2 3 fo Bansal Classes Fluid Mechanics [10]

Q.18 A jug contains 15 glasses of orange juice. When you open the tap at the bot tom it takes 12sectofilla glass with juice. If you leave the tap open, how long will it take tofillthe remaining 14 glasses and thus empty the jug? Q.19 An inte rstellar explorer discovers a remarkable planet made entirely of a uniform incom pressiblefluidon density p. The radius ofthe planet is R and the acceleration of gravity at its surface is g. What is the pressure at the center of the planet. Q.20 A cylindrical rod oflength I=2m & density —floatsvertically in a liquid B of density p as shown in Fig (a). A (a) Show that it performs SHM when pulled sligh tly up & released & (b) (a) find its time period. Neglect change in liquid level . (b) Find the time taken by the rod to completely immerse when releasedfromposi tion shown in (b). Assume that it remains vertical throughout its motion, (take g = % m/s ) 2 2 Q.21 A uniform cylinder of a light material of length /=0.8m and radius of cross section r = 0.01 m floats on a liquid of specific density p = 0.9 upto half its length. The container of the liquid is a long cylindrical beaker of radius R = 0.04m. Another perfectly immiscible liquid of specific density o = 0.6 is now sl owly poured all along the inner periphery of the beaker at a uniform rate of v = 0.25x 10 m /s and it spreads itself uniformly over the first liquid. Find the v elocity with which the cylinder will rise or sink in the liquid. 4 3 Q.22 Auniform rod of length b capable oftuning about its end which is out ofwate r, rests inclined to the vertical. Ifits specific gravity is 5/9,findthe length immersed in water. Q.23 An open cylindrical vessel of large cross-section A cont ains liquid upto a height H = 120cm. After an orifice of area A/1000 at a height ofh = 20cm is opened. (a) Calculate liquid heights above orifice for which it f alls on both ends of H horizontal plate. (b) How long will the liquid be falling on the plate. Given: plate AB is of length 60cm. (g = 10m/s ) 2 2 T 20cm A B Q.24 A cylindrical vessel ofheightH = 4m&area of cross section 1 m filled with w ater rests on a stand of same height H. It has a small plugged hole near its bot tom. When plug is removed the liquid starts to come out. (a) Find the range ofth e liquid as a function ofinstantaneous height ofthe liquid in the upper vessel ( b) Find the volume of liquid collected in a large initially empty vessel lying o n floor at a distance Hfromthe stand. Assume that water falling on to thefloordo es notflowinto the vessel. Q.25 A cube with a mass'm' completely wettable by wat er floats on the surface of water. Each side of the cube is 'a'. What is the dis tance h between the lower face of cube and the surface of the water if surface t ension is S. Take density ofwater as p Take angle of contact in zero. ZZ3 fo Bansal Classes Fluid Mechanics [10]

EXERCISE # III Q.l A horizontal pipe line carries water in a streamline flow. At a point along the pipe where the cross-sectional area is 10 cm , the water velocity is 1 ms" & the pressure is 2000 Pa. The pressure of water at another point where the cross sectional area is 5 cm , is pa. [ Density ofwater = 10 kg. mr ] [ JEE '94,2 ] 2 1 2 3 3 Q.2 A container oflarge uniform cross-sectional area A resting on a horizontal surfa ce, holds two immiscible, non-viscous & incompressible liquids of densities d & 2d, each of height — as shown in figure. The lower density liquid is open to the a tmosphere having pressure P . 0 (a) A homogeneous solid cylinder oflength L fI < ) cross-sectional area A,is immerse d such that it floats with its axis vertical at the liquid-liquid 5 (i) 00 (b) CO (ii) (iii) Q.3 (0 (ii) (iii) Q.4 interface with the length — in the denser liquid. Determine: The density D ofthe s olid & The total pressure at the bottom of the container. The cylinder is remove d and the original arrangement is restored. A tiny hole of area s (s « A) is punch ed on the vertical side of the container at a height h j^fc < -^J . Determine Th e initial speed of efflux of the liquid at the hole ; The horizontal distance x travelled by the liquid initially & The height h at which the hold should be pun ched so that the liquid travels the maximum distance x initially. Also calculate x . [Neglect the air resistance in these calculations]. [JEE '95,10] m m m Q.5 (0 (ii) A cylindrical tank 1 m in radius rests on a platform 5 m high. Initially the tan k isfilledwith water to a height of 5m. A plug whose area is 1CT m is removed fr om an orifice on the side ofthe tank at the bottom. Calculate the following: ini tial speed with which the water flows from the orifice ; initial speed with whic h the water strikes the ground & time taken to empty the tank to halfits origina l value. [ REE '95,5] A thin rod of length L & area of cross-section S is pivote d at its lowest point P inside a stationary, homogeneous & non-viscous liquid (F igure). The rod is free to rotate in a vertical plane about a horizontal axis pa ssing through P. The density d, of the material of the rod is smaller than the e ntity d ofthe liquid. The rod is displaced by a small angle 9 from its equilibri um position and then released. Show that the motion of the rod is simple harmoni c and determine its angular frequency in terms ofthe given parameters. [ JEE '96 , 5 ] A large open top container of negligible mass & uniform cross-sectional ar ea A has a small hole of cross-sectional area A/100 in its side wall near the bo ttom. The container is kept on a smooth horizontal floor and contains a liquid o f density p and mass m.. Assuming that the liquid starts flowing out horizontall y through the hole at t = 0, calculate the acceleration ofthe container and its velocity when 75 % ofthe liquid has drained out. [ JEE 97 , 5 ] 4 2 2 ^Bansal Classes Fluid Mechanics [12]

Q.6 A nonviscous liquid of constant density 1000 kg/m flows in a streamline motion a long a tube of variable cross section. The tube is kept inclined in the vertical plane as shown in the figure. The area of cross section of the tube at two poin ts P and Q at heights of 2 meters and 5 meters are respectively 4 x 10 m and 8 x 10~ m . The velocity ofthe liquid at point P is 1 m/s. Find the work done per u nit volume by the pressure and the gravity forces as the fluid flows from point P to Q. [ JEE '97] 3 3 2 3 3 Q. 7 Water from a tap emerges vertically downwards with an initial speed of 1.0 ms" . The cross-sectional area of the tap is 10~ m . Assume that the pressure is cons tant throughout the stream ofwater, and that the flow is steady. The cross-secti onal area of the stream 0.15m below the tap is [ JEE '98, 2 ] (B) 1.0 xlO" m (C) 5.0 x 10" m (D) 2.0 x l0" m (A) 5.0 x 10~ m 1 4 2 4 2 5 2 5 2 5 2 Q.8 Q.9 A wooden stick of length 1, and radius R and density p has a small metal piece o f mass m (of negligible volume) attached to its one end. Find the minimum value for the mass m (in terms of given parameters) that would make the stick float ve rtically in equilibrium in a liquid of density a (>p). [ JEE '99,10] A large ope n tank has two holes in the wall. One is a square hole of side L at a depth yfro mthe top and the other is a circular hole of radius R at a depth 4yfromthe top. When the tank is completely filled with water, the quantities ofwater flowing ou t per secondfromboth holes are the same. Then, R is equal to: (A) (B) 2nL (C)L (D) 2it [JEE 2000 (Scr.)] Q.10 A hemispherical portion of radius R is removed from the b ottom of a cylinder of radius R. The volume ofthe remaining cylinder is V and it s mass is M. It is suspended by a string in a liquid of density p where it stays vertical. The upper surface of the cylinder is at a depth h below the liquid su rface. The force on the bottom of the cylinder by the liquid is [JEE 2001 (Scr.) ] (A)Mg (B)Mg-vpg (C) Mg + tz R h p g 2 (D) pg (V + 7iR h) 2 Q.ll A wooden block, with a coin placed on its top, floats in water as shown in figure. The distances I and h are shown there. After some time the coin falls in to the water. Then [JEE 2002 (Scr.)] (A) / decreases and h increases (B) I incre ases and h decreases (C) both I and h increase (D) both / and h decrease 3 yCoin Q.12 Auniform solid cylinder of density 0.8 gm/cm floats in equilibrium in a com bination of two non mixing liquids A and b with its axis vertical. The densities

ofthe liquids A and B are 0.7 gm/cm and 1.2 g/cm , respectively. The height of liquid Ais h = 1.2 cm. The length ofthe part of the cylinder immersed in liquid B is h^ = 0.8 cm. (a) Find the toal force exerted by liquid Aon the cylinder. (b ) Find h, the length ofthe part of the cylinder in air. (c) The cylinder is depr essed in such a way that its top surface is just below the upper surface ofliqui d A and is then released. Find the acceleration of the cylinder immediately afte r it is released. [JEE 2002] 3 3 A fo Bansal Classes Fluid Mechanics [10]

Q.13 Consider a horizontally oriented syringe containing water located at a heig ht of 1.25 m above the ground. The diameter ofthe plunger is 8 mm and the diamet er of the nozzle is 2 mm. The plunger is pushed with a constant speed of 0.25 m/ s. Find the horizontal range of water stream on the ground. Take g = 10 m/s2. [J EE 2004] D=8MM ^ IP d=2mm 1.25m A\\m\\Tu\\\\\u\\»u\\\u'\ Q.14 A solid sphere of radius R is floating in a liquid of density p with half o f its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find thefrequencyof these oscillations. [JEE 2004] Q.15 Water isfilledin a container upto height 3m. A small hole of area 'a ' is punched in the wall ofthe container a at a height 52.5 cmfromthe bottom. Th e cross sectional area ofthe container is A. If — =0.1 then v is (where v is the v elocity ofwater coming out ofthe hole) (D) 51.5 (A) 48 (B) 51 (C) 50 [JEE to2005 (Scr)] z Q.16 A U tube is rotated about one of it's limbs with an angular velocity a. Fin d the difference in height H of the liquid (density p) level, where diameter oft he tube d « L. [JEE 2005] . a fo Bansal Classes Fluid Mechanics [10]

ANSWER KEY EXERCISE # I Q Q.l Q.4 Q.7 Q.9 Q.ll 45°,9600V2 (gauge)N/m 19.6 m, 4 sec hj = 3 hj 375 24 15.6cm 2 , 2 2m 11 Q.5 Q.8 3 2.79 gm/cc 37.5 N 3 Q.6 at the water surface, h/2 Q.10 V3:V2 (a) 6^2 m/s, (b)9.6V2 xlQ- M /sec, (c) 4.6 x 10 N/m 4 2 Q.12 (a)9600 ^ , (b) nH Q.13 0.75 Q.14 -0.5m„2.5 m Q.16 (a) 5, (b) 2/3 Q.17 101.8 Kgf-m Q.15 100kg 3 5000 Q.18 (i) 2500 kg/m , —— kg/m , (ii) \ = 7.5 kg, f ^ = 2.5 kg 3 Q.19 3/5 Q.23 6.43 x 10^m /s 3 Q.20 | Dist. moved H Q.21 2.5 cm Q.22 20.4 m + Ag Q.24 20cm, 60cm Q.27 54.4 cm Q.25

£(d -d ) 2 x V2 d Q.26 co= 3 1 0 5 rad/s, t a n a = Q.28 13cc Q.29 7.2gm/cm Q.30 4.5m EXERCISE # II Q.l Q 3 6N 2(3 + 7t) —li^— 0 2 2 6 3 + 71 -^"!^" 0 Q.2 4m/s , 10%, 0, 45kPa 2 1 9 5 Q 4 x = 1 / 3 Q.5 y-x+z Q.6 Q.7 Q.9 W P 7th gtana(R-|htana) (M-m)gx (a) M + m j g - (M + m)L ^ v 7C L 2]l g M - m j v Q.8 18g 19a 80 % rrr Q.10 h ^ h j

Q.ll 80V5 sec, 40V5 sec Classes Fluid Mechanics [15]

Q.12 (a)33.2s,(b) 64.6 s Q.13 y = 4xl0~ x Q.14 431 sec 3 4 Q.15 (i) 4m/s, (ii) F = 7.2N, (iii) F = 0, F = 52.2 N, (iv) both 4 x 10~ m/s mm max 3 Q16 (a) dHS 2 2 = 32 K g f - m , (b) - S H ( 1 - d) = 2 2 2 v, Kgf-m Q17 163 388 Q-" " ^ T J T a 12Vl4 Q 1 9 pgR 2 Q.20 2 sec., 1 sec Q.21 1/90TC Q.22 b/3 3 Q.23 (a) 80cm, 5cm; (b) 300sec. Q.24 2VhH, 3m Q.25 h= mg + 4sa Pw^ § EXERCISE # III Q.l 500 Pa 5 1 / 3 Q.2 (a)(i) P = - d, (ii) p=P + - (6H+L)dg ; (b)(i) v = J % H - 4 h ) , (ii) = Vh(3H-4h) (iii) x = 7 H 0 x

Q.3 (i) 10 m/s, (ii) 14.1 m/s, (iii) 2.5 hr m 0 fd2~dl^ 2L I dl J 3 3 Q.5 (i) 0.2 m/s , (ii) Q.6 + 29625 J/m , - 30000 J/m Q.7 C Q.8 m = 7ir l (^pc p); if tilted then it's axis should become vertical. C.M. should be lower than c entre of bouyancy. Q.9 A Q.10 D Q.ll D Q.12 (a) 0, (b) h = 0.25 cm, (c) a= g/6 ( upward) 2 min 2 Q.13 x = 2m * 1 [37 Q.14 ^ = Q.15 C 271 V 2R L© Q.16H= — — 2g 2 2 fo Bansal Classes Fluid Mechanics [10]

BANSAL CLASSES TARGET IIT JEE 2007 XI (PQRS & J) WESTIMMM. QR FL UID MECHANICS Time Limit: 2 Sitting Each of 90 minutes, duration approx.

Q. 1 There are 58 questions in this question bank. QUESTION BANK ON FLUID MECHANICS Q2 Two cubes of size 1.0m sides, one of relative density 0.60 and another of relati ve density = 1.15 are connected by weightless wire and placed in a large tank of water. Under equilibrium the lighter cube will project above the water surface t o a height of (A) 50 cm (B) 25 cm (C)10cm (D) zero A rectangular tank is placed on a horizontal ground and is filled with water to a height H above the base. A small hole is made on one vertical side at a depth D below the level of the wate r in the tank. The distance xfromthe bottom ofthe tank at which the water jetfro mthe tank will hit the ground is (A) 2VD(H-D) (B) 2 VDH (C) 2VD(H7D) 2 -1 (D) | VDH Q.3 Q4 Q.5 Abeaker isfilledin with water is accelerated a m/s in+x direction. The surface o fwater shall make on angle (A) tan (a/g) backwards (B) tan (a/g) forwards (C) co t (g/a) backwards (D) cot (g/a) forwards A jet ofwater with cross section of 6 c m strikes a wall at an angle of 60° to the normal and rebounds elastically from th e wall without losing energy. If the velocity of the water in the jet is 12 m/s, the force acting on the wall is (A) 0.864 Nt (B) 86.4 Nt (C)72Nt (D)7.2Nt The v ertical limbs of a U shaped tube arefilledwith a liquid of density p upto a heig ht h on each side. The horizontal portion ofthe U tube having length 2h contains a liquid of density 2p. The U tube is moved horizontally with an accelerator g/ 2 parallel to the horizontal arm. The difference in heights in liquid levels in the two vertical limbs, at steady state will be 2h 8h 4h (D) None of these (A) ( B) ( Q T -1 -1 -1 2 Q.6 The cross sectional area of a horizontal tube increases along its length linearl y, as we move in the direction of flow. The variation of pressure, as we move al ong its length in the direction offlow (x-direction), is best depicted by which ofthe following graphs (A) (B) 2 (C) 2 (D) 2 Q.7 A cylindrical tank of height 1 m and cross section area A= 4000 cm is initially empty when it is kept under a tap of cross sectional area 1 cm . Water starts fl owing from the tap at t = 0, with a speed = 2 m/s. There is a small hole in the

base of the tank of cross-sectional area 0.5 cm . The variation of height of wat er in tank (in meters) with time t is best depicted by (A) (B) (C) Q.8 A bucket contains waterfilledupto a height = 15 cm. The bucket is tied to a rope which is passed over africtionlesslight pulley and the other end ofthe rope is tied to a weight of mass which is half ofthat of the (bucket+water). The water p ressure above atmosphere pressure at the bottom is (A) 0.5 kPa (B)lkPa (C) 5 kPa (D) None of these % Bansal Classes Question Bank on Fluid Mechanics [2]

Q.9 A cubical box ofwine has a small spout located in one ofthe bottom corners. When the box is full and placed on a level surface, opening the spout results in a f low of wine with a initial speed of v (see figure). When the box is half empty, someone tilts it at 45° so that the spout is at the lowest point (see figure). Whe n the spout is opened the wine willflowout with a speed of •.(A) v (B)V /2 ' (Qvo/ V2 (D) v M Q.10 A cone ofradius R and height H, is hanging inside a liquid of de nsity p by means of a string as shown in thefigure.The force, due to the liquid acting on the slant surface ofthe cone is , 0 0 mn 0 0 (A) prtgHR 2 (B) rcpHR 2 (C) ~ 7tpgHR 2 7TT77T7T77T7 TT (D) -TipgHR 2 Q.ll A cuboidal piece ofwood has dimensions a, b and c. Its relative density is d. It isfloatingin a large body of water such that side a is vertical. It is pus hed down a bit and released. The time period of SHM executed by it is : [be ( D ) 27T (C) 2tc (B) 2ti ^ da \'dg Q.12 Water is flowing steadily through a horizon tal tube ofnonuniform cross-section. If the pressure ofwater is 4 x 10 N/m at a point where cross-section is 0.02 m and velocity of flow is 2 m/s, what is press ure at apoint where cross-section reduces to 0.01 m . (A) 1.4 x 10 N/m (B) 3,4 x 10 N/m (C) 2.4 x 10" N/m (D)noneofthese Q.13 A vertical cylindrical container o f base area Aand upper cross-section A, area Aj making an angle 30° with the. hori zontal is placed in an open rainy field as shown near another cylindrical contai ner having same base area A. The ratio of rates of collection ofwater inthe two containers A iimvnrfTii riffTTTrrTfnTr will be (D) None (A) 2/73 (B) 4/V3 (C)2 Q .14 The area of cross-section of the wider tube shown in figure is 121cg 800 cm . If a mass of 12 kg is placed on the massless piston, the difference in heights h in the level ofwater in the two tubes is : (A) 10 cm (B) 6 cm (C) 15 cm (D)2c m Q.15 A slender homogeneous rod of length 2L floats partly immersed in water, b eing supported by a string fastened to one ofits ends, as shown. The specific gr avity of the rod is 0.75. The length of rod that extends out ofwater is : i 4 2 2 2 4 2 4 2 4 2 2 (A)L (B)-L (D) 3 (Q 4 L is accelerating upward on a as shown. Then the ) tan" (B) tan -l gcosa (l-cosa) 1

Q.16 Afluidcontainer is containing a liquid of density p with acceleration a along the inclined place ofinclinati angle of inclination 9 offree surface is: a + g sin a (A gcosa a-gsma (D) tan" a - g sin a (C) tan" g(l + cosa) g

L % Bansal Classes Question Bank on Fluid Mechanics [3]

Q.17 A dumbbell is placed in water of density p. It is observed that by attachin g a mass m to the rod, the dumbbell floats with the rod horizontal on the surfac e of water and each sphere exactly half submerged as shown in the figure. The vo lume of the mass m is negligible. The value of length I is d(V -3M) d(V -2M) (B) 2(V 3M) ( ) 2(V -2M) d(V -2M) d(V + 2M) (D) 2(V +3M) ( ) 2(V - 3M) p p A p P p p C p P Q.18 Figure shows a three arm tube in which a liquid is filled upto levels of he ight I. It is now rotated at an angular frequency co about an axis passing throu gh arm B. The angular frequency co at which level of liquid in arm B becomes zer o. (B) (C) (D) 3 Q.19 Two bodies having volumes Y and 2V are suspended from the two arms of a com mon balance and they are found to balance each other. If larger body is immersed in oil (density dj = 0.9 gm/cm ) and the smaller body is immersed in an unknown liquid, then the balance remain in equilibrium. The density of unknown liquid i s given by: (A) 2.4 gm/cm (B) 1.8 gm/cm (C) 0.45 gm/cm (D) 2.7 gm/cm Q.20 A tube is attached as shown in closed vessel containing water. The velocity of water c oming out from a small hole is : 20cm (A) ^ m/s (B)2m/s (C) depends on pressure of air inside vessel (D) None ofthese Q.21 Alarge tank is filled with water to a height H. Asmall hole is made at the base ofthe tank. It takes Tj time to decre ase the height of water to H/r|, (r| > 1) and it takes T time to take out the re st of water. If Tj = T , then the value ofr\ is : (A) 2 (B) 3 (C)4 (D) V2 i Q.22 A container oflarge surface arpa isfilledwith liquid of density p. Acubical blo ck of side edge a and m iss M is floating in it with four-fifth of its volume su bmerged. If a coin ofmass m is placed gently on the top surface ofthe block is j ust submerged. M is (A) 4m/5 (B)m/5 (C)4m (D)5m Q.23 The weight of an empty ball oon on a spring balance is Wj. The weight becomes w when the balloon is filled w ith air. Let the weight ofthe air itselfbe w .Neglect the thickness ofthe balloo n when it isfilledwith air. Also neglect the difference in the densities of air inside & outside the balloon. Then: (A) w w. (B) w Wj + w (C) w < Wj + w (D) w > Wj Q.24 In the case ofa fluid, Bernoulli's theorem expresses the application of the principle of conservation of : (A) linear momentum (B) energy (C)mass (D) an gular momentum Q.25 Fountains usually seen in gardens are generated by a wide pi pe with an enclosure at one end having many small holes. Consider one such fount

ain which is produced by a pipe of internal diameter 2 cm in which waterflowsat a rate 3 ms~ . The enclosure has 100 holes each of diameter 0.05 cm. The velocit y ofwater coming out ofthe holes ids (in ms ): (A) 0.48 (B) 96 (C) 24 (D)48 3 3 3 3 2 2 2 2 2 : 2 : 2 2 ! -1 % Bansal Classes Question Bank on Fluid Mechanics [4]

Q.26 Water flows through afiictionlessduct with a cross-section varying as shown in figure. Pressure p at points along the axis is represented by (A) (B) (C) (D) Q.27 A boy carries a fish in one hand and a bucket(not full) of water in the oth er hand . If he places the fish in the bucket, the weight now carried by him (as sume that water does not spill) : (A) is less than before (B) is more than befor e (C) is the same as before (D) depends upon his speed Q.28 A cubical block ofwo od ofedge 10cmandmass0.92kgfloatsonatankofwaterwithoilofrel. density0.6 to a dep th of 4cm above water. When the block attains equilibrium with four ofits sides edges vertical (A) 1 cm of it will be above the free surface of oil. (B) 5cm of it will be under water. (C) 2cm ofit will be above the common surface of oil and water. (D) 8cm of it will be under water. Q. 29 The spring balance Areads 2 kg with a block m suspended from it. A balance B reads 5 kg when a beaker with liqu id is put on the pan ofthe balance. The two balances are now so arranged that th e hanging mass is inside the liquid in the beaker as shown in thefigurein this s ituation: (A) the balance A will read more than 2 kg (B) the balance B will read more than 5 .kg (C) the balance A will read less than 2 kg and B will read more than 5 kg (D) the balances A and B will read 2 kg and 5 kg respectively. Q.30 A n open cubical tank was initially fullyfilledwith water. When the tank was accel erated on a horizontal plane along one ofits side it was found that one third of volume ofwater spilled out. The acceleration was (A) g/3 (B) 2g/3 (C) 3g/2 (D)No ne Q.31 Acork of density 0.5gcm floats on a calm swimming pool. The fraction oft he cork's volume which is under water is (A) 0% (B) 25% (C)10% (D) 50% Q.32 A cy lindrical vesselfilledwith water upto the height H becomes empty in time t due t o a small hole at the bottom of the vessel. Ifwater isfilledto a height 4H it wi llflowout in time (A) to (B)4t (C)8t (D)2t Q.33 Acylindrical vessel open at the top is 20cm high and 1 Ocmin diameter. A circular hole whose cross-sectional are a 1 cm is cut at the centre ofthe bottom ofthe vessel. Waterflowsfroma tube abov e it into the vessel at the rate 100 cm s"'. The height ofwater in the vessel un der steady state is (Take g=1000 cms ) (A) 20 cm (B) 15 cm (C)10cm (D) 5 cm -3 0 0 0 0 2 3 -2 Q.34 A fire hydrant delivers water of density p at a volume rate L. The water tr avels /-— vertically upward through the hydrant and then does 90° turn to emerge hor izontally at speed V. The pipe and nozzle have uniform crosssection throughout. The force exerted by the water on the corner ofthe hydrant is (A)pVL (B) zero (C )2pVL (D)V2 VL P v % Bansal Classes Question Bank on Fluid Mechanics [5]

Q.35 Avertical tank, open at the top, isfilledwith a liquid and rests on a smoot h horizontal surface. A small hole is opened at the centre of one side ofthe tan k. The area of cross-section ofthe tank is N times the area of the hole, where N is a large number. Neglect mass ofthe tank itself. The initial acceleration oft he tank is Q.36 A body of density p' is droppedfromrest at a height h into a lak e of density p, where p > p'. Neglecting all dissipative forces, calculate the m aximum depth to which the body sinks before returning to float on the surface. ( A) A (B, V (C) J ^ P-P p P-P P-P Q.37 A Newtonianfluidfillsthe clearance between a shaft and a sleeve. When aforce of800N is applied to the shaft, parallel to t he sleev,e, the shaft attains a speed of 1.5 cm/sec. If a force of 2.4 kN is app lied instead, the shaft would move with a speed of (A) 1.5 cm/sec (B) 13.5 cm/se c (C) 4.5 cm/sec (D) None Q.38 A solid metallic sphere ofradius r is allowed to fallfreelythrough air. Ifthefrictionalresistance due to air is proportional to t he cross-sectional area and to the square of the velocity, then the terminal vel ocity of the sphere is proportional to which of the following? (A) r (B)r (C)r ( D)r Q.39 Two water pipes P and Q having diameters 2 x 10" m and 4x10" m, respect ively, are j oined in series with the main supply line ofwater. The velocity ofw aterflowingin pipe P is (A) 4 times that of Q (B) 2 times that of Q (C) 1/2 time s of that of Q (D) 1/4 times that of Q Q. 40 Waterflowsinto a cylindrical vessel oflarge cross-sectional area at a rate of 10~ m /s. Itflowsoutfroma hole of are a 10^ m , which has been punched through the base. How high does the water rise in the vessel? (A) 0.075 m (B) 0.051m (C) 0.031m (D) 0.025 m Q.41 Two cyllinders of same cross-section and length L but made of two materi al of densities d j a nd d are cemented together to form a cylinder oflength 2L, The combinationfloats in a liquid of density d with a length L/2 above the surface of the liquid. If d j > d then: ( D ) 2 3/2 1/2 2 2 4 3 2 2 2 (A) dj > d (B)|>d 1 (C) - > d , 4 (D) d < d j Q.42 There is a horizontalfilmof soap solution. On it a thread is placed in the form of a loop. Thefilmis pierced inside the loop and the thread becomes a circu lar loop of radius R. If the surface tension of the loop be T, then what will be the tension in the thread? (A) 7cR /T (b) TIR T (C) 2TTRT (D)2RT Q.43 S ome liq uid isfilledin a cylindrical vessel ofradius R. Let F j be the force applied by the liquid on the bottom ofthe cylinder. Now the same liquid is poured into a ve ssel ofuniform square crss-section of side R. Let F be the force applied by the liquid on the bottom of this new vessel. Then. 2 2 2 (A) Fj = 7tF. (6) ^ = 571" '' (C) F, = VttF (D)F,=F Q. 44 A tank isfilledup to a height 2H with a liquid and is placedon a platform ofheight Hfromthe ground. Th e distance xfromthe ground where a small hole is punched to get the maximum rang e R is: (A)H (B) 1.25 H (C) 1.5 H (D)2H 2 2

2 % Bansal Classes Question Bank on Fluid Mechanics [6]

Q.45 Acontainer, whose bottom has round holes with diameter 0.1 mm isfilledwith water. Themaximum height in cm upto which water can befilledwithout leakage will be what? Surface tension = 75 x 10 N/m and g = 10 m/s : (A) 20 cm (B) 40 cm (C) 30cm (D)60cm Q.46 In a cylindrical vessel containing liquid of density p, there are two holes in the side walls at heights ofhj and h respectively such that the range of efflux at the bottom ofthe vessel is same. The height of a hole, for w hich the range of efflux would be maximum, will be ( B ^ + hj (A) - hj h -h, (C) (D) h +hj 3 2 2 0 2 Q.47 Apiece of steel has a weight Win air, Wj when completely immersed in water and W when completely immersed in an unknown liquid. The relative density (speci fic gravity)ofliquid is: W-W, w -w W, - w W-Wo (A) W - W , (C) W-W, (D) w - w (B ) W -T , wW Q.48 Alarge tank isfilledwith water (density = 10 kg/m ). Asmall hol e is made at • 10m a depth 10 m below water surface. The range ofwater issuing out ofthe hole is Ron ground. What extra pressure must be applied on the water surf ace so that the range becomes 2R (take 1 atm = 10 Pa and g = 10 m/s ): (A) 9 atm (B) 4 atm (C) 5 atm (D) 3 atm R Q.49 Two drops of same radius are falling throu gh air with steady velocity of v cm/s. If the two drops coalesce, what would be the terminal velocity? (A) 4 v (C)2v (D) 64 v (B) (4) 1/3, Q.50 A ball ofrelativ e density 0.8 falls into waterfroma height of 2m. The depth to which the ball wi ll sink is (neglect viscous forces): (A) 8 m (B)2m (C)6m (D)4m i Q.51 A liquid o f mass 1 kg isfilledin a flask as shown infigure.The force exerted by the flask on the liquid is (g = 10 m/s ): (A) ION (B) greater than ION (C) less than 1 ON (D)zero 2 2 ; 2 2 3 3 5 2 2 Q.52 Figure shows a siphon. Choose the wrong statement: (A) Siphon works when h > 0 (B) Pressure at point 2 is P = P - pgh h=0 (C) Pressure at point 3 is P (D) None of the above X 1(P = atmospheric pressure) Q.53 If two soap bubbles of diff erent radii are connected by a tube, (A) air flowsfromthe bigger bubble to the s maller bubble till the sizes become equal (B) air flowsfrombigger bubble to the smaller bubble till the sizes are interchanged (C) airflowsfromthe smaller bubbl e to the bigger (D) there is noflowof air. i 3 2 0 3 0 0 % Bansal Classes Question Bank on Fluid Mechanics m

Q.54 A cubical block of side 'a' and density 'p' slides over a fixed inclined pl ane with constant velocity V . There is a thinfilmof viscousfluidofthickness't' between the plane and the block. Then the coefficient ofviscosity ofthe thinfilm will be \B=37° 3pagt pagt 4pagt (D) none ofthese (A) 5v (C) v (B) 5v Q.55 Which of the following graphs best represents the motion of a raindrop? (A) (B) 5 2 (C) CD) Q.56 Two soap bubbles with radii r and (r > r ) come in contact. Their common su rface has a radius of curvature r. r, + r, (C)r (D)r = ^ 7 (B)r = l 2 r,+r (A) negligible AW a-fiii Cv^GQe eJ comJc(*Mo ego c oOken er^t r^ t ^ (B) to make the image les s bright than before (C) to make the upper half of the image disappear (D) to ma ke the lower half of the image disappear Q.20 Aconvex mirror of focal length 'f is placed at the origin with its reflecting surface towards the negative x-axis. Choose the correct graphs between V and 'u' for u < 0. D i (D) fo o Q.21 In the figure shown, the image of a real object is formed at point I. AB is the principal axis ofthe mirror. The mirror must be: T2 l I 1 (A) concave & p laced towards right I (B) concave & placed towards left of I (C) convex and plac ed towards right of I (D) convex & placed towards left of I. Q.22 An infinitely long rod lies along the axis of a concave mirror of focal length f. The near end ofthe rod is at a distance u > f from the mirror. Its image will have a length (A) u - f (D) Uuf f (B) uuff (C) U + f ~/ + Q.23 Apoint source is situated at a distance x < ffromthe pole ofthe concave mirror of focal length f. At time t = 0 , the point source starts moving awayfromthe mirror with constant velocity. Whic h ofthe graphs below represents best, variation ofimage distance j v | with the distance x between the pole ofmirror and the source. |V| M M It d >d V & Bansal Classes Xo f Xo f Xo f Xo f Q.24 Apoint object is between the Pole and Focus of a concav e mirror, and moving away from the mirror with a constant speed. Then, the veloc ity of the image is: (A) awayfrommirror and increasing in magnitude (B) towards mirror and increasing magnitude (C) awayfrommirror and decreasing in magnitude ( D) towards mirror and decreasing in magnitude Q.25 An object is placed in front of a convex mirror at a distance of 50 cm. A plane mirror is introduced covering the lower half of the convex mirror. Ifthe distance between the object and the plane mirror is 30 cm, it is found that there is no gap between the images forme d by the two mirrors. The radius ofthe convex mirror is: (A) 12.5 cm (B) 25 cm ( C)50cm (D) 100 cm (A) (B) (C) (D)

£ Question Bank on Geometrical Optics [61

Q.26 A concave mirror is placed on a horizontal table, with its axis directed ve rtically upwards. Let O be the pole of the mirror and C its centre of curvature. Apoint object is placed at C. It has a real image, also located at C (a conditi on called auto-collimation). Ifthe mirror is nowfilledwith water, the image will be: (A) real, and will remain at C (B) real, and located at a point between C a nd oo (C) virtual, and located at a point between C and 0. (D) real, and located at a point between C and O. Q.27 Aray oflight is incident on a concave mirror. It is parallel to the principal axis and its heightfromprincipal axis is equal t o the focal length of the mirror. The ratio of the distance of point B to the di stance of the focusfromthe centre of curvature is (AB is the reflected ray) < (B) s cof (D) Q.28 A luminous point object is moving along the principal axis ofa concave mirr or offocal length 12 cm towards it. its distancefrommirror 4 velocity e Whentowa rds the mirror is 20 cm its velocity is(B)cm/s. Thefromthe ofthe image in cm/s a t that instant is: (A) 6 6 away mirror (C) 9 awayfromthe mirror (D) 9 towards th e mirror Q.29 When an object is placed at a distance of 25 cmfroma concave mirro r, the magnification is m . The object is moved 15 cm farhter away with respect to the earlier position, and the magnification becomes m . If m,/m = 4 the focal length of the mirror is (Assume image is real m,, m are numerical values) (A) 1 0 cm (B) 30 cm (C) 15 cm (D) 20 cm 2L . | 71X Q.30 A reflecting surface is repre sented by the equation Y = — j ~L~ 0 < x < L. A ray travelling y horizontally beco mes vertical after reflection. The coordinates of the point (s) where this ray i s incident is t 2 2 2 s m 1s (A) 1,4' (L -J2L) % Q.31 The origin ofx and y coordinates is the pole of a concave mirror of focal l ength 20 cm. The x-axis is the optical axis with x > 0 being the real side of mi rror. A point object at the point (25 cm, 1 cm) is moving C with a velocity 10 c m/s in positive x-direction. The velocity of the image in cm/s is approximately (A) - 80 i + 8 j (B) 160 i + 8 j (C) - 160 i + 8 j (D) 1 6 0 i - 4 j Q.32 In the figure shown ifthe object 'O' moves towards the plane mirror, then the image (f I (which is formed after successive reflectionsfromMj & M respectively) -- '* w ill move: -r* : Mi (A) towards right (B) towards left (C) with zero velocity (D) cannot be determined 2 M J (B) I ' 3 (L V3L^ 71

J 2L V 3 L " ] (C) ^ 4 ' t t J (D) [ 3 ' t t J r 3L V2L N r Q.33 All ofthe following statements are correct except (for real object): (A) th e magnification produced by a convex mirror is always less thenor equal to one ( B) a virtual, erect, same sized image can be obtained using a plane mirror (C) a virtual, erect, magnified image can be formed using a concave mirror (D) a real , inverted, same sized image can be formed using a convex mirror. Q.34 The dista nce of an objectfromthe pole of a concave mirror is equal to its radius of curva ture. The image must be : (A) real (B) inverted (C) same sized (D) erect £ Bansal Classes Question Bank on Geometrical Optics m

r Q.35 A straight line joining the object point and image point is always perpendi cular to the mirror (A)ifmirrorisplaneonly (B) ifmirror is concave only (C) if m irror is convex only (D) irrespective ofthe type of mirror. Q.36 A concave mirro r forms a real image three times larger than the object on a screen. Object and ( screen are moved until the image becomes twice the size of object. If the shif t of object is 6 cm. The - shift of the screen & focal length of mirror are (A) 36 cm, 36cm (B) 36cm, 16cm (C) 72cm, 36cm (D) none of these Q.37 A point source oflight is 60 cmfroma screen and is kept at the focus of a concave mirror which reflects • \ light on the screen. The focal length ofthe mirror is 20 cm. The rati o ofaverage intensities ofthe illumination on the screen when the mirror is pres ent and when the mirror is removed is : (A) 36:1 (B) 37 : 1 (C)49:l (D)10:l Q.38 The distance of a real object from the focus of a convex mirror of radius of cu rvature 'a' is 'b'. Then the distance ofthe image from the focus is i2 ^ ^2 (A)— ( B) - (C) — (D) none ofthese 4a b 4b Q.39 Choose the correct statement(s) related t o the motion of object and its image inthe case ofmirrors (A) Object and its ima ge always move along normal w.r.t. mirror in opposite directions (B) Only in the case of convex mirror, it may happen that the object and its image move in the same direction (C) Only in the case of concave mirror, it may happen that the ob ject and its image move in the same direction (D) Only in case of plane mirrors, object and its image move in opposite directions Q.40 A point source oflight is placed at a distance h below the surface of a large deep lake. What is the perc entage oflight energy that escapes directlyfromthe water surface is p ofthe wate r=4/3 ? (neglect \\ partial reflection) (A) 50% (B) 25% (C) 20% (D) 17% Q.41 The x-z plane separates two media Aand B with refractive indices p,j and P2 respect ively. Aray oflight travels from A to B. Its directions in the two media are giv en by the unit vectors, r = a i + b j & C r = a i + p j respectively where i & ] are unit vectors in the x and y directions. Then (A)pja = p a (B) PjOC ~ p a (C )pjb = p P (D)pjp = p b Q.42 A ray Rj is incident on the plane surface ofthe gla ss slab (kept in air) ofrefractive index -J2 at angle ofincident equal to the cr itical angle for this air glass system. The refracted ray R2 undergoes partial r eflection & refraction at the other surface. t The angle between reflected ray R and the refracted ray R at that surface is: (A)45° (B)135° (C) 105° (D) 75° Q.43 A ray oflight from a denser medium strike a rarer medium. The angle of reflection is r and that of refraction is r'. The reflected and refracted rays make an angle of 90° with each other. The critical angle will be: (A) sin (tan r) (B) tan (sin r) (C) sin" (tan r') (D) tan" (sin r') Q. 44 A tiny air bubble in a glass slab (p, = 1.5) appears from one side to be 6 cm from the glass surface and 9 from other side, 4 cm. The thickness ofthe glass slab is (A) 10 cm (B) 6.67 cm (C)15cm (D) one of these £ 2 A B 2 2 2 2 3

4 -1 1 1 1 d) are constants. l-(x/r) Y I (A) The incident ray travels in parabolic ally inside the slab. (B) The incident ray travels in hyperbolic path inside the slab. (C) The incident ray travels in circular path inside the slab. (D) The in cident ray travels in elliptical path inside the slab. Q.52 A ray oflight travel sfroman optical denser medium to rarer medium. The critical angle for the two me dia is C. The maximum possible deviation ofthe refracted light ray can be : 71 £ ( A) 7t - C (B)2C (C) it. - 2C (D)--C Q.53 A microscope is focused on a point obje ct and then its objective is raised through a height of 2cm. If a glass slab ofr efractive index 1.5 is placed over this point object such that it is focused aga in, the thickness of the glass slab is: (A) 6 cm (B) 3 cm (C)2cm (D) 1.5 cm Q.54 Aparaxial beam oflight is converging towards a point P on the screen. Aplane pa rallel sheet of glass of thickness t and refractive index p is introduced in the path ofbeam. The convergence point is shifted by: (A) t (1 - 1/p) away (B) t (1 + 1/p) away (C) t (1 - 1/p) nearer (D) t (1 + 1/p) nearer Q.55 A bird is flying 3 m above the surface ofwater. If the bird is diving vertically down with speed = 6 m/s, r his apparent velocity as seen by a stationaryfishunderwater is : X, (A) 8 m/s (B)6m/s (C) 12 m/s (D)4m/s £ 2 3 3 3 2 3 3 2 > 0

£ (feBansal Classes Question Bank on Geometrical Optics [12]

Q.56 A flat glass slab of thickness 6 cm and index 1.5 is placed in front of a p lane mirror. An observer is standing behind the glass slab and looking at the mi rror. The actual distance of the observer from the mirror is 50 cm. The distance of his image from himself, as seen by the observer is : (A) 94 cm (B) 96 cm (C) 98cm (D) 100 cm sini Q.57 In the figure shown is equal to: C Vh 1*3 Hi Q.58 A ra y oflight moving along the unit vector (- i - 2j) undergoes refraction at an int erface oftwo media, which is the x-z plane. The refractive index for y > 0 is 2 while for y < 0, it is -J5 j2 • The unit vector along which the refracted ray move s is: ( a ) MM) (D) None of these Q. 5 9 An object is placed 20 cm infrontof a 4 cm thick plane mirror. The image ofthe obj ect finally is formed at 45 cmfromthe obj ect itself. The refractive i ndex of the material ofthe unpolished side of the mirror C is (considering near normal incidence) (A) 1.5 (B) 1.6 (C) 1.4 (D) none of these Q.60 A ray oflight i s incident on a parallel slab of thickness t and refractive index n. If the angl e of incidence 9 is small than the displacement in the incident and emergent ray will be : tOCn-1) t9 t9n (A) (B) ™ (C) — (D) none Q.61 A ray oflight is incident at an angle of 75° into a medium having refractive index p. The reflected and the re fracted rays are found to suffer equal deviations in opposite direction p equals c r?^^' V3+1 V3+1 2V2 (A) ^ 2 ( C ) ^ (D) None of these Q.62 A small source ofl ight is 4m below the surface of a liquid ofrefractive index 5/3. In order to cut off all the light coming out of liquid surface, minimum diameter ofthe disc pla ced on the surface of liquid is: (A) 3m (B)4m (C)6m (D)oo m \ ^ IvT Q. 63 From t he figure shojvn establish a relation between, Pj, p , p . (A)pjX ph (D) ( p - l ) A (C) (p-l)A Q.75 A ray of sunlight enters a spherical water droplet (n=4/3) at an angle of incidenc e 53° measured with respect to the normal to the surface. It is reflected from the back surface ofthe droplet and re-enters into air. The angle between the incomi ng and outgoing ray is [Take sin 53° = 0.8] (A) 15° (B) 34° (C) 138° (D)30° Q.76 A concave spherical surface ofradius of curvature 10cm separates two medium x & y of refr active index 4/3 & 3/2 respectively. If the object is placed along principal axi s in medium X then (A) image is always real (B) image is real ifthe object dista nce is greater than 90cm (C) image is always virtual (D) image is virtual if the object distance is less than 90cm Q.77 The correct conclusion that can be drawn fromthesefiguresis £ ih. Al * , \/ \ £ Q.78 £• Q.79 e-. Q.80 & Q.81 £ (b) (A) p,jp but p< p (C) Pj= p but p< p (D) p, = p, but p < p Afishis near the centre of a spherical waterfilled( p = 4/3)fishbowl. Achild s tands in air at a distance 2R (R is the radius of curvature of the sphere) from the centre ofthe bowl. At what distance from the centre would the child nose app ear to thefishsituated at the centre: (A) 4R (B)2R (C)3R (D)4R A spherical surfa ce of radius of curvature R separates air (refractive index 1.0) from glass (ref ractive index 1.5). The centre of curvature is in the glass. Apoint object P pla ced in air is found to have a real image Q in the glass. The lime PQ cuts the su rface at the point O, and PO = OQ. The distance PO is equal to: (A) 5R (B) 3 R ( C)2R (D)1.5R A spherical surface of radius of curvature 10 cm separates two medi a X and Y ofrefractive indices 3/2 and 4/3 respectively. Centre of the spherical surface lies in denser medium. An object is placed in medium X. For image to be real, the object distance must be (A) greater than 90 cm (B) less than 90 cm. ( C) greater than 80 cm (D) less than 80 cm. A beam of diameter' d' is incident on a glass hemisphere as shown. Ifthe radius of curvature ofthe hemisphere is very large in comparison to d, then the diameter of the beam at the base of the hemi sphere will be: d (B)d (D)|d 3 () 4 2 2 2 2 A d (a) V\ (feBansal Classes Question Bank on Geometrical Optics [12]

Q. 82 A concave spherical refracting surface separates two media glass and air ( p = 1.5). Ifthe image is to be real at what minimum distance u should the object be placed in glass if R is the radius of curvature? (A)u>3R (B) u > 2R (C)u y > z (B) x > z > y (C)y>z>x (D)None Q. 95 Abi-concave glass lens having refractive index 1.5 has both surfaces of same radius of curvature R. On immersion in a medium of refract ive index 1.75, it will behave as a (A) convergent lens of focal length 3.5 R (B ) convergent lens of focal length 3.0 R (C) divergent lens of focal length 3.5 R (D) divergent lens of focal length 3.0 R Q. 96 The power (in diopters) of an eq uiconvex lens with radii of curvature of 10 cm and refractive index ofl,6is: (A) - 1 2 (B) +12 (C) +1.2 (D) -1.2 Q.97 The focal length ofa lens is greatest for which colour? (A) violet (B)red (C) yellow (D) green Q.98 A converging lens form s an image of an object on a screen. The image is real and twice the size ofthe object. If the positions of the screen and the object are interchanged, leaving the lens in the original position, the new image size on the screen is (A) twice the obj ect size (B) same as the object size (C) halfthe object size (D) can't say as it depends on the focal length of the lens. Q. 99 An object is placed inf rontofa symmetrical convex lens with refractive index 1.5 and radius of curvatur e 40 cm. The surface ofthe lens further awayfromthe object is silvered, Under au to-collimation condition, the object distance is (A) 20 cm (B) 10 cm (C)40cm (D) 5cm Q. 100 When the object is at distances u and u the images formed by the same lens are real and virtual respectively and ofthe same size. Then focal length o f the lens is: ] 2 (B)|(U!+U ) 2 ( O ^ T (D) 2 (u, + u ) 2 Q. 101 A planoconvex lens, when silvered at its plane surface is equivalent to a concave mirror of focal length 28cm. When its curved surface is silvered and th e plane surface not silvered, it is equivalent to a concave mirror of focal leng th 10cm, then the refractive index of the material of the lens is: (A) 9/14 (B) 14/9 (C) 17/9 (D)none Q. 102 The height of the image formed by a converging lens on a screen is 8cm. For the same position ofthe object and screen again an imag e of size 12.5cm is formed on the screen by shifting the lens. The height M ofth e object: (A) 625/32cm (B)64/12.5cm (C) 10cm (D)none 4f (D)u t 2 2 30 (C) < 30° c Q.131 The dispersive powers oftwo lenses are 0.01 and0.02. Iffocai length of one lens is + 10cm, then what should the focal length ofthe second lens, so that th ey form an achromatic combination? (A) Diverging lens having focal length 20 cm. (B) Converging lens having focal length 20 cm (C) Diverging lens having focal l ength 10 cm. (D) Converging lens having focal length 10 cm i c ' Xi 10cm i 120cm c Q.6 A ray oflight is incident normally on one face of 30° - 60° - 90° prism \P/ of refract ive index 5/3 immersed in water of refractive index 4/3 as shown in figure. 1 \ (A) The exit angle 0 ofthe ray is sin (5/8) (B) The exit angle 9 of the ray is s in (5/4J3) (C) Total internal reflection at point P ceases ifthe refractive inde x of water is increased to 5/2V3 by dissolving some substance. (D) Total interna l reflection at point P ceases if the refractive index ofwater is increased to 5 /6 by dissolving some substance. 2 -1 2 -1

p (C)P?-P >P^ Q.10 In the figure shown a point object O is placed in air on the principal axis. Th e radius of curvature of the spherical surface is 60 cm. I is thefinalimage form ed after all the refractions and reflections. (A) If dj = 120 cm, then the T ' i s formed on 'O' for any value of d . (B) If dj = 240 cm, then the T ' is formed on 'O' only if d = 360 cm. (C) If dj = 240 cm, then the T ' is formed on 'O' for all values of d.2(D) If dj = 240 cm, then the T ' cannot be formed on 'O'. 2 3 3 2 2 2 f f 2 f 2 f f Q.9 ~n H3 n =3/2 g 2 Q.ll Two refracting media are separated by a spherical interface as shown in the figure. PP' is the principal axis, Pj and P2 are the refractive indices of medi um of incidence and medium of refraction respectively. Then: (A) if P2 > pj, the n there cannot be a real image of real obj ect. (B) if pj > pj, then there canno t be a real image ofvirtual object. (C) if Pj > P2, then there cannot be a virtu al image ofvirtual object. (D) if pj > p , then there cannot be a real image of real object. 2 p, (B) Real image is formed only when x > R (C) Real image is formed due to the convex nature of the interface ir respective of Pj and p^ (D) None of these Q.13 Choose the correct statement(s) r elated to the virtual image formed by obj ect O placed at a distance x, as shown infigureA (A) Virtual image is formed for any position of O if p < (B) Virtual image can be formed ifx > R and p < Pj (C) Virtual image is formed if x < R and p > p, (D) None of these 2 2 2 2 R Fig. A x Fig. B Q.14 Identify the correct statement(s) related to the formation of images of a r eal obj ect O placed at x from the pole ofthe concave surface, as shown infigure B (A) I f p > p j , then virtual image is formed for any value ofx 2 (B)If \u< p., then virtual image is formed if x< Hi H (C) If p < Pj, then real i mage is formed for any value ofx (D) none ofthese 2 2 Q.15 Which of the following can form diminished, virtual and erect image ofyour face. (A) Converging mirror (B) Diverging mirror (C) Converging lens (D) Divergi ng lens Q.16 A convex lens forms an image of an object on a screen. The height o f the image is 9 cm. The lens is now displaced until an image is again obtained on the screen. The height ofthis image is 4 cm. The distance between the object and the screen is 90cm. (A) The distance between the two positions of the lens i s 3 0cm. (B) The distance of the obj ect from the lens in itsfirstposition is 3 6cm. (C) The height of the object is 6cm. (D) The focal length of the lens is 21 .6 cm. Q. 17 A diminished image of an object is to be obtained on a large screen 1 mfromit. This can be achieved by (A) using a convex mirror of focal length le ss than 0.25 m (B) using a concave mirror offocal length less than 0.25 m (C) us ing a convex lens offocal length less than 0.25 m (D) using a concave lens of fo cal length less than 0.25 m p . s j 1 I 0 has refractive index V2 and medium - 2 with z < 0 has a refractive index V3 .Aray oflight in medium -1 given by the vector A = 6^3 i + 8^3 j — 10k is incident on the plane of separation. Find the unit vector i n the direction of refracted ray in medium -2. [JEE '99] Q.12 A quarter cylinder of radius R and refractive index 1.5 is placed on a table. A point object P is kept at a distance ofmRfromit. Find the value ofm for which a ray from P will em erge parallel to the table as shown in the figure. [JEE '99] 2 T p Q.13 Two symmetric double-convex lenses L, and L with their radii of curvature 0 .2m each are made from glasses with refractive index 1.2 and 1.6 respectively. T he lenses with a separation of0.345 m are submerged in a transparent liquid medi um with a refractive index of 1.4. Find the focal lengths of lens L, and L An ob ject is placed at a distance of 1.3 mfromL find the location of its image while the whole system remains inside the liquid. [REE' 99] r p Q. 14 Select the correct alternative. [JEE '2000 (Scr)] (a) A diverging beam ofl ightfroma point source S having divergence angle a, falls symmetrically on a gla ss slab as shown. The angles of incidence of the two extreme rays are equal. If the thickness of the glass slab is t and the refractive index n, then the diverg ence angle ofthe emergent beam is (A) zero (B) a (C) sin (l/n) (D) 2sin~ (l/n) s _1 i (b) A rectangular glass slab ABCD, of refractive index nj, is immersed in water fractive index n^n > r^). Aray oflight is incident at the surface AB of the as shown. The maximum value ofthe angle of incidence a , such that the ray s out onlyfromthe other surface CD is given by -1 n. (B) sin" n, cos sm (A) . -cos sin n ni ) 1y max

ofre slab come sm n

"2 c -1 n 2 (C) sin" (c) n, V2 ) n (D) sin -1 n. v iy n A point source oflight B is placed at a distance L in front ofthe centre ofa mir ror ofwidth d hung vertically on a wall. A man walks infrontof the mirror along a line parallel to the mirror at a distance 2Lfromit as shown. The greatest dist ance over which he can see the image ofthe light source in the mirror is (A)d/2

(B)d (C) 2d (D) 3d B. i< >1 L 2L (!%Bansal Classes Geometrical Optics [10]

(d) A hollow double concave lens is made ofvery thin transparent material. It can be filled with air or either oftwo liquids L, or L having refractive indices n, an d n, respectively (n >n > 1). The lens will diverge a parallel beam oflight if i t isfilledwith (A) air and placed in air. (B) air and immersed in L,. (C) L, and immersed in L (D) L and immersed inL 2 2 ) r 2 r Q.15 A convex lens of focal length 15 cm and a concave mirror of focal length 30 cm are kept with their optic axes PQ and RS parallel but separated in vertical direction by 0.6 cm as shown. The distance between the lens and mirror is 30 cm. An upright object AB ofheight 1.2 cm is placed on the optic axis PQ of the lens at a distance of 20 cmfromthe lens. IfA' B' is the image after refraction from the lens and reflectionfromthe mirror,findthe distance A' B' from the pole ofthe mirror and obtain its magnification. Also locate positions of A' and B' with re spect to the optic axis RS. [JEE 2000] Q.16 A thin equi biconvex lens ofrefracti ve index 3/2 is placed on a horizontal plane mirror as shown in thefigure.The sp ace between the lens and the mirror is then filled with water of refractive inde x 4/3. It is found that when a point object is placed 15cm above the lens on its principal axis, the object coincides with its own image. On repeating with anot her liquid, the object and the image again coincide at a distance 25cm from the lens. Calculate the refractive index ofthe liquid. [JEE 2001 ] viiiiTiiriniirminiiin; Q.17 The refractive indices ofthe crown glass for blue and red lights are 1.51 a nd 1.49 respectively and those of the flint glass are 1.77 and 1.73 respectively . An isosceles prism of angle 6° is made of crown glass. A beam of white light is incident at a small angle on this prism. The other flint glass isosceles prism i s combined with the crown glass prism such that there is no deviation ofthe inci dent light. Determine the angle of the flint glass prism. Calculate the net disp ersion ofthe combined system. [JEE 2001 ] Q.18 An observer can see through a pin -hole the top end of a thin rod of height h, placed as shown in thefigure.The be aker height is 3h and its radius h. When the beaker isfilledwith a liquid up to a height 2h, he can see the lower end of the rod. Then the refractive index ofth e liquid is (A) 5/2 (B)V572 (Q JJ/2 (D) 3/2 [JEE 2002 (Scr)] Q.19 Which one of t he following spherical lenses does not exhibit dispersion? The radii of curvatur e of the surfaces ofthe lenses are as given in the diagrams. [JEE 2002 (Scr)] (A ) R R,*R 2 (B)R (C)R (D) (!%Bansal Classes Geometrical Optics [10]

Q.20 Two plane mirrors A and Bare aligned parallel to each other, as shown in th e figure. A light ray is incident at an angle of 30° at a pointjust inside •"""uiunu iiiainiu one end ofA. The plane of incidence coincides with the plane of the 0.2 m ,0 3 figure. The maximum number oftimes the ray undergoes reflections ,, (inc luding thefirstone) before it emerges out is [JEE 2002 (Scr)] (A) 28 (B)30 (C) 3 2 (D)34 |c Q.21 Aconvex lens of focal length 30 cm forms an image of height 2 cm for an obj ect situated at infinity. If a convcave lens of focal length 20 cm is placed coa xially at a distance of 26 cm in front ofconvex lens then size image would be [J EE 2003 (Scr)] (A) 2.5 cm (B)5.0 (C) 1.25 (D)None Q.22 A meniscus lens is made o f a material of refractive index Both its surfaces have radii of curvature R. It has two different media of refractive indices (ij and |x respectively, on its t wo sides (see figure). Calculate its focal length for jx j < \x2 < |a , when lig ht is incident on it as shown. [JEE 2003] 3 3 HI j R ft ft/ VR / ft R (ii) Show that the gravitationalfieldin side the hole is uniform,findits magnitude and direction. Q.18 A body moving rad ially awayfroma planet ofmass M, when at distance rfromplanet, explodes in such a way that two ofits manyfragmentsmove in mutually perpendicular circular orbits around the planet. What will be (a) then velocity in circular orbits. (b) maxim um distance between the twofragmentsbefore collision and (c) magnitude oftheir r elative velocity just before they collide. Q.19 The fastest possible rate ofrota tion of a planet is that for which the gravitational force on material at the eq uator barely provides the centripetal force needed for the rotation. (Why?) (a) Show then that the corresponding shortest period ofrotation is given by 0 t' - VGp fWhere p is the density of the planet, assumed to be homogeneous. (b) E valuate the rotation period assuming a density of 3.0 gm/cm , typical of many pl anets, satellites, and asteroids. No such object is found to be spinning with a period shorter than found by this analysis. Q.20 Athin spherical shell of total mass M and radius R is heldfixed.There is a small hole in the shell. Amass m is released from rest a distance R from the hole along a line that passes through t he hole and also through the centre ofthe shell. This mass subsequently moves un der the gravitational force ofthe shell. How long does the mass take to travelfr omthe hole to the point diametrically opposite. 2 List of recommended questions from LE. Irodov. 1.213,1.216 to 1.220,1.224 to 1.2 27,1.229 ^Bansal Classes Gravitation [3]

EXERCISE-III Q. 1 A satellite P is revolving around the earth at a height h = radius of earth (R) above equator. Another satellite Q is at a height 2h revolving in opposite direction. At an instant the two are at same vertical line passing through centr e of sphere. Find the least time of after which again they are in this situation . Q.2 A certain triple-star system consists of two stars, each of mass m, revolv ing about a central star, mass M, in the same circular orbit. The two stars stay at opposite ends ofa diameter ofthe circular orbit, seefigure.Derive an express ion for the period of revolution of the stars; the radius of the orbit is r. Fin d the gravitational force of interaction between the mass m and an infinite rod of varying mass density X such that A(x)= X/x, where x is the distance from mass m. Given that mass m is placed at a distance d from the end of the rod on its a xis as shown in figure. Inside an isolatedfixedsphere of radius R and uniform de nsity r, there is a spherical cavity of radius R/2 such that the surface of the cavity passes through the centre ofthe sphere as infigure.Aparticle ofmass m is released from rest at centre B ofthe cavity. Calculate velocity with which parti cle strikes the centre Aof the sphere. In a certain double star system the two s tars rotate in circular orbits about their common centre ofmass. The stars are s pherical, they have same density p and their radii arc R and 2 R. Their centres are 5 R apart. Find the period T of stars in terms of p, R & G. Aring ofradius R is madefroma thin wire ofradius r. If p is the density ofthe material ofwire th en what will be the gravitational force exerted by the ring on the material part icle ofmass m placed on the axis of ring at a distance x from its centre. Show t hat the force will be maximum when x = R/V2 and the maximum value of force will be given as 471 Gr pm F max = (3) R 3/2 _1Q Q.3 0< m X(x)= X x TT Q.4 Q.5 Q.6 Q7 (a) (b) Q. 8 In a particular double star system, two stars ofmass 3.22 x 10 kg each revolve a bout their common center of mass, 1.12 x 10 m away. Calculate their common perio d of revolution, in years. Suppose that a meteoroid (small solid particle in spa ce) passes through this centre of mass moving at right angles to the orbital pla ne ofthe stars. What must its speed be ifit is to escapefromthe gravitational fi eld of the double star? 30 11 A man can jump over b=4m wide trench on earth. Ifmean density of an imaginary pl anet is twice that of the earth, calculate its maximum possible radius so that h e may escape from it by jumping. Given radius of earth = 6400 km. (!%Bansal Classes Gravitation [2]

Q.9 A launching pad with a spaceship is moving along a circular orbit of the moon, w hose radius R is triple that of moon Rm. The ship leaves the launching pad with a relative velocity equal to the launching pad's initial orbital velocity v and the launching pad then falls to the moon. Determine the angle 0 with the horizon tal at which the launching pad crashes into the surface if its mass is twice tha t of the spaceship m. Q Q.10 A small satellite revolves around a heavy planet in a circular orbit. At ce rtain point in its orbit a sharp impulse acts on it and instantaneously increase s its kinetic energy to' k' (< 2) times without change in its direction ofmotion . Show that in its subsequent motion the ratio ofits maximum and minimum distanc es from the planet is k , assuming the mass ofthe satellite is negligibly small as compared to that ofthe 2 k planet. Q.ll A satellite of mass m is in an ellipt ical orbit around the earth of mass M ( M » m ) The speed of the 6GM satellite at its nearest point to the earth (perigee) is J ^ where R=its closest distance to the earth. It is desired to transfer this satellite into a circular orbit around the earth of radius equal its largest distance from the earth. Find the increas e in its speed to be imparted at the apogee (farthest point on the elliptical or bit). Q.12 Abody is launchedfromthe earth's surface a an angle a=3 0° to the horiz ontal at a speed v - 1.5GM R Neglecting air resistance and earth's rotation,find (a) the height to which the body will rise, (ii) The radius of curvature oftraje ctory at its top point. 0 Q.13 Assume that a tunnel is dug across the earth (radius = R) passing through i ts centre. Find the time a particle takes to reach centre of earth if it is proj ected into the tunnel from surface of earth with speed needed for it to escape t he gravitationalfieldof earth. ^Bansal Classes Gravitation [3]

EXERCISE-III Q. 1 If the distance between the earth and the sun were half its present value, the number of days in a year would have been [JEE 96] (A) 64.5 (B) 129 (C) 182.5 (D)730 Q. 2 Distance between the centres of two stars is 10 a. The masses of th ese stars are M and 16 M and their radii a and 2a respectively. Abody of mass m isfiredat nightfromthe surface of the larger star towards the smaller star. What should be its minimum initial speed to reach the surface ofthe smaller star ? O btain the expression in terms of Q M and a. [JEE' 96] An artificial satellite mo ving in a circular orbit around the earth has a total (K.E. + P.E.) E . Its pote ntial energy is [JEE 97] (A)-E (B) 1.5 E (C) 2 E (D)E 0 0 0 0 0 Q. 3 Q.4 A cord of length 64 m is used to connect a 100 kg astronaut to spaceship whose m ass is much larger than that of the astronaut. Estimate the value of the tension in the cord. Assume that the spaceship is orbiting near earth surface. Assume t hat the spaceship and the astronaut fall on a straight linefromthe earth centre. The radius of the earth is 6400 km. [REE 98] In a region of only gravitationalf ieldof mass 'M' a particle is shifted from A to B via three different paths in t hefigure.The work done in different paths are Wj, W , W respectively then 2 3 Q.5 (A)W!=W = W3 ( B ) W ! > W > W 3 2 2 (C)Wj=W >W 2 3 (D)W! ( b ) v = i ~ Q8 V ^ k m Q.9 cos0: Vio v 2 Q.ll GM' R 2 _ _8_ 3 V15 Q.13 T = sin" Q.12 (a)h = , R, (b) 1.13R ' N ir T = 3 x io~2 N Q.l B Q2 Q.5 A vmm .

3 Q.6 2 V a 5GM EXERCISE-III Q3 Q7 C D Q.4 h = 99R ^Bansal Classes Gravitation [3]

BANSALCLASSES TARGET IIT JEE 2007 I XII (ALL) C JZT TZ ? ^ Al 1 / JL 7 L3S 4 l 7 J t v v r v i OMMMIQEIMK ON J L TX T O I •V ^ ^

QUESTIONS FOR SHORT ANSWER Q. 1 .Two satellites move along a circular orbit in the same direction at a smal l distance from each other. A container has to be thrownfromthefirstsatellite on to the second one. When will the container reach the second satellite faster: if its is thrown in the direction of motion of the first satellite or in the oppos ite direction ? The velocity ofthe container with respect to the satellite u is much less than that ofthe satellite v. Q.2 Because the Earth bulges near the equ ator, the source ofthe Mississippi River (at about 50°N latitude), although high a bove sea level, is about 5 km closer to the centre of the Earth than is its mout h (at about 30°N latitude). How can the river flow "uphill" as it flows south? Q.3 Use qualitative arguments to explain why the following four periods are equal ( all are 84 min, assuming a uniform Earth density): (a)' time of revolution of a satellite just above the Earth's surface (b) period of oscillation of mail in a tunnel through the Earth (c) period of a simple pendulum having a length equal t o the Earth's radius in a uniformfield9.8 m/s (d) period of an infinite simple p endulum in the Earth's real gravitational field. Q. 4 After Sputnik I was put in to orbit, it was said that it would not return to Earth but would burn up in its descent. Considering the fact that it did not burn up in its ascent, how is thi s possible ? Q.5 An artificial satellite is in a circular orbit about the Earth. How will its orbit change if one ofits rockets is momentarily fired, (a) toward s earth, (b) away from the Earth, (c) in a forward direction, (d) in a backward direction, and (e) at right angles to the plane of the orbit? Q.6 A stone is dro pped along the centre of a deep vertical mine shaft. Assume no air resistance bu t consider the Earth's rotation. Will the stone continue along the centre ofthe shaft ? Ifnot, describe its motion. Q.7 An iron cube is placed near an iron sphe re at a location remotefromthe Earth's gravity. What can you say about the locat ion of the centre of gravity ofthe cube? Of the sphere ? In general, does the lo cation ofthe centre of gravity of an object depend on the nature of the gravitat ionalfieldin which the object is placed? / Q. 8 Figure shows a particle ofmass m that is movedfroman infinite distance to the # centre of a ring of mass M, alon g the central axis of the ring. For the trip, how | does the magnitude ofthe gra vitational force on the particle due to the ring \ ' \ change. % i 2 m / X W / M Q.9 (a) (b) (c) (d) Infigure,a particle ofmass m is initially at point A, at distance dfromthe centr e of one uniform sphere and distance 4dfromthe centre of another uniform sphere, both of mass M » m. State whether, if you moved the particle to point D, the foll owing would be positive, negative, or zero: the change in the gravitational pote ntial energy of the particle, the work done by the net gravitational force on th e particle, the work done by your force. What are the answers if, instead, the m ove were from point B to point C ? B C ^T; D Q.10 Reconsider the situation of above questioa Would the work done by you be po sitive, negative, or zero ifyou moved the particle (a)fromAto B, (b)fromAto C, ( c) from B to D ? (d) Rank those moves accroding to the absolute value ofthe work

done by your force, greatest first. 1*1' Q.5 Let co be the angular velocity of the earth's rotation about its axis. A ssume that the acceleration due to gravity on the earth's surface has the same v alue at the equator and the poles. An object weighed at the equator gives the sa me reading as a reading taken at a depth d below earth's surface at a pole ( d « R ) The value of d is co R OR 2O R jRg ,(A) g ( B ) ^2g ~ r (C) g (D) — ^ g 7 1/2 v 7 2 W 2 2 1/2 2 2 2 2 2 2 w W Q.6 A spherical hole of radius R/2 is excavated from the asteroid of mass M as shown in fig. The gravitational acceleration at a point on the surface ofthe asteroid just above the excavation is (A) GM/R (B) GM/2R (C) GM/8R (D) 7GM/8R 2 2 2 2 Q.7 Q.8 If the radius of the earth be increased by a factor of 5, by what factor its den sity be changed to keep the value of g the same? (A) 1/25 (C) 1/V5 (D) 5 A man o f mass m starts falling towards a planet of mass M and radius R. As he reaches n ear to the surface, he realizes that he will pass through a small hole in the pl anet. As he enters the hole, he sees that 2M the planet is really made oftwo pie ces a spherical shell of negligible thickness of mass —— and a point M mass — at the c entre. Change in the force of gravity experienced by the man is 2 GMm (A) 3 - ^ 1 GMm 4 GMm 3l^~ 1*1 Q.18 . , ( M ! + M 2 > and G ^ g 1 + 2 2 f f l a n d ( C ) G M^ (M M ) q p ; G G zero ^ Q. 19 ( d ) ( m 1 + M 2 )

W ; G M ^ zero QO 2 Q. 12 A satellite ofthe earth is revolving in circular orbit with a uniform velo city V. If the gravitational force suddenly disappears, the satellite will (A) c ontinue to move with the same velocity in the same orbit. (B) move tangentially to the original orbit with velocity V. (C) fall down with increasing velocity. ( D) come to a stop somewhere in its original orbit. Q. 13 A newly discovered plan et has a density eight times the density of the earth and a radius twice the rad ius ofthe earth. The time taken by 2 kg mass to fallfreelythrough a distance S n ear the surface of the earth is 1 second. Then the time taken for a 4 kg mass to fall freely through the same distance S near the surface of the new planet is ( A) 0.25 sec. (B) 0.5 sec (C) 1 sec. (D) 4 sec. Q. 14 Four particles of equal mas ses M move along a circle of radius R under the action of their mutual gravitati onal attraction maintaining a square shape. The speed of each particle is (A)Q \ 2 q 22 GM 2V2+1 R (B) 1 GM V R 4 GM (D) 4GM R(V2+l) & Bansal Classes Question Bank on Gravitation [4] 4lBan

height from the Q.15 At whatsurface?above the earth's surface does the accelerat ion due to gravity fall to 1 % of its value at the earth's (A) 9R (B)10R (C) 99R (D) 100R Q.16 Find the distance between centre of gravity and centre of mass of a two particle system attached to the ends ofa light rod. Each particle has sam e mass. Length ofthe rod is R, where R is the radius of earth (A) R (B) R/2 (C) zero (D) R/4 Q.17 The radius of a planet is R. A satellite revolves around it in a circle of radius r with angular velocity co . The acceleration due to the gra vity on planet's surface is 3„3 rco ra r_3 „ 2 M (D) (B) (C) (A) R R R R 0 3 0 2 Q.18 A solid sphere of uniform density and radius R applies a gravitational forc e of attraction equal to F on a particle placed at a distance 3R from the centre ofthe sphere. A spherical cavity ofradius R/2 is now made in the sphere as show n in the figure. The sphere with cavity now applies a gravitational force F on t he same particle. The ratio F / F j is: 22 41 (A) 50 (C) (B) 25 24R7 e e e A C Q.47 A particle is projected from the mid-point of the line joining two fixed pa rticles each of mass m. If the distance of separation between thefixedparticles is /, the minimum velocity of projection of the particle so as to escape is equa l to GM GM , 2GM , 2GM (A)J— (B ) J — (C)J — (D)2/ V2/ I I v x W Q.48 The escape velocity for a planet is v . Atunnel is dug along a diameter oft he planet and a small body is dropped into it at the surface. When the body reac hes the centre of the planet, its speed will be e (A) v e (B)^ (C) y (D)zero & Bansal Classes Question Bank on Gravitation [280] 4lBan

nanimpulse Q.49 A per son brings a mass of 1 kg from infinity to a point A. Initially the m ass was at rest but it moves at a speed of 2 m/s as it reaches A. The work done by the person on the mass is -3 J. The potential at Ais: (A) -3 J/kg (B) -2 J/kg (C) -5 J/kg (D)-7 J/kg Q.50 A small ball of mass'm' is released at a height'R' above the earth surface, as shown in thefigureabove. Ifthe maximum depth ofthe b all to which it goes is R/2 inside the earth through a narrow grove before comin g to rest momentarily. The grove, contain an ideal spring of spring constant K a nd natural length R, find the value of K if R is radius of earth and M mass of e arth 3 GMm 6GMm (A) R (B) R 7 GMm 9GMm (C) " R ^ (D) R 3 3 3 equal to the iystem. e to gravity o that from where the Q.51 The magnitude of the potential energy per unit mass ofthe object at the sur face of earth is E. Then the escape velocity ofthe object is: (A)V2E (B)4E (C)VE (D)2E 2 Q.52 Suppose a smooth tunnel is dug along a straight line joining two points on the surface ofthe earth and a particle is dropped from rest at its one end. Assu me that mass of earth is uniformly distributed over its Volume. Then (A) the par ticle will emerge from the other end with velocity GM where M and R^. are earth' s mass 2R„ and radius respectively, (B) the particle will come to rest at centre o fthe tunnel because at this position, particle is closest to earth centre. (C) p otential energy of the particle will be equal to zero at centre oftunnel if it i s along a diameter. (D) acceleration of the particle will be proportional to its distancefrommidpoint ofthe tunnel. r e 1 e tl . Energy e Q.53 A hollow spherical shell is compressed to half its radius. The gravitationa l potential at the centre (A)increases (B) decreases (C) remains same (D) during the compression increases then returns at the previous value. Q.54 A body is pr ojection horizontallyfromthe surface of the Earth (radius = R) with a velocity e qual to 'n' times the escape velocity. Neglect rotational effects of the earth. The maximum height attained by the body from the Earth's surface is R/2. Then, ' n' must be (A) V06 (B) (V3J/2 (C)V04 (D)None Q.55 Consider two configurations of a system ofthree particles of masses m, 2m and 3m. The work done by external ag ent in changing the configuration of the systemfromfigure(i) tofigure(ii) is 6Gm .2 /1 + J J (A)zero (B) (C)6Gm' , V2, (D)6Gm V2, 2 s m. If the he particle 3M all body is nr a 1

figure(i) —5" •2m m« a„ v>3 figure(ii) A 2m m 1*1 . In particular, = - Two special points along the orbit are singled out by astronomers. Perig ee is the point at which the companion star is closest to the black hole, and ap ogee is the point at which it is furthestfromthe black hole. Q. 74 At which poin t in the elliptical orbit does the companion star attain its maximum kinetic ene rgy? (A) Apogee (B) Perigee (C) The point midwayfromapogee to perigee (D) All po ints in the orbit, since the kinetic energy is a constant of the motion. Q.75 Fo r circular orbits, the potential energy ofthe companion star is constant through out the orbit. If the radius ofthe orbit doubles, what is the new value of the v elocity of the companion star? (A) It is 1/2 ofthe old value (B) It is 1/V2 ofth e old value (C) It is the same as the old value. (D) It is double the old value Q. 76 Which ofthe following prevents the companion starfromleaving its orbit and falling into the black hole? (A) The centripetal force (B) The gravitational fo rce (C) The companion star's potential energy (D) the companion star's kinetic e nergy Q. 77 The work done on the companion star in one complete orbit by the gra vitational force ofthe black hole equals (A) the difference in the kinetic energ y ofthe companion star between apogee and perigee. (B) the total mechanical ener gy ofthe companion star (C)zero (D) the gravitational force on the companion sta r times the distance that it travels in one orbit. Q.78 For a circular orbit, wh ich ofthe following gives the correct expression for the total energy? (A) - (1/ 2) mv (B)mv (C)-(GmM)/r (D)(GmM)/2r 2 2 Q. 79 What is the ratio of the acceleration of the black hole to that ofthe comp anion star? (A) M / m (B)m/M (C)mM/r (D) 1 /1 1*1 air of stars ianion star, i escape its on star, the ar with the re G is the c h ole, and Since the ant of the ige kinetic =- .ompanion >le. y? ee -bit. Ift he ONE OR MORE THAN ONE OPTION MAY BE CORRECT Take approx. 3 minutes for answering each question. Q. 1 Assuming the earth to be a sphere ofuniform density the acceleration due to gravity (A) at a point outside the earth is inversely proportional to the squar e of its distancefromthe centre (B) at a point outside the earth is inversely pr oportional to its distancefromthe centre (C) at a point inside is zero (D) at a point inside is proportional to its distancefromthe centre. Q2 Mark the correct statement/s (A) Gravitational potential at curvature centre of a thin hemispheri cal shell of radius R and mass M is equal to GM R (B) Gravitationalfieldstrength at a point lying on the axis of a thin, uniform circular ring ofradius R and GM x mass M is equal to (K 2+x 2x3/2 where x is distance of that pointfromcentre of the ring. ,T> ) (C) Nekton's law of gravitation for gravitational force between two bodies is applicable only when bodies have spherically symmetric distributi on of mass. (D) None of these. Three particles are projected vertically upward f rom a point on the surface of the earth with velocities V(2gR/3), V(gR), V(4gR/3 ) respectively where R is the radius ofthe earth and g is the acceleration due t o gravity on the surface ofthe earth. The maximum heights attained are respectiv ely h,,!^,!^. (A) hj: h = 2 : 3 (B) h^: h = 3 :4 (C)h,: 1^=1:4 (D) h ^ R 2 3 Q.3 lack hole? ole equals Q4 A geostationary satellite is at a height h above the surface of earth. If earth radius is R (A) The minimum colatitude q on earth upto which the satellite can b e used for communication is sin- (R/R + h). (B) The maximum colatitudes q on ear th upto which the satellite can be used for communication is sin" (R/R + h). (C) The area on earth escapedfromthis satellite is given as 2pR (1 + sinq) (D) The area on earth escapedfromthis satellite is given as 2pR (1 + cosq) 1 1 2 2 Q5 & Gravitational potential at the centre of curvature of a hemispherical bowl of ra dius R and mass M is V. (A) gravitational potential at the centre of curvature o f a thin uniform wire of mass M, bent into a semicircle of radius R, is also equ al to V. (B) In part (A) if the same wire is bent into a quarter of a circle the n also the gravitational potential at the centre of curvature will be V. (C) In part (A) if the same wire mass is nonuniformly distributed along its length and it is bent into a semicircle of radius R, gravitational potential at the centre is V. (D) none ofthese Q.6 In a solid sphere two small symmetrical cavities are created whose centres lie on a diameter AB of sphere on opposite sides of the ce ntre. (A) The gravitationalfieldat the centre of the sphere is zero. (B) The gra vitational potential at the centre remains unaffected if cavitiesare not present (C) A circle at which all points have same potential is in the plane of diamete r AB. (D) A circle at which all points have same potential is in the plane perpe ndicular to the diameter AB. Ban sal Classes

Question Bank on Gravitation[285]4lBan ~P~31

Q.7 The spherical planets have the same mass but densities in the ratio 1:8. For the se planets, the (A) acceleration due to gravity will be in the ratio 4:1 (B) acc eleration due to gravity will be in the ratio 1:4 (C) escape velocitiesfromtheir surfaces will be in the ratio V2 : 1 (D) escape velocitiesfromtheir surfaces wi ll be in the ratio 1 : V2 When a satellite in a circular orbit around the earth enters the atmospheric region, it encounters small air resistance to its motion. Then (A) its kinetic energy increases (B) its kinetic energy decreases (C) its angular momentum about the earth decreases (D) its period ofrevolution around th e earth increases A communications Earth satellite (A) goes round the earthfrome ast to west (B) can be in the equatorial plane only (C) can be vertically above any place on the earth (D) goes round the earth from west to east Q. 8 Q.9 Q. 10 An earth satellite is movedfromone stable circular orbit to another larger and stable circular orbit. The following quantities increase for the satellite as a result ofthis change (A) gravitational potential energy (B) angular vleocit y (C) linear orbital velocity (D) centripetal acceleration Q. 11 Two satellites of same mass of a planet in circular orbits have periods of revolution 32 days a nd 256 days. If the radius of the orbit of the first isx, then the (A) radius of the orbit of the second is 8x (B) radius ofthe orbit of the second is 4x (C) to tal mechanical energy ofthe second is greater than that of the first (D) kinetic energy of the second is greater than that of the first. Q. 12 Two satellites Sj & s of equal masses revolve in the same sense around a heavy planet in coplanar circular orbit of radii R & 4R (A) the ratio of period of revolution Sj & s is 1 : 8. (B) their velocities are in the ratio 2 : 1 (C) their angular momentum ab out the planet are in the ratio 2 : 1 (D) the ratio of angular velocities of s w .r.t. s, when all three are in the same line is 9 : 5. 2 2 2 Q. 13 A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth (A) the acceler ation of S is always directed towards the centre ofthe earth (B) the angular mom entum of S about the centre ofthe earth changes in direction, but its magnitude remains constant (C) the total mechanical energy of S varies periodically with t ime (D) the linear momentum of S remains constant in magnitude 1*1x ^Bansal Classes (e) Rm if a = 45° ' Note that for a given velocity ofprojection & a given horizont al range there are in general two directions of proj ection which are complement ofeach other and are equally inclined to the direction ofthe maximum range. Kinematics [2]

(F) VELOCITY & DIRECTION O F MOTION A T A GIVEN TIME : VcosB =ucosa Squaring & adding these 2 equations we will get the velocity of the VsinB =usina-gt projectile. Dividing the velocities in y and x directions gives the direction of motion. V cos 0 =u cos a on adding V = u - 2 gh V sin 0 =u sin a-2gh_ 2 2 2 2 2 2 2 2 2 2 ( g ) VELOCITY & DIRECTION O F MOTION A T A GIVEN HEIGHT H : ( h ) EQUATIONS O F MOTION IN VECTOR NOTATION : (i) V=u+ gt (ii) S=ut+—gt (iii) V = -=u+-gt (V = average velocity vector) 2 2 av t 2. av (i) EQUATION O F TRAJECTORY : gx - x tan a 2u cos a v Ry dy Note that — represent the direction of motion dx 7. PROJECTILE UP AN INCLINED PLANE : (a) Total time of flight onthe inclined plane —2 / (aT _ 2u sincosp P ) // \ s g Oblique Proj ection (refer fig-1) y = x tan a 2 2 / — (b) Range PQ on the inclined plane PQ 2u cosa . sin(a-P) g cos p 2 2 (c) (d) (e) W 71 ForMaxmimumrange 2 a - P = — =>a= u— Z*— +p ^ T " Hence the direction for maximum r ange bisects the angle between the vertical and the inclined plane. R = u 2 max gcos (3 [sin (2 a - P) - sinP] kf N Greatest distance ofthe projectile from the inclined plane; u sin (a-p) 2 2

g(l+sinP) S = 2g cosp when the projectile is at H, its velocity perpendicular to the plane is zero. 8. PROJECTILE DOWN AN INCLINED PLANE: (a) Time offlight= ' ( P) (b) (c ) (d) gcosp Range OP 2u sin(a + p). cosa g cos p u Maximum range= g(l-sinp) 2 2 2 u s n a + 7 _p C Angle ofproj ection a for maximum range= 4 2 faBansal Classes Kinematics [3]

Q.l Q. 2 Q.3 Q.4 Q.5 A butterfly is flying with velocity 10 i +12 j m/s and wind is blowing along x a xis with velocity u. If butterfly starts motionfromA and after some time reaches point B,findthe value of u. EXERCISE - / y B 37° Find the change in velocity of the tip of the minute hand (radius =10 cm) of a c lock in 45 minutes. A,B&Care threeobjects each movingwith constant velocity. A's speed is lOm/sec in a direction pQ. The velocity of B relative to A is 6 m/sec at an angle of, cos (15/24) to PQ. The velocity of C relative to B is 12 m/sec i n a direction Qp, thenfindthe magnitude of the velocity of C. Rain is falling ve rtically with a speed of 20 ms" relative to air. A person is running in the rain with a velocity of 5 ms and a wind is also blowing with a speed of 15 ms (both towards east). Find the angle with the vertical at which the person should hold his umbrella so that he may not get drenched. -1 1 -1 -1 The velocity-time graph ofthe particle moving along a straight line is shown. Th e rate of acceleration and deceleration is constant and it is equal to 5 ms"" . If the s average velocity during the motion is 20 ms , thenfindthe value of t. 2 -1 25 sec Q.6 The fig. shows the v-t graph of a particle moving in straight line. Find the tim e when particle returns to the starting point. v Q.7 Q.8 Q.9 A particle is proj ected in the X-Y plane. 2 sec after proj ection the velocity of the particle makes an angle 45° with the X - axis. 4 sec after projection, it m oves horizontally. Find the velocity of projection (use g = 10 ms ). -2 A small ball rolls off the top landing of a staircase. It strikes the mid point of the first step and then mid point of the second step. The steps are smooth & identical in height & width. Find the coefficient of restitution between the bal l & the first step. A stone is dropped from a height h. Simultaneously another s tone is thrown up from the ground with such a velocity that it can reach a heigh t of 4h. Find the time when two stones cross each other. 2 Q.10 A particle is proj ected upwards with a velocity of 100 m/sec at an angle o f 60° with the vertical. Find the time when the particle will move perpendicular t o its initial direction, taking g=10 m/sec . Q.ll A particle is moving on a stra ight line. Its displacementfromthe initial position |s„j is plotted against time i n the graph shown. What will be the velocity of the particle at 2/3 sec? Assume the graph to be a sine curve. - /

\time T = 2s~ faBansal Classes Kinematics [4]

Q.12 A large number of bullets are fired in all direction with the same speed v. What is the maximum area on ground on which these bullets can spread? Q.13 A bo at starts from rest from one end of a bank of a river of width d flowing with ve locity u. The boat is steered with constant acceleration a in a direction perpen dicular to the bank. Ifpoint of start is origin, direction ofbank is x axis and perpendicular to bank is y axis. Find the equation oftrajectory ofthe boat. Q.14 A ball is thrown horizontallyfroma cliff such that it strikes ground after 5 se c. The line of sightfromthe point of projection to the point ofhitting makes an angle of 37° with the horizontal. What is the initial velocity ofprojection. Q.15 A ball is proj ected on smooth inclined plane in direction perpendicular to line of greatest slope with velocity of 8m/s. Find it's speed after 1 sec. Q.16 A gl ass wind screen whose inclination with the vertical can be changed, is mounted o n a cart as shown infigure.The cart moves uniformly along the horizontal path wi th a speed of 6 m/s. At what maximum angle a to the vertical can the wind screen be placed so that the rain drops falling vertically downwards with velocity 2 m /s, do not enter the cart? Q.17 A particle is proj ectedfrompoint P with velocit y 5 A/2 m/s perpendicular to the surface of a hollowrightangle cone whose axis i s vertical. It collides at Q normally. Find the time ofthe flight ofthe particle . Q.18 Find range ofproj ectile on the inclined plane which is proj ected perpen dicular to the incline plane with velocity 20m/s as shown in figure. Q.19 AB and CD are two smooth parallel walls. A child rolls a ball along ground from A towa rds point P find PD so that ball reaches point B after striking the wall CD. Giv en coefficient of restitution e = 0.5 0 8 m/s 777777777777777777/7777777777 mmn o y o t-u = 20ms-' 37°X C- P«—X- -D A; 1.5m -B Q.20 Initial acceleration of a particle moving in a straight line is a and initi al velocity is zero. The acceleration reduces continuously to half in every t se conds as a =a . Find the terminal velocity of the particle. 2— ta 0 mvuuuuummuwmv Q.21 Find the acceleration of movable pulley P and block B if rH rK acceleration of block A = 1 m/s 4-. Q 2 El Q.22 The velocities of Aand B are marked inthefigure.Find the velocity of blo ck C (assume that the pulleys are ideal and string inextensible). ^777777777777777777777777 3m/s lm/s )£ m 777777777777777777777777 J3 B

faBansal Classes Kinematics [5]

Q.23 A particle is moving in x-y plane such that x = t + sin(t) meter, y = cos ( t) meter, t is the time in sec. Find the length of the path taken by the particl e from t = 0 to t = 2n sec. Q.24 The speed of a particle when it is at its great est height ^2/5 is of its speed when it is at its half the maximum height. The a ngle ofproj ection is and the velocity vector angle at half the maximum height i s . Q.25 A weightless inextensible rope on a stationary wedge forming angle a wi th the horizontal. One end of the rope is fixed to the wall at point A. A small load is attached to the rope at point B. The wedge starts moving to therightwith a constant acceleration. Determine the acceleration a, ofthe load when it is st ill on the wedge. 777777777777/ Q.26 The horizontal range of a projectiles is R and the maximum height attained by it is H. A strong wind now begins to blow in the direction of motion of the p rojectile, giving it a constant horizontal acceleration = g/2. Under the same co nditions ofproj ection, find the horizontal range of the proj ectile. Q .27 Cons ider the acceleration of a particle for a given time't' at 'a' m/s followed imme diately by retardation at the same rate of'a' m/s for time 't/2', as one cycle. If the particle startedfromrest,findthe distance travelled by it after 'n' such cycles in succession. 2 2 Q. 2 8 A particle is thrown horizontally with relative velocity 10 m/sfroman inc lined plane, which is also moving with acceleration 10 m/s vertically upward. Fi nd the time after which it lands on the plane (g = 10 m/s ) ^ 2 2 10m/s 3 0 2 faBansal Classes Kinematics [6]

Q. 1 A steel ball bearing is releasedfromthe roof of a building. An observer sta nding infrontof a window 120 cm high observes that the ball takes 0.125 sec to f all from top to the bottom of the window. The ball continutes to fall & makes a completely elastic collision with side walk & reappears at the bottom of the win dow 2 s after passing it on the way down. How tall is the building ? Q. 2 A trai n takes 2 minutes to acquire its full speed 60kmphfromrest and 1 minute to come to restfromthe full speed. If somewhere in between two stations 1 km ofthe track be under repair and the limited speed on this part be fixed to 20kmph, find the late running ofthe train on account of this repair work, assuming otherwise nor mal at running of the train between the stations. Q. 3 A speeder in an automobil e passes a stationary policeman who is hiding behind a bill board with a motorcy cle. After a 2.0 sec delay (reaction time) the policeman accelerates to his maxi mum speed of 150 km/hr in 12 sec and catches the speeder 1.5 km beyond the billb oard. Find the speed of speeder in km/hr. Q. 4 Q.5 Q. 6 Q.7 Q. 8 (a) (b) Q. 9 Q. 10 Q. 11 (i) (ii) (iii) Aballoon is ascending vertically with an acceleration o f 0.2m/s , Two stones are droppedfromit at an interval of 2 sec. Find the distan ce between them 1.5 sec after the second stone is released.(use g=9.8m/s ) 2 2 EXERCISE # III A ship steaming north at the rate of 12 km/h observes a ship due east to itself and distant 10 km, which steaming due west at the rate of 16 km/h. After what ti me they are at least distancefromone another and what is this least distance. An aeroplane is observed by two persons travelling at 60 km/hr in two vehicles mov ing in opposite directions on a straight road. To an observer in one vehicle the plane appears to cross the road track at right angles while to the observer in the other vehicle the angle appears to be 45°. At what angle does the plane actual ly cross the road track and what is its speed relative to the ground. A girl can paddle her canoe at 5m/sec. in still water. She wishes to cross a straight rive r which is flowing at 3m/sec. At what angle to the river bank should she steer t o cross, (a) as quickly as possible, (b) by the shortest route. How long will ap lane take to fly around a square with side a with the wind blowing at a velocity u, in the two cases the direction ofthe wind coincides with one ofthe sides the direction ofthe wind coincides with one diagonal ofthe square. The velocity oft he plane in still air is v > u. Two ships A and B originally at a distance d fro m each other depart at the same time from a straight coastline. Ship A moves alo ng a straight line perpendicular to the shore while ship B constantly heads for ship A, having at each moment the same speed as the latter. After a sufficiently great interval oftime the second ship will obviously follow thefirstone at a ce rtain distance. Find the distance. The slopes of the wind-screen of two motorcar s are p = 3 0° and p = 15° respectively. The first car is travelling with a velocity of v horizontally. The second car is travelling with a velocity v in the same d irection. The hail stones are falling vertically. Both the drivers observe that the hail stones rebound vertically after elastic collision with the wind-screen. Find the ratio of v,/v A rocket is launched at an angle 53° to the horizontal wit h an initial speed of 100 ms . It moves along its initial line of motion with an acceleration of 30 ms~ for 3 seconds. At this time its engine falls & the rocke t proceeds like afreebody. Find : the maximum altitude reached by the rocket tot al time of flight. the horizontal range . [ sin 53° = 4/5 ] 2 t 2 r _1 2 ^Bansal Classes Kinematics [7]

Q.12 A small ball is thrown between two vertical walls such that in the absence of the wall its range would have been 5d. The angle of projection is a. Given th at all the collisions are perfectly elastic, find (a) Maximum height attained by the ball. \u\uu\uuvwu\ (b) Total number of collisions before the ball comes bac k to the ground, and d/2 (c) Point at which the ball fallsfinally.The walls are supposed to be very tall. Q.13 A hunter is riding an elephant ofheight 4m moving in straight line with uniform speed of 2m/sec. A deer running with a speed V in frontat a distance of 4V5m moving perpendicular to the direction of motion of th e elephant. If hunter can throw his spear with a speed of 1 Om/sec. relative to the elephant, then at what angle 0 to it's direction of motion must he throw his spear horizontally for a successful hit. Find also the speed 'V' ofthe deer. Q. 14 A perfectly elastic ball is thrownfromthe foot of a smooth plane inclined at an angle a to the horizontal. If after striking the plane at a distance Ifromthe point of projection, it rebounds and retraces its former gl (1 + 3 sin a) path, show that the velocity of projection is 2 sin a Q.15 A particle is proj ectedfr omthe foot of an inclined plane at an angle a in the vertical plane through the line of greatest slope & hits the plane at right angles. If p be the angle the d irection of projection makes with the plane & if the particle returns to the poi nt of proj ection in two jumps,findthe value ofthe coefficient ofrestitution. Q. 16 A projectile is to be thrown horizontallyfromthe top of a wall of height 1.7 m. Calculate the initial velocity ofprojection if it hits perpendicularly an inc line of angle 37° which startsfromthe ground at the bottom of the wall. The line o f greatest slope of incline lies in the plane ofmotion of projectile. Q.17 Two i nclined planes OA and OB having inclination (with horizontal) 30° and 60° respective ly, intersect each other at O as shown infig.Aparticle is projected from point P with velocity u = \ 0^3 m s along a direction perpendicular to plane OA. Ifthe particle strikes plane OB perpendicularly at Q, calculate velocity with which pa rticle strikes the plane OB, (a) (b) time offlight, (c) vertical height h of Pfr omO, (d) maximum heightfromO attained by the particle and (e) distance PQ Q.18 A particle is projected with a velocity 2 ^/ag so that it just clears two walls o f equal height 'a' which are at a distance '2a' apart. Show that the time of pas sing between the walls is 2-JaJg • Q.19 A stone is projected from the point of a g round in such a direction so as to hit a bird on the top of a telegraph post of height h and then attain the maximum height 2h above the ground. If at the insta nt of projection, the bird were to fly away horizontally with a uniform speed, f ind the ratio between the horizontal velocities ofthe bird and the stone, if the stone still hits the bird while descending. Q.20 Two persons Ram and Shyam are throwing ball at each other as shown in thefigure.The maximum horizontal distanc efromthe building where Ram can stand and still throw a ball at Shyam is dj. The maximum ^ horizontal distance of Ramfromthe building where Shyam can throw a ba ll is d . If both of them can throw ball with a velocity of ^2gk, find -nn mm' m u u u m m fc m ufl the ratio of dj/d . Neglect the height of each person. 2 _1 Shyam 2 2 faBansal Classes Kinematics [8]

EXERCISE # III Q. 1 The motion of a body is given by the equation = 6 . 0 - 3 v(t) ; where v (t ) is the speed in m/s & t in sec., if the body has v = 0 at t = 0 then (A) the t erminal speed is 2.0 m/s (B) the magnitude of the initial acceleration is 6.0 m/ s (C) the speed varies with time as v(t) = 2(l -e~ )m/s (D) the speed is 1.0 m/s when the acceleration is halfthe initial value. [JEE' 1995] 2 3t Q.2 Two guns, situated at the top of a hill of height 10 m, fire one shot each with the same speed 5 yfs m/s at; some interval oftime. One gun fires horizontally an d other fires upwards at an angle of 60° with the horizontal. The shots collide in air at a point P. Find (a) the time interval between thefirings,and (b) the coo rdinates ofthe point P. Take origin of the coordinates system at the foot ofthe hill right below the muzzle and traj ectories in X-Y plane. [JEE' 1996] The traj ectory ofa proj ectile in a vertical plane is y = ax - bx , where a, b are cons tants & x and y are respectively the horizontal & vertical distances ofthe proje ctilefromthe point ofprojection. The maximum height attained is & the angle of p rojectionfromthe horizontal is . [JEE' 1997] 2 Q. 3 Q.4 (a) (b) Q.5 (i) (ii) Q.6 Q.7 A large heavy box is sliding without friction down a smooth plane of inclination 9. From a point P on the bottom ofa box, a particle is proj ected inside the bo x. The'initial speed ofthe particle with respect to box is u and the direction o f projection makes an angle a with the bottom as shown in figure. ithe particle lands. (Assume that the particle does not litany other surface of the box. Negle ct air resistance). , ' Ifthe horizontal displacement ofthe particle as seen by an observer on the ground is zero,findthe speed of the box with respect to the g round at the instant when the particle was projected. [JEE' 1998] A particle of mass 10~ kg is moving slong the positive x-axis under the influence of a force —K F(x)= whereK= 10 Nm .Attimet = 0itisatx-1.0m&itsvelocityisv = 0. Find: 2x its ti me at when it reaches x = 0.25 m. thevelocitywhich it reaches x = 0.5 0 m [JEE' 1998] In 1.0 sec. a particle goesfrompoint Ato point B moving in a semicircle of radius 1.0 m. The magnitude of average velocity is: [JEE '99] (A) 3.14 m/sec (B) 2.0 m/sec im (C) 1.0 m/sec (D) zero BThe co-ordinates of a particle moving in a plane are given by x (t) = a cos (7it) and y (t) = b sin (rat) where a, b ( current 0 11. MAGNETIC INDUCTION DUE TO TOROID B = p nl N where n = —— (no. of turns per m) 2tcR N = total turns R»r 12. MAGNETIC INDUCTION DUE TO CURRENT CARRYING SHEET where I = Linear current density (A/m) 13. MAGNETIC INDUCTION DUE TO THICK SHEET At point P At point P j 2 1 B = ~ u ld B = p Jx out f) in 0 V ;2 p ' * • JA/ni

2 x 14. MAGNETIZATION INTENSITY ( H ) : B The magnetic intensity (H) at any point in a magnetic field is defined as H = — , where M B = magnetic induction at the point ; p = permeability of the medium ( a) The line of earth's magnetic induction lies in a vetical plane coinciding wit h the magnetic North South direction at that place. This plane is called the MAG NETIC MERIDIAN. Earth's magnetic axis is slightly inclined to the geometric axis of earth and this angle varies from 10.5° to 20°. The Earth's Magnetic poles are op posite to the geometric poles i. e. at earth's north pole, its magnetic south po le is situated and vice versa. Magnetics Effect of Current ^Bansal Classes [3] 15. GILBERT'S MAGNETISM ( E A R T H ' S MAGNETIC F I E L D ) :

(b) On the magnetic meridian plane , the magnetic induction vector of the earth at any point, generally inclined to the horizontal at an angle called the MAGNET IC DIP at that place , such that B = total magnetic induction of the earth at th at point. B = the vertical component of B in the magnetic meridian plane = B sin 9 . B = the horizontal component of B in the magnetic meridian plane = B cos 9 . = tan 9 . B (c) At a given place on the surface of the earth, the magnetic mer idian and the geographic meridian may not coincide. The angle between them is ca lled "DECLINATION AT THAT PLACE" . (d) Lines drawn on earth at different places having same declination angle are called as "isogonic lines" and line ofzero dec lination is called as "agonic lines". (e) Lines drawn on earth at different plac es having same dip angle are called as "isoclinic lines" and line of zero dip is called as "aclinic lines". v H H 16. When more than one magnetic fields are suspended at a point and the vector sum o f the magnetic inductions due to different fields , equal to zero, the point is a magnetic neutral point. AMPERES LAW J B . DF = > NEUTRAL POINT IN SUPERPOSED MAGNETIC FIELDS : 17 21 = algebric sum of all the currents . 18. LORENTZ FORCE : An electric charge 'q' moving with a velocity V through a magnetic field of magn etic induction B experiences a force F, given by F = qVxB There fore, if the cha rge moves in a space where both electric and magnetic fields are superposed . F = nett electromagnetic force on the charge = q E + q V x B This force is called the LORENTZ FORCE . 19. MOTION OF A CHARGE IN UNIFORM MAGNETIC FIELD : (a) When v is || to B : Motion will be in a st. line and F = 0 (b) When v is to B : Motion will be in circular path with radius R = velocity co = — and F = qvB. m mv and angular (c)When v is atZG to B : Motion will be helical with radius R, = - -------- and pitch qB P = 27tmv cos 6 qB H a n d F = q v B s i n 0 20. MAGNETIC FORCE O N A STRAIGHT CURRENT CARRYING W I R E : F = I (L x B) I = current in the straight conductor L - length of the conductor in the direction of the current in it B = magnetic induction. (Uniform throughou t the length of conduction) Note : In general force is F = JI (d£ x B) ^Bansal Classes Magnetics Effect of Current

[4]

21. (i) (ii) When two long straight linear conductors are parallel and carry a current in eac h , they magnetically interact with each other, one experiences a force. This fo rce is of : Repulsion if the currents are anti-parallel (i.e. in opposite direct ion) or Attraction if the currents are parallel (i.e. in the same direction) Thi s force per unit length on either conductor is given by F = . Where r - perpendi cular r distance between the parallel conductors When a plane closed current cir cuit of'N' turns and of area 'A' per turn carrying a current I is placed in unif orm magnetic field , it experience a zero nett force , but experience a torque g iven b y i = N I A x B = MxB = BINA sin 9 When A = area vector outward from the face of the circuit where the current is anticlockwise, B = magnetic induction o fthe uniform magnetic feild. M = magnetic moment of the current circuit = IN A N ote : This expression can be used only if B is uniform otherwise calculus will b e used. It consists of a plane coil of many turns suspended in a radial magnetic feild. when a current is passed in the coil it experiences a torque which produ ces a twist in the suspension. This deflection is directly proportional to the t orque .'. NIAB = KG I= ( ( K \ MOVING COIL GALVANOMETER : MAGNETIC TORQUE O N A CLOSED CURRENT CIRCUIT : MAGNETIC INTERACTION FORCE BETWEEN T w o PARALLEL LONG STRAIGHT CURRENTS : 22. 23. 9 K = elastic torsional constant of the suspension I=C 0 C = —-7— = GALVANOMETER CONST ANT. NAB 24. FORCE EXPERIENCED B Y A MAGNETIC DIPOLE IN A N O N - U N I F O R M MAGNETIC FIEL D : SB F = M dr where M = Magnetic dipole moment. 25. FORCE ON A RANDOM SHAPED CONDUCTOR IN MAGNETIC FIELD , 1. 2. 26. Magnetic force on a loop in a uniform B is zero b* Force experienced by a wire o f any shape is equivalent to force on a wire joining points A & B in a uniform m agnetic field . If a charge q is rotating at an angular velocity co, . qco its e quivalent current is given as I 271 & its magnetic moment is M = l7tR - ~qcoR . 2 2 __ J MAGNETIC MOMENT OF A ROTATING CHARGE: A NOTE: The rate of magnetic moment to Angular momentum of a uniform rotating object whi ch is charged M 2m uniformly is always a constant. Irrespective of the shape of conductor — - —— L q

f§,Bansa!Classes Magnetics Effect of Current [11]

1 Q.2 Figure shows a straight wire of length / carrying a current i. Find the magnitud e of magneticfieldproduced by the current at point P. - — . 5 5 Two circular coils A and B of radius cm and 5 cm respectively carry current 5 Amp and ^ Amp respec tively. The plane ofB is perpendicular to plane ofAand their centres coincide. F ind the magnetic field at the centre. Find the magneti cfieldat the centre P of square of side a shown in figure / EXERCISE # I Q.4 What is the magnitude ofmagneticfieldat the centre 'O' ofloop ofradius V2 m made of uniform wire when a current of 1 amp enters in the loop and taken out of it by two long wires as shown in the figure. Find the magnetic induction at the ori gin in thefigureshown. Find the magnetic induction at point 0, ifthe current car rying wire is in the shape shown in the figure. Find the magnitude ofthe magneti c induction B of a magneticfieldgenerated by a system of thin conductors along w hich a current /' is flowing at a point A (0, R, O), that is the centre of a cir cular conductor of radius R. The ring is in yz plane. ^ f 00 1 amp\ 0 H--' yT ° I 90 00 Ti airip v/ -X 5 /.I Q. 6 Q. 7 / B.d l around the closed path. 2 3 4 5 6 ^ 5 f§, Bansa! Classes

Q. 10 Electric charge q is uniformly distributed over a rod oflength /. The rod is placed parallel to along wire carrying a current i. The separation between th e rod and the wire is a. Find the force needed to move the rod along its length with a uniform velocity v. Q/i 1 An electron moving with a velocity 5 x 10 ms" i in the uniform electricfieldof 5 x 10 Vm j . Find the magnitude and direction o f a minimum uniform magneticfieldin tesla that will cause the electron to move u ndeviated along its original path. 6 1 7 1 Magnetics Effect of Current [11]

(X I2 A charged particle (charge q, mass m) has velocity v at origin in +x direc tion. In space there is a uniform magnetic field B in - z direction. Find the y coordinate of particle when is crosses y axis. 0 Q. 13/ A conducting circular loop of radius r carries a constant current i. It i s placed in a uniform magnetic field B o such that B is perpendicular to the pla ne ofthe loop. Find the magnetic force acting on the loop is 0 Q . \ y A rectangular loop ofwire is oriented with the left corner at the origin , one edge along X-axis and the other edge along Y-axis as shown in thefigure.A magnetic field is into the page and has a magnitude that is given by (3 = ay whe re a is contant. Find the total magnetic force on the loop if it carries current i. B -»x Q.15 Two coils each of 100 turns are held such that one lies in the vertical pla ne with their centres coinciding. The radius ofthe vertical coil is 20 cm and th at ofthe horizontal coil is 3 0 cm. How would you neutralize the magneticfieldof the earth at their common centre ? What is the current to be passed through eac h coil ? Horizontal component of earth's magnetic induction-3.49 x 10" T and ang le of dip = 30°. 5 Q.16 Find the ratio of magneticfieldmagnitudes at a distance 10 m along the axis and at 60° from the axis, from the centre of a coil of radius 1 cm, carrying a cu rrent 1 amp. Q.17 A particle of charge +q and mass m moving under the influence of a uniform electricfieldE i and a magneticfieldB k enters in I quadrant of a c oordinate system at a point (0, a) with initial velocity v i and leaves the quad rant at a point (2a, 0) with velocity - 2v j. Find (a) Magnitude of electric fie ld (b) Rate ofwork done by the electricfieldat point (0, a) (c) Rate of work don e by both the fields at (2a, 0). Q.18 A system of long four parallel conductors whose sections with the plane of the drawing lie at the vertices of a square the re flow four equal currents. The directions of these currents are as follows : t hose marked ® point away from the reader, while those marked with a dot point towa rds the reader. How is the vector of magnetic induction directed at the centre o f the square? Ij l 2 ©© Q.19 A cylindrical conductor ofradius R carries a current along its length. The current density J, however, it is not uniform over the cross section of the cond uctor but is a function ofthe radius according to J = br, where b is a constant. Find an expression for the magneticfieldB. r^ (a) at T j < R & (b) at distance r > R, mesured from the axis l[ ( I 2 R Q . 20 A square current carrying loop made of thin wire and having a mass m =1 O g can rotate withoutfrictionwith respect to the vertical axis 0 0 , passing thro ugh the centre of the loop at right angles to two opposite sides of the loop. Th e loop is placed in a homogeneous magneticfieldwith an induction B = 10" T direc ted at right angles to the plane of the drawing. Acurrent I = 2Ais flowing in th e loop. Find the period of small oscillations that the loop performs about its p osition of stable equilibrium. } 1

O^B O, [11] f§,Bansa!Classes Magnetics Effect of Current

Q.21 A charged particle having mass m and charge q is accelerated by a potential difference V, it flies through a uniform transverse magneticfieldB. Thefieldocc upies a region of space d. Find the time interval for which it remains inside th e magnetic field. Q. 22 A proton beam passes without deviation through a region of space where there are uniform transverse mutually perpendicular electric and magneticfieldwith E and B. Then the beam strikes a grounded target. Find the for ce imparted by the beam on the target ifthe beam current is equal to I. Q.23 An infinitely long straight wire carries a conventional current I as shown in the f igure. The rectangular loop carries a conventional current I in the clockwise di rection. Find the net force on the rectangular loop. 1 Q.24 An arc of a circular loop of radius R is kept in the horizontal plane and a constant magneticfieldB is applied in the vertical direction as shown in the fi gure. If the arc carries current I thenfindthe force on the arc. Q.25 Two long s traight parallel conductors are separated by a distance of r = 5cm and carry cur rents i = 10A&i = 20A. What work per unit length of a conductor must be done to increase the separation between the conductors to r„ = 10 cm if, currents flow in the same direction? 1 1 2 List of recommended questions from I.E. Irodov. 3.220, 3.223, 3.224, 3.225, 3.226, 3.227, 3.228, 3.229, 3.230, 3.234, 3.236, 3.2 37, 3.242 3.243, 3.244, 3.245, 3.251, 3.252, 3.253,3.254,3.257, 3.258, 3.269, 3. 372, 3.373, 3.383, 3.384, 3.386, 3.389, 3.390, 3.391, 3.396 f§,Bansa!Classes Magnetics Effect of Current [11]

Q. 1 Three infinitely long conductors R, S and T are lying in a horizontal plane as shown in thefigure.The currents in the respective conductors are R S T T • • 2-K I = I Sin(Qt+y) I = I sin (©t) I = I sin (®t —) Find the amplitude of the vertical co mponent of the magneticfieldat a point P, distance 'a' away from the central con ductor S. x R 0 s 0 T 0 EXERCISE # II Q. 2 Four long wires each carrying current I as shown in the figure are placed at the points A, B, C and D. Find the magnitude and direction of (i) magnetic field at the centre of the square. (ii) force per metre acting on wire at point D. 2 D(-a,a)© C(-a,-a)0 ffi A(a, a) © B(a.-a) Q. 3 Q.4 An infinite wire, placed along z-axis, has current I, inpositive z-direction. Ac onducting rod placed in xy plane parallel to y-axis has current I in positive ydirection. The ends of the rod subtend + 30° and - 60° at the origin with positive x -direction. The rod is at a distance afromthe origin. Find net force on the rod. A square cardboard of side / and mass m is suspendedfroma horizontal axis XY as shown infigure.A single wire is wound along the periphery of board and carrying a clockwise current I. At t = 0, a vertical downward magneticfieldof inductionB is switched on. Find the minimum value ofB so that the board will be able to ro tate up to horizontal level. A straight segment OC (of length L meter) of a circ uit carrying a current I amp is placed along the x-axis. Two infinitely ling str aight wires A and B ,each extending form z = - oo to + oo, are fixed at y = - a metre and y = +a metre respectively, as shown in the figure. If the wires A and B each carry a current I amp into plane of the paper. Obtain the expression for the force acting on the segment OC. What will be the force OC if current in the wire B is reversed? A very long straight conductor has a circular cross-section of radius R and carries a current density J. Inside the conductor there is a cyl indrical hole of radius a whose axis is parallel to the axis of the conductor an d a distance bfromit. Let the z-axis be the axis of the conductor, and let the a xis of the hole be at x=b. Find the magnetic field on the x = axis at x = 2R on the y = axis at y = 2R. Q charge is uniformly distributed over the same surface of a right circular cone of semi-vertical angle 9 and height h. The cone is unif ormly rotated about its axis at angular velocity co. Calculated associated magne tic dipole moment. *y &B O C Q.5 Q. 6 y f 'in (a) (b) Q.7 I °

WfH b / , \Lf>« f§,Bansa!Classes Magnetics Effect of Current [11]

Q.8 A wire loop carrying current I is placed in the X-Y plane as shown in the figure (a) If a particle with charge +Q and mass m is placed at the centre P and given a velocity along NP (fig). Find its instantaneous acceleration (b) If an extern al uniform magnetic induction field B = B f is applied, find the torque acting o n the loop due to the field. A long straight wire carries a current of 10 A dire cted along the negative y-axis as shown infigure.Auniform magneticfieldB ofmagni tude 10~ T is directed parallel to the x-axis. What is the resultant magneticfie ldat the following points? (a) x = 0 , z-2m; (b)x=2m, z = 0; (c)x = 0 , z = - 0 . 5 m 0 6 Q.9 Q.10 A stationary, circular wall clock has a face with a radius of 15 cm. Six tu rns of wire are wound around its perimeter, the wire carries a current 2.0 A in the clockwise direction. The clock is located, where there is a constant, unifor m external magneticfieldof 70 mT (but the clock still keeps perfect time) at exa ctly 1:00 pm, the hour hand of the clock points in the direction of the external magnetic field (a) After how many minutes will the minute hand point in the dir ection of the torque on the winding due to the magneticfield? (b) What is the ma gnitude of this torque. Q.ll A U-shaped wire ofmass m turn length / is immersed with its two ends in mercury (seefigure).The wire is in a homogeneousfieldofmagn etic B' X X X the wire, the wire will jump up. Calculate, from the height h that the wire reac hes, the size of the charge or current pulse, assuming that the time of the curr ent pulse is very small in comparision with the time of flight. Make use of the fact that impulse of force equals j F dt,which equals mv. Evaluate q for B = 0.1 Wb/m , m = 1 Ogm, t = 20cm & h = 3 meters, [g = 10 m/s ] — 1 1 — 2 2 ¥ / X X Q.l 2 A current i, indicated by the crosses infig.is established in a strip of c opper X of height h and width w. Auniformfieldof magnetic induction B is applied m at right angles to the strip. B X (a) Calculate the drift speed v for the ele ctrons. : X.(b) What are the magnitude and dirction of the magnetic force F acti ng on the electrons? (c) What would the magnitude & direction of homogeneous ele ctricfieldE have to be in order to counter balance the effect of the magnetic fi eld ? (d) What is the voltage V necessary between two sides of the conductor in order to create thisfieldE? Between which sides of the conductor would this volt age have to be applied ? (e) If no electricfieldis applied form the outside the electrons will be pushed somewhat to one side & thereforce will give rise to a u niform electricfieldE across the conductor untill the force ofthis electrostatic field E balanace the magnetic forces encountered in part (b). What will be the magnitude and direction of thefieldEH? Assume that n, the number of conduction e lectrons per unit volume, is 1. Ixl0 /m & that h = 0.02 meter, w = 0.1cm , i = 5 0 amp, & B = 2 webers/meter . d H h 29 3 2

f§,Bansa!Classes Magnetics Effect of Current [11]

Q. 13(a) A rigid circular loop of radius r & mass m lies in the xy plane on a fl at table and has a current I flowing in it. At this particular place, the earth' s magneticfieldis B = B 1 + B j . How large must I be before one edge of the loo p will lift from table ? (b) Repeat if, B = B 1 + B k. x y x z Q. 14 Zeeman effect. In Bohr's theory of the hydrogen atom the electron can be t hought of as moving in a circular orbit of radius r about the proton . Suppose t hat such an atom is placed in a magnetic field, with the plane of the orbit at r ight angle to B. (a) If the electron is circulating clockwise, as viewed by an o bserver sighting along B, will the angular frequency increase or decrease? (b) W hat if the electron is circulating counterclockwise? Assume that the orbit radiu s does not change. Q.15 In above problem show that the change in frequency of ro tation caused by the magnetefieldis given Be approximately by Av = ± . Such freque ncy shifts were actually observed by Zeeman in 1896. 4um A Q.16 A square loop ofwire of edge a carries a current i. (a) Show that B for a p oint on the axis of the loop and a distance xfromits centre is given by, ia B= 7 1 (4x + a ) (4x + 2a\ )1/2 (b) Can the result of the above problem be reduced to givefieldat x = 0 ? (c) Does the square loop behave like a dipole for points su ch that x » a ? If so, what is its dipole moment? 2 2 2 2 2 1 Q.17 A conductor carrying a current i is placed parallel to a current per unit w idth j and width d, as shown in the figure. Find the force per unit lenght on th e coductor. 0 z. / / \Z 0 /' r y; A il "B Q. 18 Find the work and power required to move the conductor of length / shown i n thefig.one full turn in the anticlockwise direction at a rotational frequency of n revolutions per second ifthe magneticfieldis of magnitude B everywhere and points radially outwards from Z-axis. The figure shows the surface traced by the wire AB. Q.19 The figure shows a conductor of weight 1.0 N and length L = 0.5 m placed on a roughinclined plane making an angle 30° with the horizontal so that c onductor is perpendicular to a uniform horizontal magneticfieldof induction B = 0.10 T. The coefficient of staticfrictionbetween the conductor and the plane is 0.1. A current of I = 10 A flows through the conductor inside the plane of this paper as shown. What is the force needed to be the applied parallel'to the incli ned plane to sustaining the conductor at rest? Q.20 An electron gun G emits elec tron of energy 2kev traveling in the (+)ve x-direction. The electron are require d to hit the spot S where GS = 0. lm & the line GS makes an angle of 60° with the x-axis, as shown in the fig. Auniform magnetic field B parallel to GS exists in the region outsiees to electron gun. Find the minimum value of B needed to make the electron hit S . f ^

/)60° Gun X §, Bansa! Classes Magnetics Effect of Current [11]

Q. 1 Abattery is connected between two points Aand B the circumference of a unif orm conducting ring of radius r and resistance R. One of the arcs AB of the ring subtends an angle 0 at the centre. The value of the magnetic induction at the c entre due to the current in the ring is : [ JEE '95, 2] (A) zero, only if 9 = 18 0° (B) zero for all values of 0 (C) proportional to 2(180°-0) (D) inversely proporti onal to r Q. 2 Two insulated rings, one slightly smaller diameter than the other , are suspended along their diameter as shown, initially the planes of the rings are mutually' perpendicular when a steady current is set up in each of them: [I IT '95, 1] (A) The two rings rotate to come into a common plane (B) The inner ri ng oscillates about its initially position (C) The outer ring stays stationary w hile the inner one moves into the plane of the outer ring (D) The inner ring sta ys stationary while the outer one moves into the plane of the inner ring An elec tron in the ground state of hydrogen atom is revolving in anticlock-wise directi on in a circular orbit of radius R . Obtain an expression for the orbital magnet ic dipole moment of the electron The atom is placed in a uniform magnetic. Induc tion B such that the plane normal of the electron orbit makes an angle of 30° with the magnetic induction. Find the torque experienced by the orbiting electron. [ JEE'96, 5] A proton, a deuteron and an a-particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field . If r r & r d enote respectively the radii of the trajectories of these particles then: " ° [JEE '97, 1] rP < r, (B) r a > rd > rp (C) ra = rd > rp (D)P r = rd = a r (A) ' ' d 3 infinitely long thin wires each carrying current /' in the same direction , are in the x-y plane of a gravity free space . The central wire is along the y-axis while the other two are along x = ±d. Find the locus of the points for which the magnetic field B is zero . If the central wire is displaced along the z-directio n by a small amount & released, show that it will execute simple harmonic motion . If the linear density ofthe wires is X,findthefrequencyof oscillation. [JEE '9 7, 5] d a r a v v V / EXERCISE # III Q. 3 (i) (ii) Q.4 Q.5 (i) Cii) Q.6 C O (ii) Select the correct alternative(s). [ JEE '98, 2 + 2 + 2 ] Two very long, straigh t, parallel wires carry steady currents I & - I respectively. The distance betwe en the wires is d. At a certain instant of time, a point charge q is at a point equidistant from the two wires, in the plane of the wires. Its instantaneous vel ocity v is perpendicular to this plane. The magnitude of the force due to the ma gnetic field acting on the charge at this instant is : (D) 0 (A) ^o iqv (B) Ho I qv (C) 2^0 2nd 7td rcd Let [ e ] denote the dimensional formula ofthe permittivi ty ofthe vaccum and [|i ] that ofthe permeability of the vacuum . If M = mass, L = length, T = time and I = electric current, (A) [e ] = M L T 1 (B) [ e j = M" L" T I (C) [^ ] = M E T ! (D) [ n j = ML T-'I 0 0 _1 -3 2 1 3 4 2 0 2 2 2 f§,Bansa!Classes Magnetics Effect of Current [11]

(iii) Q.7 Two particles, each of mass m & charge q, are attached to the two ends of a ligh t rigid rod of length 2 R. The rod is rotated at constant angular speed about a perpendicular axis passing through its centre. The ratio of the magnitudes of th e magnetic moment of the system & its angular momentum about the centre of the r od is : ( A )2m f (B) — ( C )m ^ (D) ran — m w w 0 A particle of mass m & charge q is moving in a region where uniform, constant el ectric and magnetic fields E & B are present, E & B are parallel to each other. At time t = 0 the velocity v of the particle is perpendicular to E . (assume tha t its speed is always « c, the speed oflight in vacuum). Find the velocity v of th e particle at time t. You must express your answer in terms of t, q, m, the vect ors v , E & B and their magnitudes v , E & B. [JEE '98, 8] 0 0 Q.8 (a) (b) A uniform, constant magneticfieldB is directed at an angle of 45° to the x-axis 'V lo in the xy-plane, PQRS is a rigid square wire frame carrying a steady current y I (clockwise), with its centre at the origin O. At time t = 0, the frame is a t rest in the position shown in thefigure,with its sides parallel to the x & y a xes. Each side of the frame is of mass M & length L. What is the torque t about 0 acting on the frame due to the magnetic field ? Find the angle by which the fr ame rotates under the action of this torque in a short interval of time At, & th e axis about which this rotation occurs (At is so short that any variation in th e torque during this interval may be neglected) Given the moment of inertia of t he frame about an axis through its centre perpendicular to its plane is 4/3 ML . [JEE '98, 2 + 6] 0 / / / 2 Q9 A charged particle is released from rest in a region of steady and uniform elect ric and magnetic fields which are parallel to each other. The particle will move in a (A) straight line (B) circle (C) helix (D) cycloid [JEE'99,2] 0 rj Q.10 The region between x = 0 and x=L isfilledwith uniform, steady magneticfield B k. Aparticle of mass m, positive charge q and velocity v T travels along x-axi s and enters the region ofthe magnetic field. Neglect the gravity throughout the question. (a) Find the value of L ifthe particle emergesfromthe region of magne ticfieldwith itsfinalvelocity at an angle 30° to its initial velocity. (b) Find th efinalvelocity of the particle and the time spent by it in the magneticfield,if the magnetic field now extendsupto2.IL. [JEE '99, 6 + 4] Q. 11 (i)Aparticle of c harge q and mass m moves in a circular orbit of radius r with angular speed co. The ratio of the magnitude ofits magnetic moment to that of its angular momentum depends on (A) co and q (B) co, q and m (C) q and m (D) co and m (ii) Two long parallel wires are at a distance 2d apart. They carry steady equal currentsflowi ngout of the plane of the paper, as shown. The variation of the magneticfieldB a long the XX' is given by

(A) (B) (C) (D) f§,Bansa!Classes Magnetics Effect of Current [11]

(iii) An infinitely long conductor PQR is bent to form a right angle as shown. A M cur rent I flows through PQR. The magneticfielddue to this current at the point M is H Now, another infinitely long straight conductor QS is P Qn o § connected at Q s o that the current in PQ remainingunchanged. The magnetic field at M is now H Th e ratio H /H is given by R (A) 1/2 (B)l ~ (C) 2/3 (D) 2 (iv) An ionized gas cont ains both positive and negative ions. If it is subjected simultaneously to an el ectric field along the +x direction and a magneticfieldalong the +z direction, t hen (A) positive ions deflect towards +y direction and negative ions towards -y direction (B) all ions deflect towards +y direction. (C) all ions deflect toward s -y direction (D) positive ions deflect towards -y direction and negative ions towards +y direction. [JEE 2000 (Scr)] Q.12 A circular loop of radius R is bent along a diameter and given a shape as shown in the figure. One of the semicircle s (KNM) lies in the x - z plane and the other one (KLM) in the y-z plane with th eir centers at the origin. Current I is flowing through each ofthe semicircles a s shown in figure. (i) A particle of charge q is released at the origin with a v elocity v ^ o Find the instantaneous force f on the particle. Assume that space is gravity free. (ii) If an external uniform magneticfieldB j is applied, determ ine the forces F and F on the semicircles KLM and KNM due to thisfieldand the ne t force F on the loop . [JEE 2000 Mains, 4 + 6] r 9 0 r ] 2 : 1 : 2 Q.13 A current of 1 OA flows around a closed path in a circuit which is in the h orizontal plane as shown in thefigure.The circuit consists of eight alternating arcs ofradii ^ = 0.08 m and r = 0.12 m. Each arc subtends the same angle at the centre. (a) Find the magneticfieldproduced by this circuit at the centre. (b) An infinitely long straight wire carrying a current of 1 OA is passing through the centre of the above circuit vertically with the direction of the current being into the plane of the circuit. What is the force acting on the wire at the centr e due to the current in the circuit? What is the force acting on the arc AC and the straight segment CD due to the current at the centre? [JEE 2001, 5 + 5] Q.14 Two particles A and B of masses m and m respectively and having the same charge are moving in a plane. Auniform magneticfieldexists perpendicular to this plane . The speeds of the particles are v and v respectively and the trajectories are as shown in the figure. Then (A) m v < m v (B) m v > m v (C) m < m and v < v (D) m = n^ and v = v [JEE, 2001 (Scr)] A B A B A A B B A A B B A B A B A A B Q.15 A non-planar loop of conducting wire carrying a current I is placed as show n inthefigure.Each ofthe straight sections ofthe loop is oflength2a. The magneti c field due to this loop at the point P (a, 0, a) points in the direction 1 - ,H 1+ k) Ts 1 H+k+i) < A

> 7 T (i + j + k) (i+k) [JEE, 2001 (Scr)] f§,Bansa!Classes Magnetics Effect of Current [11]

Q . 16 A coil having N turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively. When a current 1 passes through the coil, the magneticfieldat the centre is [JEE, 2001 Screening] N H NI 2^i NI IV , b (C ) — In(A) (B) (D) /n2(b - a) a > 2(b - a) a 0 n T [ ) 0 K Q.17 A particle of mass m and charge q moves with a constant velocity v along th e positive x direction. It enters a region containing a uniform magneticfieldB d irected along the negative z direction, extending from x = a to x = b. The minim um value ofv required so that the particle can just enter the region x > b is (A ) q b B./m (B)q(b-a)B/m (C)qaB/m (D) q(b + a) B/2m [JEE 2002 (screening), 3] Q. 18 A long straight wire along the z-axis carries a current I in the negative z d irection. The magnetic vector field B at a point having coordinates (x, y) in th e z = 0 plane is [JEE 2002 (screening), 3] n i (xj-yi) ji I (yi - xj) (xi+yj) ( y (A) 2n (x +y ) (B) M (x + y ) ( Q 2n (x +y ) (D) M (x -+yj )) 2n 2n 0 0 X1 2 2 2 2 2 2 2 2 Q. 19 The magneticfieldlines due to a bar magnet are correctly shown in N V. ^—. N [JEE 2002 (screening), 3 ] ^ N , Q.20 A rectangular loop PQRS madefroma uniform wire has length a, width b and ma ss m. It isfreeto rotate about the arm PQ, which remains.hinged along a horizont al line taken as the y-axis (seefigure).Take the vertically upward direction as the z-axis. Auniform magneticfieldB = (3 i + 4 k) B exists in the region. The lo op is held in the x-y plane and a current I is passed through it. The loop is no w released and is found to stay in the horizontal position in equilibrium. R Wha t is the direction of the current I in PQ? (a) (b) Find the magnetic force on th e arm RS. [JEE 2002, 1+1+3] (c) Find the expression for I in terms of B a, b and m. 0 Q. 21 A circular coil carrying current I is placed in a region of uniform magnet icfieldacting perpendicular to a coil as shown in the figure. Mark correct optio n [JEE 2003 (Scr)] * (A) coil expands (B) coil contracts x (C) coil moves left ( D) coil moves right x x Q.22 Figure represents four positions ofa current carrying coil is a magneticfie lddirected towards right, h represent the direction of area ofvector of the coil . The correct order ofpotential energy is: [JEE 2003 (Scr)] (A) I > III > II > I V (B) I < III < II < IV (C) IV < I < II < II (D) II > II > IV > I f§,Bansa!Classes Magnetics Effect of Current [11]

Q.23 A wheel of radius R having charge Q, uniformly distributed on the rim of th e wheel is free to rotate about a light horizontal rod. The rod is suspended by light inextensible stringe and a magneticfieldB is applied as shown inthe figure . The 3Tn initial tensions in the strings are T . Ifthe breaking tension ofthe s trings are find the maximum angular velocity co with which the wheel can be rota te. [JEE 2003] 0 0 Q.24 A proton and an alpha particle, after being accelerated through same potent ial difference, enter a uniform magneticfieldthe direction ofwhich is perpendicu lar to their velocities. Find the ratio of radii ofthe circular paths ofthe two particles. [JEE 2004] Q.25 In a moving coil galvanometer, torque on the coil can be expressed as T = ki, where i is current through the wire and k is constant. The rectangular coil of the galvanometer having numbers of turns N, area A and m oment of inertia I is placed in magneticfieldB. Find (a) k in terms of given par ameters N, I, Aand B. (b) the torsional constant of the spring, if a current i p roduces a deflection of %!2 in the coil in reaching equilibrium position. (c) th e maximum angle through which coil is deflected, id charge Q is passed through t he coil almost instantaneously. (Ignore the damping in mechanical oscillations) [JEE 2005] 0 Q.26 An infinite current carrying wire passes through point O and in perpendicul ar to the plane containing a current carrying loop ABCD as shown in thefigure.Ch oose the correct option (s). (A) Net force on the loop is zero. (B) Net torque o n the loop is zero. (C) As seen from O, the loop rotates clockwise. (D) As seen from O, the loop rotates anticlockwise f§,Bansa!Classes Magnetics Effect of Current [11]

ANSWER KEY EXERCISE # I x 10-5 T Q.l Q.4 8 Til Q.2 JL M — 2V2 3 Q.3 (2V2-l)jl /' 7ta 3 1 ^o — 7t +1 47tr 2 1 zero 05 ^ 4RU f rk + —"1 1 k J Q.6 Q.7 Q.8 (i) 1.3 x 10 4T, (ii)zero Q.9 ^ weber.nr1 Q. 10 Q.14 1W 2na Q 11 Q.15 10k Q. 12 2mvc Q. 13 Q.16

zero 4/^ F = aa2ij /' = 0.1110A, i2 = 0.096A Q 1 ? (a) 3mv2 3mv3 4qa ' ( b ) - ^ ~ , ( c ) z e r o Q 18 In the plane of the drawing from right to left Q.19 B I = Hobrf 3 fl = 2 H bR 0 3 Q 20 3r2 ' 0.57 s Q.21 a t - m—~ , wherea = shr qB m EI Be v V2mV y HoII'C 0 2% Q.22 \ Q.23 a b to the left

I I Q.24 V2IRB n Q 2 5 _ W - Ml 2 J / n r 2 Q.l Mo 2* (a +b 2 ) 2 V3b EXERCISE # IT Q 2 ® 471^ a y along Y-axis, 4n v 2 a y Jfo fr 2 ^ V10 tan + 7t with positive axis f§,Bansa!Classes Magnetics Effect of Current [11]

Q.3 Q.5 Q.6 Q.7 Q.9 Q.ll Li I I " T ^ /n(3) along-vez direction F 'Ml 2n 0 Q.4 m § in rj2 + a* L , „2 V a-2 j , zero \l-k), ' 2 'l a ^ R p J ( a ) B = ~ y 2R - b 2 , (b) B = M R 4 4R + b 2 2 p J' 0 v ab ^ 4R + b 2 2 2 y -^h tan e 2 2 Qs 8 rz QVp i /_3^3 - 1 \ 71 , (b) x=BI v y v 0 3 -6 a 2J• (a) 0 (b) 1.41 x 10~ T, 45° in xz-plane, (c) 5 x 10 T, +x-direction] 2 Q. 10 (a) 20 min. (b) 5.94 x 10" Nm Vl5 C 4 23 4 6 Q. 12 (a) 1.4 x 10~ m/s (b) 4.5 x 10~ N (down) (c) 2.8 x 10" V/m (down) (d) 5.7 x 10~ V (top +, bottom-) (e) same as (c) Q.13 (a) I = Q.17 7tr ( x 4- Ry)\ + B B 2 7 m g (h\

(b)T I mg 7crB v Q. 14 (a) increase, (b) decrease Q.18 - 2 re r B / / , - 2 7t r B z / « 0 0 ^-tan 2hy H O 7t 1 V r a \ Q.19 0.62 N < F < 0.88 N Q-20 B eh ehB Q.3(i)m=^;N min = 4.7X10- T 3 EXERCISE # III Q.l B Q.5 Q 7 Q.2 A z = 0,x = ± ^ , d t a Q.4 A ^ ^ (iO^fe 1 Q.6 (i) D (ii) B, C (iii) A £ - (v x g ) / | v x g 0 0 v=^ E l + v o coscot + [v sin rat] k, where co = 0 f§,Bansa!Classes 4i Magnetics Effect of Current 3 BIo At 4 M 2

Q.9 A [11]

mv 7im Q 10 ( ) 2qB^ (b)velocity=-v, time= — 0 a Q.ll (i) C (ii) B (iii) C (iv) C (V Q.12 (i) q v j; (ii) F = 2 I R B F , = 2 I R B , Net force = F , + F = 4 I R B 1 0 5 2 5 6 Q. 13 (a) 6.6 x 10~ T, (b) 0, 0, 8 x io~ Nt Q.14B Q. 15 D Q.16 C Q.17 B Q.18 A Q.19 D • 6bB a 1 Q.20 (a) current in loop PQRS is clockwise from P to QRS., (b) p = BI b (3k-4i), (c) I = Q.21 A Q.22 A Q.23 ©= dT P p q 0 QR B 2 Q.24 — J — — a y a q V2 r m = a p 2i„NAB NAB 71 Q.25 (a) k = NAB, (b) C = — 7C , (c) Q x V— — 0 Z11 Q.26 A,C f§,Bansa!Classes Magnetics Effect of Current [11]

I BANSALCLASSES TARGET IIT JEE 2007 XII (ALL) MA GNETIC EFFECT OF CURRENT QUESTION BANK ON

QUESTION FOR SHORT ANSWER Q. 1 Consider a magneticfieldline. Is the magnitude of B constant or variable al ong such a line? Can you give an example of each case? Q. 2 Q. 3 A current is se nt through a vertical springfromwhose lower end a weight is hanging. What will h appen? B= fx i/ 2nd suggets that a strong magneticfieldis set up at points near a long wire carrying a current. Since there is a current i and magneticfieldB, w hy is there not a force on the wire in accord with the equation F = iL x B ? 0 0 Q.4 Twofixedwires cross each other perpendicularly so that they do not actually touc h but are close to each other, as shown infigure.Equal currents i exist in each wire in the directions indicated. In what region(s) will there be some points of zero net magnetic field? £ II III I- I IV 3 Q.5 A messy loop of limp wire is placed on a frictionless table and anchored at poin ts a and b as shown infigure.If a current i is now ' passed through the wire, wi ll it try to form a circular lo op i or will it try to bunch up further? Q..£L A v ery long conductor has a square cross section and contains a coaxial cavity also with a square cross section. Current is distributed uniformly over the material cross section ofthe conductor. Is the magnetic field in the cavity equal to zer o? Justify you answer. Two long solenoids are nested on the same axis, as infigure.They carry identical currents but in opposite directions, Ifthere is no magnetic field inside the in ner solenoid, what can you say about n, the number ofturns per unit length, for the two solenoids? Which one, if either, has the larger value? Q. 8 The magnetic fieldat the center of a circular current loop has the value B = M-i / 2R . Howev er, the electricfieldat the center of a ring of charge is zero. Why this differe nce? 0 Q. 7 Q. 9 A steady current is set up in a cubical network of resistive wires, as in figure . Use symmetry arguments to show that the magneticfieldat the v'J center of the cube is zero J A

P Q. 10 A copper pipefilledwith an electrolyte. When a voltage is applied, the cur rent in the electrolyte is constituted by the movement of positive and negative ions in opposite directions. Will such a pipe experience a force when placed in a magnetic field perpendicular to the current. Q. 11 Magnetic moments arise due to charges. Can a system have magnetic moments even though it has no charge. Q. 12 Imagine that the room in which you are seated is fillie with a uniform magnet icfieldwith B pointing vertically upward. A circular loop of wire has its plane horizontal. For what direction of current in the loop, as viewed from above, wil l the loop be in stable eqiulibrium with respect to forces & torques of magnetic origin ? (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

Q .13 Two current-carrying wires may attract each other. In absence of other for ces, the wires will move towards each other increasing the kinetic energy. From where does this energy come? Q.14 In order to have a current in a long wire, it should be connected to a battery or some such device. Can we obtain the magnetic fielddue to a straight, long wire by using Ampere's law without mentioning this other part ofthe circuit. Q.15 A uniform magnetic field fills a certian cubical region of space. Can an electron be fired into this cube from the outside in suc h a way that it will travel in a closed circular path inside the cube? Q. 16 In Ampere's law | B.dl - \i0 i the current outside the curve is not included on the right hand side. Does it mean that the magnetic field B calculated by using Amp ere's law, gives the contribution of only the currents crossing the area bounded by the curve ? Q.17 A magnetic field that varies in magnitude form point to poi nt, but has constant direction (East to West) is set up in a chamber . A charged particle enters the chamber and travels undeflected along a straight path with constant speed . What can you say about the initial velocity of the particle? Q. 18 A charged particle enters an environment ofa strong & non-uniform magneticfie ldvarying from point to point both in magnitude and direction and comes out of i t following a complicated trajectory. Would its final speed equal the initial sp eed , if it suffered no collisions with the environment. Q.19 A straight wire ca rrying on electric current is placed along the axis of a uniformly charged ring. Will there be a magnetic force on the wire ifthe ring starts rotating about the wire ? If yes, in which direction ? Q.20 An electron travelling West to East en ters a chamber having a uniform electrostatic field in North to South direction . Specify the direction in which a uniform magnetic field should be set up to pr event the electron from deflecting from its straight line path . Q.21 The magnet ic field inside a tightly wound, long solenoid is B = ju0 ni. It suggests that t hefielddoes not depend on the total length of the solenoid, and hence if we add more loops at the ends ofa solenoid the field should not increase. Explain quali tatively why the extra-added loops do not have a considerable effect on the fiel d inside the solenoid. Q . 22 A lightening conductor is connected to the earth b y a circular copper pipe. After lightning strikes, it is discovered that the pip e has turned into a circular rod. Explain the cause of this phenomenon. Q.23 We know that the work required to turn a current loop end for end in an external ma gnetic field is 2pB. Does this hold no matter what the original orientaion of th e loop was ? (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

Q.l ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question. A current of i ampere is flowing through each ofthe bent wires as shown the magn itude and direction of magneticfieldat 0 is 1 3^ Poi_fj_ _2_ (B) fV R R' (A) 4 ^ R R' \ l^o 1 M-oM (C) v.R 2R' j (D) 8 l R R' y + + 1 1 + Q. 2 Net magneticfieldat the centre ofthe circle O due to a current carrying loop as shown infigureis (9 < 180°) /k \ (A) zero il>i 8^>0 ; (B) perpendicular to paper i nwards V' J (C) perpendicular to paper outwards (D) is perpendicular to paper in wards if 9 < 90° and perpendicular to paper outwards if 90° VT Q.37 A particle having charge of 1 C, mass 1 kg and speed 1 m/s ent ers a uniform magneticfield,having magnetic induction of 1T, at an angle 9 = 30° b etween velocity vector and magnetic induction. The pitch of its helical path is (in meters) ~ (C) * (D) 71 (A) 2 (B)V3tt "" 2 Q.38 A charged particle is released from rest in a region ofuniform electric and magneticfields,which are parallel to each oth er. The locus of the particle will be (A) helix of constant pitch (B) straight l ine (C) helix ofvarying pitch (D) cycloid v v v _ / (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

Q.39 A particle of specific charge (charge/mass) a starts movingfromthe origin u nder the action ofan electric field E = E i and magnetic field B = B k. Its velo city at (x , y ,0) is (4i + 3 j). The value of x is: 25 5a 16aB 13 aE (C) 2aE (B ) (A) 2 B Q.40 A particle of specific charge (q/m) is projected from the origin of coordinates with initial velocity [ui - vj ]. Uniform electric magneticfields exist in the region along the +y direction, ofmagnitude E and B. The particle wi ll definitely return to the origin once if (A) [VB/2TCE] is an integer (B) (u + v ) [B/7tE] is an integer (C) [VB/' TIE] in an integer (D) [uB/TTE] is an intege r Q.41 An electron moving with a velocity V, = 2i m/s at a point in a magneticfi eldexperiences a force F, = - 2 j N. 0 0 0 0 0 c f r r 2 2 1/2 _ 2 A 2 A Ifthe electron is moving with a velocity V = 2 j m/s at the same point, it exper iences a force F = +2i N. The force the electron would experience ifitweremoving withavelocity V = 2k m/s at the same point is (A) zero (B) 2kN (C) - 2 k N (D) i nformation is insufficient Q. 42 Two particles of charges +Q and -Q are proj ect edfromthe same point with a velocity v in a region of uniform magneticfieldB suc h that the velocity vector makes an angle q with the magneticfield.Their masses are M and 2M, respectively. Then, they will meet again for the first time at a p oint whose distance from the point of projection is (A) 2:tMvcos9/QB (B) 8TIMVCO S0/QB (C) 7tMvcos0/QB (D) 4TIMVCOS9/QB Q.43 A particle of charge Q and mass M mo ves in a circular path of radius R in a uniform magneticfieldof magnitude B. The same particle now moves with the same speed in a circular path of same radius R in the space between the cylindrical electrodes ofthe cylindrical capacitor. Th e radius ofthe inner electrode is R/2 while that of the outer electrode is 3R/2. Then the potential difference between the capacitor electrodes must be (A) QBR( /n3)/M (B) QB R (/n3)/2M (C) QB R (/n3)/M (D)None Y Q. 44 A particle with charge +Q and mass m enters a magneticfieldof magnitude B, B existing only to theright ofthe boundary YZ. The direction ofthe motion ofthe m particle is perpendicular to the direction of B. Let T = 2 T . The time spent T by the particle in thefiel dwill be 71-29 'tc + 29^ (A)T0 (B) 2T9 (C)T ( D ) T 271 2n 3 2 2 2 2 X Q.45 In the previous question, ifthe particle has -Q charge, the time spend by t he particle in thefieldwill be tc-29 tt + 29 (C)T ( D ) T 2TZ (B)2T9 (A) TO 27C Q.46 The direction of magnetic force on the electron as shown in the diagram is along (A) y-axis (B) -y-axis (C) z-axis (D) -z-axis ' IL Y (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

Q.47 A particle having charge q enters a region ofuniform magnetic field B (dire cted inwards) and is deflected a distance x after travelling a distance y. The m agnitude of the momentum ofthe particle is: qB qBy qBy (B) x ( C ) y X •+x (A) (D) qBy' 2x Q.48 A block of mass m & charge q is released on a long smooth inclined plane magnetic field B is constant, uniform, horizontal and parallel to surface as shown. Find the time from start when block loses contact with the surface. m cosecG mcosB (B) qB (A) qB mcotQ (D)none (C) qB Q. 49 A particle moving with velocity v having specific cha rge (q/m) enters a region of B 3mv © P' 53>" magneticfieldB having width d = "^rj^ at angle 53° to the boundary ofmagnetic X field. Find the angle 9 in the60° diagram . (A) 37° (B) (C) 90° (D) none Q. 5 0 A charged particle enters a uriferm magneticfi eldperpendicular to its initial direction travelling in air. The path of the par ticle is seen to follow the path infigure.Which of statements 1-3 is/are correct ? [1] The magneticfieldstrength may have been increased while the particle was t ravelling in air [2] The particle lost energy by ionising the air entry* [3] The particle lost charge by ionising the air (A) 1, 2, 3 are correct (B) 1,2 only a re correct (C) 2, 3 only are correct (D) 1 only Q. 51 A straight rod of mass m a nd length L is suspended from the identical spring as shown in thefigure.The spr ing stretched by a distance of x due to the weight of the wire. The circuit has total resistance RQ. When the magneticfieldperpendicular to the plane ofthe pape r is switched on, springs are observed to extend further by the same distance. T he magneticfieldstrength is mgR (A) 8 7~; directed outward from the plane of the paper L mgR (B) 2ex ; directed outwardfromthe plane of the paper mgR (C) sL ; d irected into the plane of the paper (D) ; directed into the plane of the paper 0 0 £X„ Q. 52 A conducting wire bent in the form of a parabola y = 2x carries a current i = 2 A as shown in figure. This wire is placed in a uniform magnetic field B = -4 k Tesla. The magnetic force on the wire is (in newton) (A) — 16i (B) 321 (C)-32 i (D) 16i 2 y (m) (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

Q.53 A semi circular current carrying wire having radius R is placed in Y x-y pl ane with its centre at origin' O'. There is non-uniform magnetic Bxfield B = — ^ k (here B is +ve constant) is existing inthe region. The (-R,0,0) / I+RXOT X 2R m agnetic force acting on semi circular wire will be along (A) - x-axis (B) + y-ax is (C) - y-axis (D) + x-axis Q.54 A circular current loop of radius a is placed in a radialfieldB as shown. The net force acting on the loop is (A) zero (B) 27i BaIcos9 (C) 27taBsinG (D)None Q.55 A conductor of length I and mass m is placed along the east-west line on a table. Suddenly a certain amount of charge is pass ed throughit and it is found to jump to a height h. The earth's magnetic inducti on is B. The charge passed through the conductor is: 1 V2gh gh (A) Bmgh (B) g/m (C) B/m (D) mV2gh B/ Q.56 In thefigureshown a current Ij is established in the l ong straight wire AB. Another B wire CD carrying current I is placed in the plan e of the paper. The line joining the ends ofthis wire is perpendicular to the wi re AB. The force on the wire CD is: I, (A) zero (B) towards left (C) directed up wards (D) none of these D o v y 2 Q.57 A square loop ABCD, carrying a current i, is placed near and coplanar with a long straight conductor XY carrying a current I, the net force on the loop wil l be 2ppli/ 2poIi (A) 3tt (B) Poli (C) 371 (D) M A 2tc 271 M L/2 Q.58 A metal ri ng of radius r = 0.5 m with its plane normal to a uniform magneticfieldB of indu ction 0.2 T carries a current I = 100 A. The tension in newtons developed in the ring is: (A) 100 (B) 50 (C) 25 (D)10 X Q.59 In givenfigure,X and Y are two long straight parallel conductors each carrying 2A a current of 2 A. The force on ea ch conductor is F newtons. When the current 2A in each is changed to 1A and reve rsed in direction, the force on each is now (A) F/4 and unchanged in direction ( B) F/2 and reversed in direction (C) F/2 and unchanged in direction (D) F/4 and reversed in direction Q.60 A conducting ring ofmass 2 kg and radius 0.5 m is pla ced on a smooth horizontal plane. The ring carries a current i = 4A. A horizonta l magneticfieldB = 10T is switched on at time t = 0 as shown infigure.The initia l angular acceleration of the ring will be IJIIIIlMWllllll (A) 40 7i rad/s (B) 2 0 % rad/s (C)5 7trad/s (D) 15 tc rad/s Q.61 In thefigureshown a coil of single t urn is wound on a sphere of radius R and mass | m. The plane of the coil is para llel to the plane and lies in the equatorial plane of e>\ the sphere. Current in the coil is i. The value of B ifthe sphere is in equilibrium is wwwwwwwwwwulww mg cos 8 mg sin 9 mg tan 9 mg (D) 7tiR (C) TtiR (A) 7UR (B) 7UR Y B 7 2 2 2 2 B A (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

Q. 62 The magnetic moment of a circular orbit ofradius 'r' carrying a charge' q' and rotating with velocity v is given by qvr qvr (C) qv7rr (D) qv7ir (B) (A) 27 1 E |j. S Q. 63 The dimensional formula for the physical quantity 5— is B (E = ele ctricfieldand B = magnetic field) (D) L M°T- 1/2 (A)L°M°T° (B)L M°T~ (C) L~ M°T Q. 64 A thi non conducting disc of radius R is rotating clockwise (seefigure)with an angula r velocity w about its central axis, which is perpendicular to its plane. Both i ts surfaces carry +ve charges ofuniform surface density. Halfthe disc is in a re gion of a uniform, unidirectional magneticfieldB parallel to the plane ofthe dis c, as shown. Then, (A) The net torque on the disc is zero. (B) The net torque ve ctor on the disc is directed leftwards. (C) The net torque vector on the disc is directed rightwards. (D) The net torque vector on the disc is parallel to B. Q. 65 A rectangular coil PQ has 2n turns, an area 2a and carries a current 2/, (re fer figure). The plane of the coil is at 60° to a horizontal uniform magneticfield of flux density B. The torque on the coil due to magnetic force is (A) Bna/ sin6 0° (B) 8Bna/cos60° (C)4na/Bsin60° (D)none Q. 6 6 A straight current carrying conductor is placed in such a way that the current in the conductorflowsin the direction out of the plane ofthe paper. The P R© S N conductor is placed between two poles o f two magnets, as shown. Q The conductor will experience a force in the directio n towards (A) P (B)Q (C)R (D)S Q.67 Figure shows a square current carrying loop ABCD of side lOcmand current i = 1 OA. The magnetic moment M ofthe loop is C (A) (0.05) (I - V3k)A - m (B) (0.05) (j + k)A - m ,i= 10 (C) (0.05) (V3i + k)A - m (D) (i + k)A - m 2 2 0 0 1 1 1 I 1/2 2 2 2 2 ^Q. 1 In the following hexagons, made up oftwo different material P and Q, curre nt enters and leaves from points X and Y respectively. In which case the magneti cfieldat its centre is not zero. 0 ONE OR MORE THAN ONE OPTION MAY BE CORRECT Take approx. 3 minutes for answering each question. -vp/ ? 0 p/ Q x^ . v ^V (D) P Qjy Consider the magneticfieldproduced by afinitelylong current carrying wire. j A ) the lines offieldwill be concentric circles with centres on the wire. : The re can be two points in the same plane where magneticfieldsare same. (JJ&) There can be large number of points where the magneticfieldis same. > (D) The magneti cfieldat a point is inversally proportional to the distance ofthe pointfromthe w ire. x (SS Bansal Classes Question Bank on Magnetic Effect of Current

[12]

Q.3/ Consider three quantities x = E/B, y = J l / p e andz= -. Here, I is the le ngth of a wire, Ciis a CR capacitance and R is a resistance. All other symbols h ave standard meanings. (A) x, y have the same dimensions (Wfy, z have the same d imensions (P*z, x have the same dimensions (D) none ofthe three pairs have the s ame dimensions. 0 0 Two long thin, parallel conductors carrying equal currents in the same direction arefixedparallel to the x-axis, one passing through y = a and the other through y = -a. The resultant magnetic field due to the two conductors at any point is B. Which of the following are correct? JA) B = 0 for all points on the x-axis 4B ) At all points on the y-axis, excluding the origin, B has only a z-component. " fC) At all points on the z-axis, excluding the origin, B has only a y-component. ^(D) B cannot have anx-component. T Q..5 / Currentflowsthrough uniform, square frames as shown. In which case is the magneticfieldat the centre of the frame no t zero? (A) (B) V (C) (D) Qj}/' A wire carrying I is shaped as shown. Section AB is a quarter circle ofrad ius r. The magneticfieldat C is directed i (A) along the bisector of the angle A CB, away from AB ' (B) along the bisector ofthe angle ACB, towards AB perpendicu lar to the plane of the paper, directed into the paper (D) at an angle T 4 to th e plane of the paper C / Along straight wire carries a current along the x-axis. Consider the points A(0, 1, 0), B(0, 1,1), C(1, 0,1) and D(1, 1, 1). Which of t he following pairs of points will have magnetic fields of the same magnitude (A) A andB .(B) A and C (C)BandC B and D In the previous question, if the current i s i and the magneticfieldat D has magnitude B, Ho !V B 1 9 1 (C) B is parallel to the x-axis >SB) B makes an angle of 45° with the xy plane Which ofthe following statement is correct: JjA) A charged particle enters a reg ion ofuniform magneticfieldat an angle 8 5° to magnetic lines of force. The path o f the particle is a circle. (B) An electron and proton are movingwith the same k inetic energy along the same direction. When they pass through uniform magneticf ieldperpendicular to their direction ofmotion, they describe circular path. -^(C ) There is no change in the energy ofa charged particle moving in a magneticfiel dalthough magnetic force acts on it. Two electrons enter with the same speed but in opposite direction in a uniform transverse magnetic field. Then the two desc ribe circle of the same radius and these move in the same direction. (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

Qyl 0 Two identical charged particles enter a uniform magneticfieldwith same spe ed but at angles 3 0° and 60° withfield.Let a, b and c be the ratio oftheir time per iods, radii and pitches ofthe helical paths than j/k) abc = 1 (B) abc > 1 (C) ab c < 1 0 ) a = be i Consider thefollowingstatements regarding a charged particle in amagneticfield.Which ofthe statements are true: (A) Starting with zero veloci ty, it accelerates in a direction perpendicular to the magnetic field. (B) While deflecting in magneticfieldits energy gradually increases. (Q) Only the compone nt of magnetic field perpendicular to the direction of motion of the charged ^pa rticle is effective in deflecting it. \(0) Direction of deflecting force on the moving charged particle is perpendicular to its velocity. v QA 2 A particle of charge q and velocity v passes undeflected through a space wi th non-zero electricfieldE and magneticfieldB. The undeflecting conditions will hold if. (A) signs of both q and E are reversed. (B) signs of both q and B are r eversed. (C) both B and E are changed in magnitude, but keeping the product of | B| and |E| fixed, both B and E are doubled in magnitude. . Two charged particle A and B each of charge +e and masses G X \x 12amuand 13 amu respectively follow a circular trajectory in chamber X after the velocity selector as shown in the f igure. Both particles enter the velocity selector with speed 1.5 x 10 ms . A uni form magneticfieldof strength 1.0 T is maintained within the chamber X and in th e velocity selector. (A) Electricfieldacross the conducting plate of the velocit y selector is - 10 NC i . (B) Electricfieldacross the conducting plate of the ve locity selector is 10 NC" i . si£) The ratio r /r ofthe radii of the circular path s for the two particles is 12/13. (D) The ratio r / r ofthe radii ofthe circular paths for the two particles is 13/12. Q.j/4 An electron is moving along the pos itive X-axis. You want to apply a magneticfieldfor a short time so that the elec tron may reverse its direction and move parallel to the negative X-axis. This ca n be done by applying the magneticfieldalong .AX) Y-axis ^(B) Z-axis (C) Y-axis only (D) Z-axis only y X rrcioeity * >' X X X X 6 -1 X y y X X X y y x X X X

— X X y X B y y X X y 6 -1 6 1 A B A B In a region of space, a uniform magneticfieldB exists in the y-direction. A prot on isfiredfromthe origin, with its initial velocity v making a small angle a wit h the y-direction in the yz plane. In the subsequent motion ofthe proton, JA) it s x-coordinate can never be positive (B) its x- and z-coordinates cannot both be zero at the same time (C) its z-coordinate can never be negative (D) its y-coor dinate will be proportional to the square of its time of flight Q.16 Arod AB mov es with a unifonn velocity v in a uniform magneticfieldas shown in figure. (A) T he rod becomes electrically charged. (B) The end Abecomes positively charged. (C ) The end B becomes positively charged. (D) The rod becomes hot because of Joule heating. (SS Bansal Classes ,A B Question Bank on Magnetic Effect of Current [12]

The following experiment was performed by J.J.Thomson in order to measure the ra tio of the charge e to the mass m of an electron. Figure shows a modern version ofThomson's apparatus. Electrons emittedfroma hotfilamentare accelerated by a po tential difference V. As the electrons pass through the deflector plates, they e ncounter both electric and magneticfields.When the electrons leave the plates th ey enter a field-free region that extends to the fluorescent screen. The beam of electrons can be observed as a spot of light on the screen. The entire region i n which the electrons travel is evacuated with a vacuum pump. Thomson's procedur e was to first set both the electric and magneticfieldsto zero, note the positio n ofthe undefiected electron beam on the screen, then turn on only the electricf ieldand measure the resulting deflection. The deflection of an electron in an el ectric field of magnitude E is given by dj=eEL /2mv , where L is the length of t he deflecting plates, and v is the speed of the electron. The deflection d can a lso be calculated from the total deflection of the spot on the screen, d. + d an d the geometry ofthe apparatus. In the second part ofthe experiment, Thomson adj usted the magneticfieldso as to exactly cancel the force applied by the electric field,leaving the electron beam undefiected. This gives eE = evB. By combining t his relation with the expression for d , one can calculate the charge to mass ra tio ofthe electron as a function ofthe known quantities. The result is: e _ 2d,E m BL Qyl 7 Why was it important for Thomson to evacuate the air from the appara tus? (A) Electrons travel faster in a vacuum, making the deflection d, smaller. (B) Electromagnetic waves propagate in a vacuum. (C) The electron collisions wit h the air molecules cause V them to be scattered, and a focused beam will not be produced. (D) It was not important and could have been avoided. Q.slS"' One mig ht have considered a different experiment in which no magneticfieldis needed. Th e ratio e/m can then be calculated directlyfromthe expression for d,. Why might Thomson have introduced the magneticfieldB in his experiment? (A) To verify the correctness of the equation for the magnetic force. ^ (B) To avoid having to mea sure the electron speed v. (C) To cancel unwanted effects ofthe electricfieldE. (D) To make sure that the electricfielddoes not exert a force on the electron. Q I f the electron speed were doubled by increasing the potential difference V, w hich ofthe following would have to be true in order to correctly measure e/m? Kk ) The magneticfieldwould have to be cut in halfin order to cancel the force appl ied by the electric field. (B) The magneticfieldwould have to be doubled in orde r to cancel the force applied by the electric field. (C) The length of the plate s, L, would have to be doubled to keep the deflection, dj,fromchanging. (D) Noth ing needs to be changed. Q . 2ty The potential difference V which accelerates th e electrons, also creates an electricfield.Why did Thomson NOT consider the defl ection caused this electricfieldin his experiment? (A) This electricfieldis much weaker than the one between the deflecting plates and can be neglected. (B) Onl y the deflection, d, + d caused by the deflecting plates is measured in the expe riment. ..(C) There is no deflectionfromthis electric field (D) The magneticfiel dcancels the force caused by this electric field. 2 2 t 2 } 2 2 2 Question No. 17 to 21 (5 questions) (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

k Q.21 Ifthe electron is deflected downward when only the electri c field is turne d on (as shown in figure) then in what directions do the electric and magneticfi eldspoint in the second part ofthe experiment? (A) The electricfieldpoints to th e bottom, while the magneticfieldpoints into the page. / (B) The electricfieldpo ints to the bottom, while the magneticfieldpoints out ofthe page. (C) The electr icfieldpoints to the top, while the magneticfieldpoints into the page. i(D)The e lectricfieldpoints to the top, while the magneticfieldpoints out of the page. Q/ L2 A conductor ABCDE, shaped as shown, carries a current i. It is placed in the xy plane with the ends A and E on the x-axis. Auniform magnetic field ofmagnitud e B exists in the region. The force acting on it will be Y 4JA) zero, if B is in the x-direction -(B) XQi in the z-direction, if B is in the y-direction JJ2) AB /' in the negative y-direction, if B is in the z-direction (D)2aB/',.ifB is in t he x-direction . / Q.23 A square loop of side i is placed in the neighbourhood o fan infinitely long straight wire carrying a current I j. The loop carries a cur rent I as shown in figure (A) The magnetic moment of the loop is p (B) The magne tic moment ofthe loop is p = / Lk (C) The potential energy of the loop is minimu m- / % k : (D) The torque experienced by the loop is maximum Q.24 The magnetic d ipole p is placed parallel to an infinitely long straight wire as shown in figur e (A) the potential energy of the dipole is minimum (B) the torque acting on the dipole is zero (C) the force acting on the dipole is zero (D) none of these z 2 u m m ONLY ONE OPTION IS CORRECT. ANSWER Q4 Q.ll Q.18 Q.25 Q.32 Q.39 Q.46 Q.53 Q.60 Q.67 KEY Qi Q8 Q.15 Q.22 Q.29 Q.36 Q.43 Q.50 Q.57 Q.64 D A B B C B C B A B Q.l Q.5 Q.9 Q 13 Q.17 Q.21 Q.2 Q.9 Q.16 Q.23 Q.30 Q.37 Q.44 Q.51 Q.58 Q.65 A C B,C C C D C A A D C B C A D B Q.3 Q.10 Q.17 Q.24 Q.31 Q.38 Q.45 Q.52 Q.59 Q.66 Q.2 Q.6 Q.10 Q.14 Q.18 Q.22 C B C D C B D B A B A,B,C C AD A,B B AB,C A B A B A C A A A A Q.3 Q.7 Q.ll Q.15 Q.19 Q.23 Q.5 Q.12 Q.19 Q.26 Q.33 Q.40 Q.47 Q.54 Q.61 A C B A C C C C B Q.6 Q.13 Q.20 Q.27 Q.34 Q.41 Q.48 Q.55 Q.62 A A B D B A C D B Q.7 Q.14 Q.21 Q.28 Q.35 Q.42 Q.49 Q.56 Q.63

D A A B A D C D A ONE OR MORE THAN ONE OPTION MAY BE CORRECT AB,C B,D C,D A. A A Q.4 Q8 Q.12 Q.16 Q.20 Q.24 AB,C,D AD D B C C (SS Bansal Classes Question Bank on Magnetic Effect of Current [12]

BANSAL CLASSES TARGET IIT JEE 2007 XI (PQRS & J) MECHANICAL WA VES Time Limit: 2 Sitting Each of 90 minutes, duration approx. Q UESTION BANK ON

There are 76 questions in this question Objective Question Bank On Mechanical Waves bank. Q.l Q.2 An open organ pipe oflength L vibrates in second harmonic mode. The pressure vib ration is maximum (A) at the two ends (B) at a distance L/4 from either end insi de the tube (C) at the mid-point of the tube (D) none ofthese Figure shown the s hape of part of a long string in which transverse waves are produced by attachin g one end of the string to tuning fork offrequency250 Hz. What is the velocity o f the waves? (A) 1.0 ms (B) 1.5 ms(C) 2.0 ms" (D) 2.5 ms-1 1 1 1 5cm - 5cm ' \ 0.3cm O.lcrnV / 0.5cm Q.3 A sinusoidal progressive wave is generated in a string. It's equation is given b y y = (2 mm) sin (2%x — 100 7tt + 7t/3). The time when particle at x = 4 m first p asses through mean position, will be 1 1 1 sec 1 (A) 150 sec (D) .100 sec (C) (B ) 12 sec ' 300 v Q.4 A block of mass 1 kg is hanging vertically from a string of length 1 m and mass/ length = 0.001 Kg/m. A small pulse is generated at its lower end. The pulse reac hes the top end in approximately (A) 0.2 sec (B) 0.1 sec (C) 0.02 sec (D) 0.01 s ec Find the resultant of 2 wave progressing along x-axis. Yj = 3 sin (3t - 6x) y = - 4 cos(3t - 6x) (A) 5 sin (3t- 6 x - 37°) (B) 5 sin (3t - 6x + 53°) (C) 5 sin (3 t - 6x - 53°) (D) None 2 ///////// ^^ il • Q. 5 Q. 6 A pulse shown here is reflected from the rigid wall A and then fromfreeend B. Th e shape of the string after these 2 reflection will be (A) OB (C) Q)B (B)B (D) B B Q.7 An open organ pipe of length I is sounded together with another organ pipe of le

ngth I + x in their fundamental tones (x « / ) . The beat frequency heard will be (speed of sound is v) : vx vx vx (C)JJ2 (D) ~2l (B) 2x (A) 4/ 1 4 Q. 8 Ataut string at both ends vibrates in its n overtone. The distance between adjac ent Node and Antinode is found to be'd'. If the length of the string is L, then (A) L = 2d (n +1) (B)L = d ( n + l ) (C)L = 2dn (D)L = 2 d ( n - l ) [10] s&Bans al Classes Objective Question Bank On Mechanical Waves

Q. 9 Two waves are propagating along a taut string that coincides with the x-axis. Th efirstwave has the wave function y = Acos [k(x - vt)] and the second has the wav e function y = A cos [k(x + vt) + (j)]. (A) For constructive interference at x = 0, cj) = %. (B) For constructive interference atx = 0, () = 3 T J T. (C) For de structive interference at x = 0,

r behind)^;f'=^~f Here v is the velocity of sound in air. ( s (ii) (a) (b) (c) (d) The apparent frequency = —-— f When the source is moving towards the observer and th e observer is moving awayfromthe source, the apparent frequency V-Vp t s o v. vv When the source and the observer are moving towards each other. s a s 5. 6 tilBansal Classes l±^f . _ a V-V » When the source and observer are moving awayfromeach other, ~ o fs v+v When the source is moving awayfromthe observer and the observer is moving to wards the source v+ v a V + V,. o v s v Here all velocities are relation to the medium. Loudness of sound : The loudness level B of sound is expressed in decibe ls, I B = 10 log T where I is the intensity, I is a reference intensity. Beats : When two tuning forks of close but differentfrequenciesf and f are vibrating si multaneously at nearby places, a listener observes afluctuationin the intensity of sound, called beats. The number of beats heard per second is fj - f . f = s S 0 v Vs s f = v y a s 0 •*• c 0 s 0 s 2 2 Mechanical Waves

[6]

Q. 1 Two stationary sources Aand B are sounding notes offrequency680 Hz. An obse rver movesfromAto B with a constant velocity u. If the speed of sound is 340 ms , what must be the value ofu so that he hears 10 beats per second? Q. 2 Find the intensity of sound wave whosefrequencyis 250 Hz. The displacement amplitude ofp articles of the medium at this position is 1 10 ^ m. The density of the medium i s 1 kg/m , bulk modulus of elasticity of the medium is 400 N/m . Q. 3 Two string s A and B with |i = 2 kg/m and u = 8 kg/m respectively are joined in series and kept on a horizontal table with both the endsfixed.The tension in the string is 200 N. If a pulse of amplitude 1 cm travels in Atowards the junction, thenfindth e amplitude of reflected and transmitted pulse. Q.4 A parabolic pulse given by e quation y (in cm) = 0.3 - 0. l(x- 5t) (y > 0) x in meter and t in second travell ing in a uniform string. The pulse passes through a boundary beyond which its ve locity becomes 2.5 m/s. What will be the amplitude ofpulse in this medium after transmission? Q.5 A car moving towards a vertical wall sounds a horn. The driver hears that the sound ofthe horn reflected from the cliff has a pitch half-octav e higher than the actual sound. Find the ratio ofthe velocity ofthe car and the velocity of sound. Q. 6 Thefirstovertone of a pipe closed at one end resonates w ith the third harmonic of a stringfixedat its ends. The ratio ofthe speed of sou nd to the speed of transverse wave travelling on the string is 2:1. Find the rat io ofthe length ofpipe to the length of string. Q.7 A stretched uniform wire of a sonometer between two fixed knife edges, when vibrates in its second harmonic gives 1 beat per second with a vibrating tuning fork of frequency 200 Hz. Find t he percentage change in the tension of the wire to be in unison with the tuning fork. Q. 8 A train blowing its whistle moves with a constant velocity v awayfrom an observer on the ground. The ratio of the naturalfrequencyofthe whistle to tha t measured by the observer is found to be 1.2. Ifthe train is at rest and the ob server moves awayfromit at the same velocity, thenfindthe ratio. -1 x 3 2 2 EXERCISE-I Q. 9 Q. 10 Q. 11 Q. 12 Q. 13 Tuning fork A when sounded with a tuning fork B of frequency 480 Hz gives 5 beat s per second. When the prongs of A are loaded with wax, it gives 3 beats per sec ond. Find the original frequency ofA. A sound wave offrequencyf propagating thro ugh air with a velocity C, is reflectedfroma surface whi h is moving awayfromthe fixedsource with a constant speed n. Find thefrequencyofthe reflected wave, meas ured by the observer at the position of the source. The loudness level at a dist ance Rfrom a long linear source of sound is found to be 40dB. At this point, the amplitude of oscillations of air molecules is 0.01 cm. Thenfindthe loudness lev el & amplitude at a point located at a distance' 1 OR' from the source. A sonome ter wires resonates with a given tuning fork forming standing waves withfiveanti nodes between the two bridges when a mass of 9 kg is suspendedfromthe wire. When this mass is replaced by M, the wire resonates with the same tuning fork formin g three antinodes for the same position ofbridges. Find the value of M. A car is moving towards a huge wall with a speed = d 10, where c = speed of sound in sti ll air. A wind is also blowing parallel to the velocity of the car in the same d irection and with the same speed. If the car sounds a horn of frequency f, then what is the frequency of the reflected sound of the horn heared by driver ofthe car? tilBansal Classes Mechanical Waves [6]

Q.14 A 40 cm long wire having a mass 3.2 gm and area of c.s. 1 mm is stretched b etween the support 40.05 cm apart. In its fundamental mode. It vibrate with a fr equency 1000/64 Hz. Find the young's modulus ofthe wire. 2 Q.l5 A steel rod having a length of 1 m is fastened at its middle. Assuming youn g's modulus to be 2 x 10 Pa. and density to be 8 gm/cm findthe fundamentalfreque ncyofthe longitudinal vibration and frequency offirst overtone. 11 3 Q. 16 A sound source of small size produces a spherical sound wave with a freque ncy of 3 kHz in air. At a distance r, = 100 m from the source, the sound loudnes s level is L, = 60 dB. Find the sound loudness level at a distance of r,, = 200 m dB and the distance at which the sound stops being heard km. Q.17 Two identica l sounds Aand B reach a point in the same phase. The resultant sound is C. The l oudness of C is n dB higher than the loudness ofA. Find the value of n, Q. 18 So und ofwavelength A, passes through a Quincke's tube, which is adjusted to give a maximum intensity I . Find the distance through the sliding tube should be move d to give an intensity I /2. 0 0 Q. 19 In a resonance-column experiment, a long tube, open at the top, is clamped vertically. By a separate device, water level inside the tube can be moved up o r down. The section of the tubefromthe open end to the water level act as a clos ed organ pipe. A vibrating tuning fork is held above the open end, and the secon d resonances occur when the water level is 24.1 cm and 74.1 cm repsectively belo w the open end. Find the diameter of the tube. [Hint: end correction is 0.3 d] Q . 20 In a mixture of gases, the average number of degrees offreedomper molecule is 6. The mis speed of the molecules of the gas is c. Find the velocity of sound in the gas. Q. 21 A sonometer wire of length 114 cm is stretched between twofix edpoints. Two bridges, that should be mounted to divide the wire into three segm ents, such that their fundamental frequencies are in the ratio 1 : 3 : 4 must be mounted at distance and from onefixedend of the wire. Q. 22 Afixedsource of sou nd emitting a certainfrequencyappears as f when the observer is approaching the source with speed v and frequency f when the observer recedes from the source wi th the same speed. Find the frequency of the source. Q.23 A, B and C are three t uning forks. Frequency of A is 350Hz. Beats produced by A and B are 5 per second and by B and C are 4 per second. When a wax is put on A beat frequency between A and B is 2Hz and between A and C is 6Hz. Then,findthe frequency of B and C res pectively. tilBansal Classes Mechanical Waves [6]

Q, 1 Thefigureshows a snap photograph ofa vibrating string at t = 0. The particl e P is observed moving \ up with velocity 2071 cm/s. The angle made by string wi th x-axis at P is 6°. (a) Find the direction in which the wave is moving V^(inio~ m) (b) the equation ofthe wave (c) the total energy carried by the wave per cycl e ofthe string, assuming that p, the mass per unit length of the string = 50 gm/ m, Q.2 A uniform rope oflength L and mass m is held at one end and whirled in a horizontal circle with angular velocity ©. Ignore gravity. Find the time required for a transverse wave to travelfromone end ofthe rope to the other. Q.3 A symmet rical triangular pulse of maximum height 0.4 m and total length 1 m is moving in the positive x-direction on a string on which the wave speed is 24 m/s. At t = 0 the pulse is entirely located between x = 0 and x = 1 m. Draw a graph of the t ransverse velocity of particle of string versus time at x =+1 m. 3 (in 10 m 2 EXERCISE-II Q.4 A uniform string240 cm long maintains a standing wave, with the points on the st ring at which displacements of the amplitude equalling 3 V2 mm occur at 20 cm in terval along the length of the string. Find: (a) the order ofthe overtone which these oscillations represent (b) the maximum amplitude on the wire. Q.5 A steel wire 8 x 10" m in diameter isfixedto a support at one end and is wrapped round a cylindrical tuning peg 5 mm in diameter at the other end. The length ofthe wire between the peg and the support is 0.06 m. The wire is initially kept taut but without any tension. What will be the fundamentalfrequencyof vibration ofthe wir e if it is tightened by giving the peg a quarter of a turn? Density of steel = 7 800 kg/m ,Y of steel = 20 x 10 N/m . Q. 6 The displacement ofthe medium in a sou nd wave is given by the equation ;y = Acos(ax + bt) where A a&b are positive con stants. The wave is reflected by an obstacle situated at x = 0. The intensity of the reflected wave is 0.64 times that of the incident wave. (a) what are the wa velength &frequencyofthe incident wave, (b) write the equation for the reflected wave. (c) in the resultant wave formed after reflection,findthe maximum & minim um values of the particle speeds in the medium. 4 3 10 2 1 Q.7 The harmonic wave y = (2.0 x 1Q- ) cos7C (2.Ox - 50t) travels along a string tow ard a boundary it x=0 with a second string. The wave speed on the second string is 50 m/s. Write expresions for reflected and transmitted waves. Assume SI units . Q 8 In a stationary wave pattern that forms as a result of reflection ofwavesf roman obstacle the ratio ofthe amplitude at an antinode and a node is (3= 1.5. W hat percentage ofthe energy passes across the obstacle? Q.9(a) Astanding wave in second overtone is maintained in a open organ pipe of length /. The distance be tween consecutive displacement node and pressure node is . (b) Two consecutive o vertones produced by a narrow air column closed at one end and open at the other are 750Hz and 1050Hz. Then the fundamental frequency from the column is . (c) A standing wave of frequency 1100Hz in a column of methane at 20°C produces nodes t hat are 20 cm apart. What is the ratio ofthe heat capacity at constant pressure to that at constant volume. Q.10 An open organ pipefilledwith air has a fundamen tal frequency 500Hz. Thefirstharmonic of another organ pipe closed at one end an d filled with carbon dioxide has the same frequency as that of the first harmoni c of the open organ pipe. Calculate the length of each pipe. Assume that the vel ocity of sound in air and in carbondioxide to be 330 and 264 m/s respectively. i 3 tilBansal Classes Mechanical Waves

[6]

Q. 11 A string, 25cm long, having amass of 0.25 gm/cm, is under tension. Apipe c losed at one end is 40cm long. When the string is set vibrating in its first ove rtone, and the air in the pipe in its fundamental frequency, 8 beats/sec are hea rd. It is observed that decreasing the tension in the string, decreases the beat frequency. Ifthe speed of sound in air is 320 m/s,findthe tension in the string . Q.12 A metal rod of length I - 100 cmis clamped at two points. Distance of eac h clampfromnearer end is a=30cm. If density and Young's modulus ofelasticity ofr od material are p = 9000 kg m" and Y= 144 GPa respectively, calculate minimum an d next higherfrequencyofnatural longitudinal oscillations ofthe rod. Q.13 Two sp eakers are driven by the same oscillator with frequency of 200 Hz. They are loca ted 4 m apart on a vertical pole. A man walks straight towards the lower speaker in a direction perpendicular to the pole, as shown in figure. (a) Ho w many tim es will he hear a minimum in sound intensity, and (b) how far is hefromthe pole at these moments? Take the speed of sound to be 330 m/s, and ignore any sound re flections coming off the ground. Q.14 A cylinder ABC consists of two chambers 1 and 2 which contains A B C two different gases. The wall C is rigid but the wall s Aand B are thin diaphragms. A vibrating tuning fork approaches the wall A with velocity u=30 m/s and air columns in chamber 1 and 2 vibrates with v,=1100m/s • • . . • , • • • . ,v,=300Vse • minimum frequency such that there is node (displacement) at B and ,• . • * • *, * o antinode (displacement) at A. Find (i) the fundamentalfrequencyo f air column, 0.5 m 1.0 m (ii) Find thefrequencyoftuning fork. Assume velocity o f sound in the first and second chamber be 1100 m/s and 300 m/s respectively. Ve locity of sound in air 330 m/s. Q.15 A source emits sound waves of frequency 100 0 Hz. The source moves to the right with a speed of 32 m/s relative to ground. O n the right a reflecting surface moves towards left with a speed of 64 m/s relat ive to the ground. The speed of sound in air is 332 m/s. Find (a) the wavelength of sound in air by source (b) the number ofwaves arriving per second which meet the reflecting surface, (c) the speed of reflected waves. (d) the wavelength of reflected waves. Q.16 A supersonic jet plane moves parallel to the ground at sp eed v=0.75 mach (1 mach = speed of sound). The frequency of its engine sound is v = 2 kHz and the height of the jat plane is h = 1.5 km. At some instant an obse rver on the ground hears a sound offrequencyv=2 v , Find the instant prior to th e instant of hearing when the sound wave received by the observer was emitted by the jet plane. Velocity of sound wave in the condition of observer=340 m/s. Q. 1 7 A train oflength/is moving with'a constant speed v along a circular track ofra dius R, The engine ofthe train emits a whistle offrequencyf. Find the frequency heard by a guard at the rear end of the train, Q.18 A bullet travels horizontall y at 660 m/s at a height of 5 mfroma man. How far is the bulletfromthe man when he hears its whistle? Velocity of sound in air = 340 m/s. 3 • • • • • . : . 0 0 tilBansal Classes Mechanical Waves [6]

EXERCISE-III Q.l A metallic rod of length 1 m is rigidly clamped at its mid-point. Longitudin al stationary waves are set up in the rod in such a way that there are two nodes on either side of the mid-point. The amplitude of an antinode is 2* 10 m. Write the equation of motion at a point 2 cmfromthe mid-point and those of the consti tuent waves in the rod. [Young's modulus = 2 x 10 Nm" , density = 8000 Kg m~ ]. ' [JEE'94, 6] Q. 2 A whistle emitting a sound offrequency440 Hz is tied to a str ing of 1.5 m length and rotated with an angular velocity of20 rad s in the horiz ontal plane . Calculate the range of frequencies heard by an observer stationed at a large distancefromthe whistle. [JEE '96,3 ] Q. 3 Select the correct alterna tive: [JEE ' 9 6 , 2 x 2 - 4 ] (i) The extension in a string, obeying Hooke's la w is x. The speed ofwave in the stretched string is v. If the extension in the s tring is increased to 1.5 x, the speed ofwave will be _6 11 2 3 _1 (ii) An open pipe is suddenly closed at one end with the result that the frequency of third harmonic ofthe closed pipe is found to be higher by 100 Hz than the funda mentalfrequencyofthe open pipe. The fundamentalfrequencyof the open pipe is: (A) 200 Hz (B) 300 Hz (C) 240 Hz (D) 480 Hz Q.4 A whistle giving out 450 Hz approac hes a stationary observer at a speed of 33 m/s. Thefrequencyheard by the observe r in Hz is : [JEE '97,1 ] (A) 409 (B) 429 (C) 517 (D) 500 Q. 5 The first overton e of an open organ pipe beats with the first overtone of a closed organ pipe wit h a beat frequency of 2.2 Hz. The fundamentalfrequencyofthe closed organ pipe is 110 Hz. Find the lengths of the pipes. [JEE'97, 5] Q.6 A place progressive wave offrequency 25 Hz, amplitude 2.5 * 10~ m&initial phase zero propagates along th e (-ve) x-direction with a velocity of300 m/s. At any instant, the phase differe nce between the oscillations at two points 6 m apart along the line ofpropagatio n is & the corresponding amplitude difference is m. [JEE '97, 2] Q.7 A band play ing music at afrequency/ is moving towards a wall at a speed v . A motorist is f ollowing the band with a speed v . Ifv is the speed of sound, obtain an expressi on for the beat frequency hear. by the motorist. [JEE '97,5] Q. 8 A travelling i n a stretched string is described by the equation y = A sin (kx - cot). The maxi mum particle velocity is: [JEE '97,1] (A) A© (B) 30 40 50 60 70 80 90 Wavelength (pm) Q. 10 Can atomic hydrogen be caused to emit x rays? If so, describe how. Ifnot, why not? Q.ll Why is it that B ohr theory, which does not work very well even fo r helium (Z = 2), gives such a good account ofthe characteristic x-ray spectra o fthe elements, or at least of that portion that originates deep within the atom? Q.12 The ionization potential of hydrogen is 13.6 V. Yet to obtain discharge in a cathode ray tubefilledwith hydrogen, a very high voltage ( ~10 V) has to be a pplied across the tube. Explain this clearly. Also explain why the gas must be a t low pressure to obtain discharge. 4 (fe Bansal Classes Question Bank on Modern Physics 12]

Q.13 X-rays are produced when a fast electron hits a proper target. What happens to the electron? Q.14 Why does the tail of a comet always point away from the s un? Q.15 A neutron pion at rest decays into two gamma photons. 7t° —-> y + y Why can not a single photon be born? What conservation law is in contradiction with it? Q.16 What is so special about e/m rather than e end m separately? Q.17 Why is it advisable to view a TV screen from a distance of about ten feet? Q. 18 The elec trical conductivity of a gas increases when X-rays or y-rays pass through it. Ex plain this phenomenon. Q.19 In photoelectric emission exchange of energy takes p lace among... (photon and electron/' photon, electron and lattice). Q.20 The thr eshold frequencies for photoemission for three metals numbered 1,2,3 are respect ively v v v and Vj > v > v . An incident radiation of frequency v > v ... cause photoemissionfrom3 but... cause photoemissionfrom1 (fill in the gaps with may, m ay not / will certainly). p 3 2 3 0 2 NUCLEAR PHYSICS Q. 1 Why does the relative importance ofthe Coulomb force compared to the strong nuclear force increase at large mass numbers? Q.2 Q. 3 In your body, are there more neutrons than protons? More protons than electrons? Discuss Why is the bind ing energy per nucleon (seefigure)low at low mass numbers? At high mass numbers? Jnisiqp —ii.. ~5 Br 120* f iV i j r-^stability Region of greatest Fission !H . i— 0 20 40 2 ' 1 60 80 100 120 MO 161) 180 200 220 240 Q.4 Q.5 Aradioactive nucleus can emit a positron, e . This corresponds to a proton in th e nucleus being converted to a neutron The mass ofa neutron, however, is greater than that ofa proton. How thai can positron emission occur? In beta decay the e mitted electrons form a continuous spectrum, but in alpha decay the alpha partic les form a discrete spectrum. What difficulties did this cause in the explanatio n ofbeta decay, and how were these difficultiesfinallyovercome? + Mass number, A (fe Bansal Classes Question Bank on Modern Physics 3]

Q.6 How do neutrinos differ from photons? Each has zero charge and (presumably) zero rest mass and travels at the speed oflight. Q.7 In radioactive dating with U, h ow do you get around the fact that you do not know how much U was present in the rocks to begin with? (Hint: What is the ultimate decay product of U?) Q.8 If it is so much harder to get a nucleon out of a nucleus than to get an electron out of an atom, why try? Q.9 In the generalized equation for thefissionof U by ther mal neutrons, U + n -> X+Y + bn, do you expect the Q of the reaction to depend o n the identity of X and Y? Q.10 The half-life of U is 7.0 x 10 y. Discuss the as sertion that ifit had turned out to be shorter by a factor of 10 or so, there wo uld not be any atomic bombs today. Q.ll The binding energy curve offiguretells u s that any nucleus more massive than A « 5 6 can release energy by the fission pro cess. Only very massive nuclides seem to do so, however. Why cannot lead, for ex ample, release energy by the fission process? 238 238 238 235 235 235 8 Region of greatest ^"stability J-'usiqp "'"Jr 5 Fission 7 He 4 Bp B r I20g 1 I 5 7 f l c '^Au 2 3 9 Pu • H 0 . .i 20 40 60 80 1 00 120 140 160 180 200 220 240 i — i——i——i——i— Mass number, A Q.12 Elements up to mass number w 5 6 are created by thermonuclear fusion in the cores of stars. Why are heavier elements not also created by this process? Q.13 Which would generate more radioactive waste products: - afissionreactor or a fu sion reactor? Q. 14 How can Becquerel rays, i.e., the combination of a-, P- and y-rays, be separated? Q.15 When a nucleus undergoes a-decay, is the product atom electrically neutral? In (3-decay? Q.16 Experimental results in radioactivity s how small variations from the results predicted by theory. Explain this. Q.17 If a nucleus emits only a y-rays photon, does its mass number change? Does its mas s change? (fe Bansal Classes Question Bank on Modern Physics 4]

ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question. Q. 1 Let n and n be respectively the number of photons emitted by a red bulb and a blue bulb of equal power in a given time. £ (A)n = n (B)n n (D) data i nsufficient Q.2 10~ W of 5000 A light is directed on a photoelectric cell. If th e current in the cell is 0.16 pA, the £ percentage of incident photons which produ ce photoelectrons, is (A) 0.4% (B) .04% (C) 20% (D) 10% Q.3 A proton and an elec tron are accelerated by same potential difference have de-Broglie wavelength Xp and A,. (A) Xe = Xp (B) < (C) Xe > X (D) none of these. Q ,4 Two electrons are m oving with the same speed v. One electron enters a region ofuniform electric fie ld while the other enters a region ofuniform magneticfield,then after sometime i fthe de-Broglie wavelengths of the two are X{ and X2, then: (A) = X (B)Aj > X (C ) X < X (D) X > X or X < X Q.5 In a photo-emissive cell, with exciting wavelengt h X, the maximum kinetic energy of electron is K. If the 3X exciting wavelength is changed to — the kinetic energy ofthe fastest emitted electron will be: (A) 3K/ 4 (B) 4K/3 (C) less than 4K/3 (D) greater than 4K/3 Q.6 If the frequency of ligh t in a photoelectric experiment is doubled, the stopping potential will (A) be d oubled (B) halved (C) become more than doubled (D) become less than double Q.7 A n electron with initial kinetic energy of 100 eV is acceleration through a poten tial difference of 5 0 V Now the de-Broglie wavelength of electron becomes r b r b r b r b 3 e p 2 2 l 2 1 2 l 2 ^ Q.8 (A)lA (B)VL5A 2 (C) V3 A 2 (D) 12.27 A 12 Q. 9 £ If h Planck's is SI (A) is10" h constant (B)h system, the momentum of ahphoton o fwavelength 0.01 A is: (C)10 ^(D) 10 h The stopping potential for the photo elec trons emitted from a metal surface of work function 1.7 eV is 10.4 V. Identify t he energy levels corresponding to the transitions in hydrogen atom which will re sult in emission ofwavelength equal to that ofincident radiation for the above p

hotoelectric effect (A)n = 3 to 1 (B)n = 3 to 2 (C)n=2tol (D)n = 4 t o l Q.10 When a photon oflight collides with a metal surface, number of electrons, ( if any) coming out is (A) only one (B) only two (C) infinite (D) depends upon fa ctors Q. 11 Two radioactive material Aj and ^ have decay constants of 10 X0 and X0. If initially they have same number ofnuclei, the ratio of number of their un decayed nuclei will be (1/e) after a time £ ()r A L ^ ^ 1 (> i s : 1 c 1 dl Bansal Classes Question Bank on Modern Physics i [5]

£ Q.12 The frequency and the intensity of a beam oflight falling on the surface of photoelectric material are increased by a factor of two. This will: (A) increas e the maximum energy of the photoelectrons, as well as photoelectric current by a factor of two. (B) increase the maximum kinetic energy of the photo electrons and would increase the photoelectric current by a factor of two. (C) increase th e maximum kinetic energy ofthe photoelectrons by a factor of greater than two an d will have no effect on the magnitude ofphotoelectric current produced. (D) not produce any effect on the kinetic energy ofthe emitted electrons but will incre ase the photoelectric current by a factor of two. Q Jo Light comingfroma dischar ge tubefilledwith hydrogen falls on the cathode ofthe photoelectric cell. The wo rk function ofthe surface of cathode is 4eV Which one ofthe following values of the anode voltage (in Volts) with respect to the cathode will likely to make the photo current zero. (A) - 4 (B)-6 (C) - 8 (D)-10 Q. 14 A point source of ligth is used in a photoelectric effect. Ifthe source is removed fartherfromthe emitti ng metal, the stopping potential: (A) will increase (B) will decrease (C) will r emain constant (D) will either increase or decrease. QJ/5 A point source causes photoelectric effect from a small metal plate. Which ofthe following curves may represent the saturation photocurrent as a function of the distance between the source and the metal ? (A) (B) (C) (D) Q.16 Let Kj be the maximum kinetic energy of photoelectrons emit ted by a light of wavelength A, and K corresponding to X . If = 2"k , then: 2 2 2 2 2 2 (A) 2Kj = K (B) K, - 2K (C)K, 2K Q. 17 In a photoelectric experiment, the potential difference V that must be maintained between the illuminated surf ace and the collector so as just to prevent any electron from reaching the colle ctor is determined for differentfrequenciesfofthe incident illumination. The gra ph obtained is shown. The maximum kinetic energy ofthe electrons emitted atfrequ encyf, is Vi (D)eV (f -f ) (C)h(f -f ) (A) iff. (^ )( f7fT3io ) M i-f Q.18 Radia tion oftwo photon energies twice andfivetimes the work function of metal are inc ident sucessively on the metal surface. The ratio ofthe maximum velocity of phot oelectrons emitted is the two cases will be (A) 1 :2 (B)2 . 1 (C) 1 4 (D)4: 1 Q. 19 Cut off potentials for a metal in photoelectric effect for light ofwavelength X ,X and X is found to be Vj, V and V volts if Vj, V and V are inArithmetic Pro gression and A,,, X and A will be: (A) Arithmetic Progression (B) Geometric Prog ression (C) Harmonic Progression (D) None v B 1 0 1 1 0 x 2 3 2 3 2 3 2 3 (fe Bansal Classes Question Bank on Modern Physics 6]

Q. 20 Photons with energy 5 eV are incident on a cathode C, on a photoelectric c ell. The maximum energy of the emitted photoelectrons is 2 eV. When photons of e nergy 6 eV are incident on C, no photoelectrons C will reach the anode A if the stopping potential ofA relative to C is (A)3 V (B)-3V (C)-1V (D)4 V Q.21 In a ph otoelectric experiment, the collector plate is at 2.0V with respect to the emitt er plate made of copper cp - 4.5eV). The emitter is illuminated by a source of m onochromatic light ofwavelength 200nm. (A) the minimum kinetic energy ofthe phot oelectrons reaching the collector is 0. (B) the maximum kinetic energy ofthe pho toelectrons reaching the collector is 3,7eV. p (C) if the polarity of the batter y is reversed then answer to part A will be 0. (D) if the polarity of the batter y is reversed then answer to part B will be 1,7eV. Q.22 By increasing the intens ity of incident light keepingfrequency(v > v )fixedon the surface of metal (A) k inetic energy of the photoelectrons increases (B) number of emitted electrons in creases (C) kinetic energy and number of electrons increases (D) no effect 0 Q.23 In a photoelectric experiment, electrons are ejected from metals X and Y by light of intensity I and frequency f. The potential difference V required to st op the electrons is measured for various frequencies. IfY has a greater work fun ction than X; which one ofthe following graphs best illustrates the expected res ults? Vi X V V V Y/ 4 (D) (C) < f (B) o 0 •f o X / / Q. 2,4 Monochromatic light with a frequency well above the cutoff frequency is i ncident on the emitter in a photoelectric effect apparatus. The frequency of the light is then doubled while the intensity is kept constant. How does this affec t the photoelectric current? (A) The photoelectric current will increase. (B) Th e photoelectric current will decrease. (C),The photoelectric current will remain the same. (D) None of these Q. 2 5 In a hypothetical system a particle of mass m and charge -3 q is moving around a very heavy particle having cahrge q. Assumi ng Bohr's model to be true to this system, the orbital velocity of mass m when i t is nearest to heavy particle is 3q 3q 3q 3q 2 2 Q. 26 de-Broglie wavelength of an electron in the nth B ohr orbit is \ and the a ngular momentum is J , then: n " (A) J x n (B) ln oc rt7~ ** cvr\i f (C) Xn cc j 2 (D) none ofthese q s * $$ Bansal Classes Question Bank on Modern Physics m

\ Q.27 The angular momentum of an electron in the hydrogen atom is — . Here h is Pla nck's constant. The 2tc kinetic energy ofthis electron is: (A)4.53 eV (B)1.51eV (C)3.4eV (D)6.8eV - n = oo Q.28 Consider the following electronic energy level d iagram of H-atom: A -n= 4 Photons associated with shortest and longest wavelengt hs would be D C emitted from the atom by the transitions labelled: -n = 3 B (A) D and C respectively -n = 2 (B) C and A respectively (C) C and D respectively =j (D) Aand C respectively Q.29 In a hydrogen atom, the binding energy ofthe elect ron in the n state is E , then thefrquencyofrevolutionof the electron in the nth orbits is: (A)2E /nh . (B) 2E n/h (C)E /nh (D)E n/h Q.30 Ifthe electron in a hy drogen atom were in the energy level with n=3, how much energy in joule would be required to ionise the atom? (Ionisation energy of H-atomis 2.18 10"" J): (A) 6 .54 x 10" (B) 1.43 x 10" (C) 2.42 x 10~ (D) 3.14 10" Q.31 In hydrogen and hydrog en like atoms, the ratio of difference of energies E -E and E -E varies with its atomic number z and n as: (A)z /n (B) zVn (C)z/n (D)z°n° n th n n n n n x 18 19 19 19 x 20 4n 2n 2n n 2 2 4 Q.32 In a hydrogen atom, the electron is in nth excited state. It may come down to second excited state by . emitting ten different wavelengths. What is the val ue of n: (A) 6 (B) 7 (C) 8 (D) 5 Q.33 Difference between nth and (n+1 )th Bohr's radius of'H' atom is equal to it's (n-1 )th Bohr's radius, the value ofnis: (A) 1 (B) 2 (C) 3 (D) 4 Q.34 An electron in hydrogen atom after absorbing energy ph otons can jump between energy states n and n (n, > nj). Then it may return to gr ound state after emitting six different wavelengths in emission spectrum. | the energy of emitted photons is either equal to, less than or greater than the abso rbed photons. Then nj and n are: (A) n = 4, n = 3 (B)n = 5,nj=3 (C)n = 4, n, = 2 (D) n = 4 , ^ = 1 Q.35 The electron in a hydrogen atom makes transitionfromM sh ell to L. The ratio of magnitudes ofinitial to final centripetal acceleration of the electron is (A) 9:4 (B)81:16 (C)4:9 (D)16:81 Q.36 The electron in a hydroge n atom makes a transition n, —> n whose nj and n are the principal quantum numbers of the two states. Assume the Bohr model to be valid. The frequency of orbital motion of the electron in the initial state is 1/27 of that in thefinalstate. Th e possible values of n and n are (A) n =4, n = 2 (6)^=3,^=1 (0)^ = 8,^=1 (0)^ = 6,^ = 3 Q.37 The radiu s of B ohr' sfirstorbit is a . The electron in n orbit ha s a radiu s: (A) na (B)a /n (C)n a (D)a /n t 2 2 2 } 2 2 2 2 2 t 2 t 2 0 th 0 0 2 0 0 2 (fe Bansal Classes Question Bank on Modern Physics 8]

Q.38 The ionisation potential of hydrogen atom is 13.6 volt. The energy required to remove an electron from ^ the second orbit of hydrogen is: (A) 3.4 eV (B)6.8 eV (C)13.6eV (D)27.2eV Q.39 Electron in a hydrogen atom is replaced by an identi cally charged particle muon with mass 207 times that of electron. Now the radius of K shell will be (A) 2.56 x 10~ A (B) 109.7 A (C) 1.21 x 10~ A (D)22174.4A 3 3 Q.40 Monochromatic radiation of wavelength X is incident on ahydrogen sample con taining in ground state. Hydrogen atoms absorb the light and subsequently emit r adiations of ten different wavelengths. The value of X is (A) 95 nm (B)103nm (C) 73nm (D)88nm Q.41 When a hydrogen atom, initially at rest emits, a photon result ing in transition n = 5 -> n = 1, its recoil speed is about (A) 10^ m/s (B) 2 x 10" m/s (C) 4.2 m/s (D) 3.8 x l(T m/s 2 2 Q. 42 An electron collides with afixedhydrogen atom in its ground state. Hydroge n atom gets excited and the colliding electron loses all its kinetic energy. Con sequently the hydrogen atom may emit a photon corresponding to the largest wavel ength ofthe Balmer series. The min. K.E. of colliding electron will be (A) 10.2 eV (B) 1.9 eV (C)12.1eV (D)13.6eV Q.43 Thefrequencyof revolution of electron in n Bohr orbit is v . The graph between log n and log (v / v,) may be th n n Q. 44 Consider the spectral line resulting from the transition n = 2 —» n = 1 in the atoms and ions given below. The shortest wavelength is produced by: (A) hydroge n atom (B) deuterium atom (C) singly ionized helium (D) doubly ionized lithium Q .45 In an atom, two electrons move around the nucleus in circular orbits of radi i R and 4R. The ratio of the time taken by them to complete one revolution is: ( neglect electric interaction) (A) 1:4 (B) 4 : 1 (C) 1 : 8 (D) 8 : 1 Q.46 The ele ctron in hydrogen atom in a sample is in n excited state, then the number of dif ferent spectrum lines obtained in its emission spectrum will be: (A) 1 + 2 + 3 + +(n - 1) (B) 1 + 2 + 3 + + (n) (C) 1 + 2 + 3 + +(n +1) (D) 1 2 x 3 x x ( _ l) Q .47 The total energy of a hydrogen atom in its ground state is -13,6eV. If the p otential energy in the first excited state is taken as zero then the total energ y in the ground state will be : 2Mj (C) M < 2Mj (D) M, < 10(m + m ) p n 2 2 2 2 2 n p Q.16 The decay constant of a radio active substance is 0.173 (years)" . Therefor e : (A) Nearly 63% of the radioactive substance will decay in (1/0.173) year. (B ) halflife of the radio active substance is (1/0.173) year. (C) one -forth of th e radioactive substance will be left after nearly 8 years. (D) all the above sta tements are true. 1 Bansal Classes Question Bank on Modern Physics [15]

ANSWER KEY ONLY ONE OPTION IS CORRECT. Qi C D Q.2 Q.9 B A Q.3 C Q4 D Q.5 Q.12 D Q.6 C Q.7 A Q.8 Q.10 A Q.17 C Q.24 B Q.31 D Q.38 A Q.45 C Q.52 C Q.59 D Q.66 D Q.73 C Q.ll B Q.18 A Q.25 A Q.32 A Q.39 A Q.46 B Q.53 B Q.60 B Q.67 A Q.74 A Q.13 D Q.20 B Q.27 B Q.34 C Q.41 C Q.48 A Q.55 B Q.62 C Q.69 B Q.76 B Q.14 C Q.21 B Q.28 C Q.35 D Q.42 C Q.49 A Q.56 B Q.63 C Q.70 D Q.77 A Q.15 D Q.22 B Q.29 A Q.36 B Q.43 C Q.50 B Q.57 A Q.64 B Q.71 D Q.78 C Q.l Q.5 Q. 9 Q.16 C Q.23 A Q.30 C Q.37 C Q.44 D Q.51 C Q.58 B Q.65 C Q.72 C Q.19 C Q.26 A Q.33 D Q.40 A Q.47 C Q.54 A Q.61 B Q.68 B Q.75 C A,C A,C,D AB ONE OR MORE THAN ONE OPTION MAYBE CORRECT Q2 B Q.3 B Q 4 AC,D Q.6 A Q.7 B Q.8 A Q.10 A,C Q.14 A,C Q.ll AB,D Q.15 C,D Q.12 B Q.16 A,C Q.13 A,D

TARGET IIT JEE 2007 XII (ALL) MODERN PHYSICS CONTENTS KEYCONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY

KEY (a) (c) (d) CONCEPTS l. (b) 2. 3. CATHODE RAYS : Generated in a discharge tube in which a high vaccum is maintaine d . They are electrons accelerated by high p.d. (lOto 15 K.V.) 1 = eV. K.E. of C .R. particle accelerated by a p.d. V is — mv' 2m Can be deflected by Electric & ma gnetic fields . red(7.6xl0~ m) * — vioIet(3.6*l(r m) ELECTROMAGNETIC SPECTRUM Orde red arrangement ofthe big family 3xlO" m 3*10 m 3m 3 x l 0 ^ m of electro magnet ic waves (EMW) either in ascending order of frequencies infrared Ultraviolet Gam ma rays or ofwave lengths Radio waves Speed ofE.M.W. in vacuum C = 3 x 10 m/s = v X X-rays II \ Micro waves PLANK S QUANTUM THEORY : Visible light (e.g. radar) A beam ofEMW is a stream of discrete packets of energy called PHOTONS , 10 10 10 10 10 10 10 I0 10 ° each photon having afrequencyv and Frequency (Hz) energy = E = hv . 7 7 4 l2 8 4 6 s 10 12 14 16 i8 2 4. (0 (ii) (v) (iii) (iv) h = plank's constant = 6.63 x 10" Js . PHOTO ELECTRIC EFFECT : The phenomenon of the emission of electrons , when metals are exposed to light (of a certain mini mum frequency) is called photo electric effect. Results : Can be explained only on the basis of the quantum theory (concept of photon). Electrons are emitted if the incident light hasfrequencyv > v (thresholdfrequency)emission ofelectrons is independent ofintensity. The wave length corresponding to v is called threshold wave length X0 . v is different for different metals . Number of electrons emit ted per second depends on the intensity of the incident light .

34 Q 0 0 EINSTEINS PHOTO ELECTRIC EQUATION : Photon energy = K. E. of electron + work function . h v = — mv2 + , = h v 0 2 5. The minimum value of the retarding potential to prevent electron emission is : c utofr = (KE) The number of photons incident on a surface per unit time is called photon flux. WAVE NATURE OF MATTER : Beams of electrons and other forms ofmatte r exhibit wave properties including interference and diffraction eV max STOPPING POTENTIAL O R C U T O F F PO TENTIAL : with a de Broglie wave length given by X = — P (wave length of a praticle) . _ Y + a + Energy (ii) P - emission : X > P+ Y + v (antinuetrino) (iii) y - emission : emission does not affect either the charg e number or the mass number . STASTISTICAL LAW : The disintegration is a random phenomenon. Whcih atom disintegrates first is purely a matter of chance . Number of nuclei disintegrating per second is given ; (disintegration /s/gm is called specific activity) . dN dN , (i) — a N —>—=-A,N = activity . dt dt Where N = No. of nu clei present at time t ; X - decay constant (ii) N = N e~ N = number of nuclei p resent in the beginning . Z A Z 2 A _ 4 2 4 Z A Z + 1 A 0 XT X T o 0 LAWS OF RADIOACTIVE DISINTEGRATION : 2 Mj (C) M < 2 Mj (D) M < 10 (m + m ) (ii) The half-life of I is 8 days. Given a s ample of 1 at time t = 0, we can assert that: (A) no nucleus will decay before t = 4 days (B) no nucleus will decay before t = 8 days (C) all nuclei will decay before t = 16 days (D) a given nucleus may decay at any time after t = 0. 13 2 2 20 2 p n 2 2 2 2 l n p 131 131 X _ ! + a + b (B) X +in - * - XZ - 2 „ + C (C) x — > x + / ( D ) X + e_! v -> X i + 8 Q.37 The volume and mass of a nucleus are r elated as [JEE 2003 (Scr)] (A) v qc m (B) v cc 1/m (C) v cc m (D) v oc 1/m Q.38 The nucleus of element X (A= 220) undergoes a-decay. If Q-value ofthe reaction i s 5.5 MeV, then the kinetic energy of a-particle is : [JEE 2003 (Scr)] (A) 5.4Me V (B)10.8MeV (C)2.7MeV (D)None Q.39 A radioactive sample emits n P-particles in 2 sec. In next 2 sec it emits 0.75 n P-particles, what is the mean life ofthe sa mple? [JEE 2003 ] 0 0 1 p 27 20 3 17 3 14 3 n 3 22

2 215 215 th A Z A Z A z 0 A A 3 7 A z A z A z 2 2 2 1 Xl and X2 are the de-Broglie wavelengths of the particle, when 0 < x < 1 and x > 1 respectively. If the total energy of p article is 2E , find X / X . [JEE 2005] Q. 44 Highly energetic electrons are bom barded on a target of an element containing 3 0 neutrons. The ratio of radii of nucleus to that of helium nucleus is (14) . Find (a) atomic number of the nucleu s (b) the frequency of K line ofthe X-ray produced. (R= l.lx 10 m andc = 3 x 10 m/s) [JEE 2005] Q.45 Given a sample of Radium-226 having half-life of 4 days. Fi nd the probability, a nucleus disintegrates within 2 half lives. (A) 1 (B) 1/2 ( C) 3/4 (D) 1/4 [JEE 2006] x a 0 He 1 0 l 2 1/3 a 7 _1 8 Q .46 The graph between 1IX and stopping potential (V) ofthree metals having wor k functions (j^, (J> and 3) & 2.63 eV (4 -> 3) Q 13 (a)C, (b)A Q 14 (a) n = 2, z = 4; GS.E. - 217.6 eV; Min. energy =10.58 eV; (b) 6.25x 10 per sec, 0, 5 eV Q.16 B Q.17 3,4052.3 nm Q.18 5xl0 ,2000N./C, 2 3 eV Q 15 A 1 15 15 15 5 1 least 19 7 8|IA 1=2x10- W/m 5 2 Q.19 A Q.22 C Q.26 C Q.28 Fusion, 24 Q.20 z = 42 Q.23 6 litre v Q.21 -2V 1=10 W/m -5 2 V P Q.24 (i)B,(ii)A v y Q.25 A, D Q.27 (i) t1/2 = 10 sec.,' tmeans = 14.43 s (ii) 40 seconds ^ Q.29 (a) B, (b) (i) - A, (ii) - E, (iii) -F, (iv) - C, (c) = 33.298 pW 1 Q.31 (a) N = — [a (1 X e~ )+ X N 0 e~ X t ] lt Q.30 (i) C, D (ii) D (b) 3N 2 N — 2 0 Q.32 (a) C ;(b) B ;(c) B;(d) E; (e) C Q.34 C Q.38 A Q.41 A Q.45 C Q.35 A 0 Q.33 D Q.37 A 2 4 Q.36 C In

Q.44 Q.39 1.75n-N (l e" ^, 6.95 sec, Q.42 A Q.46 A,C Q.43 V2 Q.47 n = 24 Q.40 C v = 1.546 x 10 Hz 18 Q 48 (A) P, Q; (B) P, R; (C) S, P; (D) P, Q, R d U The point C is the position odf unstable equilibrium, because —— < 0 dx (!§ Bansal Classes Particle Dynamics [6]

BANSAL C L A S S E S TARGET IIT JEE 2007 XI (P, Q, R, S) IIT-JEE SCREENING 2007 QUESTION BANK ON PARTICLE DYNAMICS Time Limit: 3 Sitting Each of 60 minutes, duration approx.

QUESTION ON PARTICLE DYNAMICS There are 81 questions in this question bank. Q.l A small block of mass m is projected horizontally with speed u where friction co efficient between block and plane is given by p = cx, where x is displacement of the block on plane. Find maximum distance covered by the block 2u u u (D) 2 V S (A) (B) V2cg < C ) V S Q.2 A body is placed on a rough inclined plane of inclin ation 0. As the angle 0 is increased from 0° to 90° the contact force between the bl ock and the plane (A) remains constant (B) first remains constant than decreases (C)firstdecreases then increases (D) first increases then decreases A block is projected upwards on an inclined plane of inclination 37° along the line of greate st slope of p = 0.5 with velocity of 5 m/s. The block 1 stops at a distance of f rom starting point (A) 1.25 m (B) 2.5 m (C)10m (D) 12.5 m g&S'S hoMihg^ j What s hould be the minimum force P to be applied to the string so that * block of mass m just begins to move up the frictionless plane. Mg cosO (D) None (A) Mg tan 0/ 2 (B) Mg cot 0/2 (C) j f — ~ st p Q.3 Q.4 Q. 5 Equal force F (> mg) is applied to string in all the 3 cases. Startingfromrest, the point of application of force moves a distance of 2 m down in all cases. In which case the block has maximum kinetic energy? (A) 1 Q.6 (B)2 (i) (2) (3) (C)3 (D) equal in all 3 cases A n=o m'urnUuuuwwuuuu Both the blocks shown here are ofmass m and are moving with constant velocity in direction shown in a resistive medium which exerts equal constant force on both blocks in direction opposite to the velocity. The tension in the string connect ing both of them will be: (Neglect friction) (A)mg (B) mg/2 (C) mg/3 (D) mg/4 Q.7 In which ofthe following cases is the contact force between A and B maximum (m = m = 1 kg) J2N A I a=2m/s A B H=0 ( D ) a=10m/s P D A L (A) 30 N 7777 77777 Tm ( B) r ^ BP n (C) rrmf A B 2 2 !iiiiuiniii/ii!i = o Wnn U / N (A) (B) (C) (D) ^ T J during the interval 0 < t < T. The velocity ofthe particle at th of the interval is: 2F T 4F T 5F T 3F T (C) 3m (D) 2m (A) 6m (B) 3m 0 0

Q. 5 9 With what minimum velocity should block be proj ected from left end A tow ards end B such that it reaches the other end B of conveyer belt moving with con stant velocity v. Friction coefficient between block and belt is p. AJM v„ B (A) V pgL (B) /2pgL (D) 2^/pgL (C) V3ugL Q. 6 0 Two masses m and M are attached to th e strings as shown in the figure. Ifthe system is in equilibrium, then 2M 2m (A) tan9 = 1 + m (B) tanB = 1 ~M 2M 2m (C) cotQ = 1 + m (D) cote = 1 + Q.61 Block B of mass 100 kg rests on a rough surface offrictioncoefficient p= 1/3. Arope is tied to block B as shown in figure. The maximum acceleration with which boy A of 25 kg can climbs on rope without making block move is: 4g 3g (A) (B) (C) (D) 4 Q.62 In the system shown in the figure there is no friction anywhere. The block C goes down by a distance x = 10 cm with respect to wedge D when system is relea sed from rest. The velocity ofA with respect to B will be (g^ 10 m/s ): (A) zero (B) 1 m/s (C)2m/s (D) None of these Q 2 100kg H=l/3 B 25kg Q.6 3 A car moves along a circular track of radius R banked at an angle of 30° to the horizontal. The coefficient of static friction between the wheels and the tr ack is p. The maximum speed with which the car can move without skidding out is y /t»\ f _ .. . /. rr. H/2 (A) [2gR(l + p)/V3j1/2 (B) [gR(l-p)/(p + V3)J (D) None (C) [gR(l + pV3)/(p + V3)^' 2 Q.64 Potential energy of a particle is related to x coordinate by equation x - 2 x. Particle will be in stable equilibrium at (A)x = 0.5 (B) x - i (C) x = 2 (D) x = 4 2 BansaI Classes Question Bank on Particle dynamics [9]

Q.65 A particle of mass m is tied to one end ofa string of length /. The particl e is held horizontal with the string mg taut. It is then projected upward with a velocity u. The tension in the string is — when it is inclined at an angle 30° to t he horizontal. The value of u is (A) fig (B)V2/J (C)j| (0)2^/5 Q.66 A force F = k[y i + x j] where k is a positive constant acts on aparticle moving in x-y plan e starting from the point (3,5), the particle is taken along a straight line to (5,7). The work done by the force is: (A) zero (B) 35 K (C) 20 K (D)15K Q.67 Wat er is pumped from a depth of 10 m and delivered through a pipe of cross section 10 nr. If it is needed to deliver a volume of 10 m per second the power required will be: (A)10kW (B) 9.8 kW (C) 15 kW (D)4.9kW Q.68 A light spring of length 20 cm and force constant 2 kg/cm is placed vertically on a table. A small block of mass 1 kg. falls on it. The length hfromthe surface of the table at which the b all will have the maximum velocity is (A) 20 cm (B) 15 cm (C)10cm (D)5cm Q.69 Th e ratio of period of oscillation of the conical pendulum to that of the simple p endulum is : (Assume the strings are of the same length in the two cases and 9 i s the angle made by the string with the verticla in case of conical pendulum) (A ) cos 9 (B)VcosO (C)l (D) none of these Q. 70 A particle is moving in a circle: (A) The resultant force on the particle must be towards the centre. (B) The cros s product of the tangential acceleration and the angular velocity will be zero. (C) The direction ofthe angular acceleration and the angular velocity must be th e same. (D) The resultant force may be towards the centre. Q. 71 The work done i n joules in increasing the extension of a spring of stiffness 10 N/cm from 4 cm to 6 cm is: (A) 1 (B) 10 (C) 50 (D)100 Q.72 A man weighing 80 kg is standing at the centre of a flat boat and he is 20 mfromthe shore. He walks 8 hi on the boat towards the shore and then halts. The boat weight 200 kg. How far is he from th e shore at the end of this time ? (A) 11.2m (B) 13.8m (C) 14.3 m (D) 15.4 m Q.73 From a circle of radius a, an isosceles right angled triangle with the hypotenu se as the diameter of the circle is removed. The distance ofthe centre of gravit y of the remaining position from the centre of the circle is 2 _1 3 (A) 3 0 . - 1 ) . ( B ) ^ (C)J~ ( D ) ™ Q.74 A sphere strikes a wall and rebounds with coefficient of restitution 1/3. I f it rebounds -with a velocity of 0.1 m/sec at an angle of 60° to the normal to th e wall, the loss of kinetic energy is 1 2 (A) 50% (B) 3 3 - % (C) 40% (D)66--% 0 /n(l + x) il + x ) Two universities A and B write questions and their correspond ing solutions for a high school mathematics tournament. University A writes 10 q uestions every hour but makes a mistake in their solutions 10% of the time. The university B writes 20 questions every hour and makes a mistake 20% of the time. Each university works for 10 hours and then sends all problems to a Miss 'C' fo r checking. However only 75% of the problems which she thinks are wrong are actu ally incorrect. Further she thinks that 20% of the questions from the university A have incorrect solutions, and that 10% of the questions from the university B have incorrect solutions. If the probability that a problem definitely written and solved correctly, randomly chosen by her, / n ( x + 2 was thought of as having incorrectly solved, is (p + q)Q.l where p and q coprimes, then find the value of PHYSICS QUESTION 1 UF 2Q 10 V In the circuit shown, the switch S is in position1 since a —I i—VvWlong time. At a certain moment t = 0, it is shifted to ,20 V posit ion-2. The 1 pF capacitor is initially uncharged. 2 (iF (a) Find the current tha t flows through the 2 Q resistor as a function of time't' for t > 0. •Wr 1D (b) Wh at percentage of the work done by the 10 V cell is lost as heat from the 2Q resi stor, from t = 0 till infinity? Q.2 A beam consisting of two wavelengths 8100 A and 4500 A is used to obtain interference fringes in a Young's double slit exper iment. The distance between the slits is 2 mm and that between the plane of the slits and the screen is 100 cm. (a) Find the least distance in millimeters from the central maxima on the screen where the bright fringes due to both the wavele ngths coincide. (b) Find the least distance in millimeters from the central maxi ma on the screen where the dark fringes due to bothrthe wavelengths coincide: Q. 3 A cylinder contains a tight fitting piston of mass 2 kg and cross-sectional ar ea 10 cm . Under the piston, there is 1 mole of a diatomic gas at 300 K initiall y. The walls of the cylinder are heat insulating and the piston is also thermall y insulating. By means of an electrical heater, the gas is slowly given a heat = 1000 Joules. The upper end of cylinder is open to the atmosphere :{H= having at mospheric pressure = 10 Pascals. Neglect any frictional loss. (a) By what distan ce does the piston shift up? (b) What is the final temperature of the gas? Q.4 A solid sphere with a hollow cavity (of radius R/2) having net mass m and radius R is resting in. equilibrium on a rough horizontal floor, as shown. The sphere i s tilted slightly and released. Find the time period of subsequent oscillations assuming that the sphere's surface does not slip over the floor. wnunWrWfuuuuu Q .5 Two monochromatic and coherent point sources oflight, S, and S of wavelength 4000 A, are placed at a distance 4 mm from each other. The line joining the two sources is perpendicular to a screen. The distance of the mid-point of S,S from the screen is D = A/2 m. Find the radius (non-zero) ofthe smallest bright ring o n the screen, using valid assumptions. 2 5 0 7 Bansal Classes PHYSICS [2]

Q.6 A glass sphere of radius R has a point isotropic source of monochromatic light o f wavelength X. The thickness of the glass wall is't' ( « R). The inner surface of the sphere is painted black so that it absorbs all the radiation incident on it . Find the maximum power of the source such that the sphere does not rupture due to the radiation pressure. Rupture stress of glass = a. In the figure shown, th e sonic source of frequency 200 Hz is moving with a speed = 10 m/s. Find the bea t frequency as heard by the listener L, who is himself moving with speed = 5 m/s . The reflecting wall is moving with a speed = 15 m/s. A wind is also blowing to the right with a speed = 5 m/s. Speed of sound in still air = 340 m/s. s Q.7 wall Q.8 A sphere of mass'm' collides elastically with another stationary sphere of mass 'm/2' obliquely. Both the spheres are smooth and there are no external forces ac ting on them. Solve the equations of collision and find the maximum angle throug h which the sphere of mass'm' can be deflected w.r.t. its original direction. A thermally insulated cylinder is divided into two parts by a heat insulating tigh t piston, which can move freely in the cylinder without friction. The left part of the cylinder contains one mole of an ideal diatomic gas and the right part is evacuated. The piston is connected to the right wall of the cylinder through a spring whose natural length is equal to the total length of the cylinder. The el ectrical heater is i •: vacuum switched on for some time so that the gas temperatu re increases and 1 mole • -mbWMmuof the piston shifts slowly to the right. What pe rcentage of the heat diatomic supplied by the heater goes in compressing the spr ing? Neglect the gas § heat capacity of the piston or the cylinder. A ball is thro wn from a point O with some speed v at an angle of 37° with the horizontal, such t hat the ball bounces from the vertical 31°( wall and returns to O. For the bounce, the coefficient of restitution 0 4m 7777777777777777777777777 7 7 7 7 is 5/8. W hat must be the value of v ? g = 10 m/s . A spherical body of mass M and radius R has a spherical cavity of radius R/2 inside it, as shown. The center of the ca vity O is displaced from the geometric center of the sphere C by a distance R/2. A tiny body of mass m ( « M) is placed at a distance 2R from the geometric center of the first body. Find the force of gravitational attraction on the tiny body. If the tiny body is released from rest, with what velocity will it hit the surf ace of the spherical body? —T'^r—nrew^— The circuit shown is fed by an a.c. source hav ing emf = (15 V) sin coil-l coil-2 200t, where time t is in seconds. Coil-1 has a resistance = 3 fl and inductance 20 mH, while coil-2 has a resistance = 6 0 an d inductance 40 mH. Find the voltages across the two coils, V, and V , as functi ons of time, t. A certain radionuclide is getting formed in some reactor at a co nstant rate = q (number per second). It undergoes alpha decay with half life T. At the moment t = 0, there are (4qT//n 2) number of radionuclide in the reactor. Find the number of radionuclide 'N' in the reactor at any later time t > 0 and plot a graph of N versus t. Find the number of alpha particles emitted till t = 2T. Q.9 Q. 10 Q.ll 0 0

2 (a) (b) Q.12 2 Q.13 (a) (b) ^Bansal Classes PHYSICS [426]

Q. 14 In a modified Young's double slit experiment, there are three identical pa rallel slits S,, S and S . A coherent monochromatic beam of wavelength 700 nm, h aving plane wavefronts, falls on the slits, as shown. The intensity of the centr al point O on the screen is found to be 7 x i(H W/m . The distance SjS = S S = 0 .7 mm. (a) Find the intensity on the screen at O if S, and S are covered. {b) Fi nd the intensity on the screen at 0 if only S is covered. (c) All three slits ar e now uncovered and a transparent plate of thickness 1.4 pm and refractive index 1.25 is placed in front of S . Find the intensity at point O. Q. 15 A jeep is m oving at a certain moment with velocity = 10 m/s. The acceleration of the jeep i s 'a'. A man sitting in the jeep throws a ball with initial velocity = 20 m/s, a t an angle of 53° with the horizontal, both w.r.t. himself. The motion of the jeep is in the same direction and vertical plane as the motion of the ball. Given: s in 53° = 4/5, cos 53° = 3/5. Neglect air resistance. (a) Find the actual initial spe ed of the ball relative to an earth observer. (b) What should be the acceleratio n 'a' of the jeep so that the man is able to catch the ball? (c) What is the far thest distance ofthe ball from the man, as perceived by him, in part (B)? Q.16 T wo blocks, 1 & 2, of masses m and 4m, interconnected by a massless spring of spr ing constant k, and are resting on africtionlesshorizontalfloor.Forces F and 2F start acting on the blocks, at t = 0, as shown. (a) Write the earth frame work-e nergy theorem for the system, in terms F 2F. \uMuuuu\uuu\fm of speeds v, and v , and displacements x, and x of the two blocks. \utMu\\uu\u\u\v,ufl\mrv» (b) Find t he maximum elongation ofthe spring during the motion of the two blocks, if F = 5 mg. (c) Find the maximum speed of block-1 in the center of mass frame, if F = 5m g, Q.17 A uniform and thin rod AB of mass 5m and length L is kept stationary on a frictionless horizontal surface. At a certain moment, a tiny ball of mass m, m oving with a horizontal velocity = v collides inelastically with the rod, at a p oint whose distance from end A of the rod is z. The direction of v is perpendicu lar to the rod, as shown. The coefficient of restitution for collision is 3/4. J ust after the collision, let v, = velocity of the center of rod (rightwards), v = velocity of the ball, assumed leftwards and co = angular velocity of the rod. (a) Write the condition for coefficient of restitution = 3/4 in terms of relevan t parameters (b) It is found that the velocity of B just after the collision is zero. Find z. (c) Assuming the condition of part (B), calculate the percentage o f B energy lost during the collision. 2 3 2 2 2 3 3 3 2 2 2 Q 0 ( 5 2 Q.18 A gaseous mixture initially at 300 K and 2 x 10 N/m pressure contains 6 g o f hydrogen and 8 gm of Helium. The m ixture is expanded to four times its ori gi nal volume, through an isobaric heating process. Then, it is isochorically coole d until its temperature again becomes 300 K. After that, the gas mixture is isot hermally compressed to its original volume. (a) Find the ratio of molar specific heats = y ofthe mixture. (b) Plot the process in P-V and P-T indicator diagrams , showing all values of P & T. (c) Find the efficiency of the entire cycle (take in 2 = 0.7) Q.19 Two radio stations broadcast their programmes at the same ampl itude A, but at slightly different frequencies ro_ and co , where o) - co = 1000 Hz. A detector receives the signals from the two stations simultaneously. It ca n only detect signals of intensity > 2A , (a) Find the time interval between the successive maxima ofthe intensity ofthe signal received by the detector. (b) Fi nd the time for which the detector remains idle in each cycle of the intensity o f the signal, Q.20 A long wire PQR is made by joining two wires PQ and QR of equ al radii. Their lengths and masses are respectively: 4.8 m and 0.06 kg; 2.56 m & 0.2 kg. The tension is 80 N. A sinusoidal wave pulse of amplitude 3.5 cm is sen t along the wire PQ from the end P. No power is dissipated during the propagatio n ofthe wave pulse. Calculate the time taken by the pulse to reach the end R and the amplitude of reflected and transmitted wave pulses at Q. 2 3 2 2 ^Bansal Classes PHYSICS

[4]

Q.21 otal that 1 Q.

In the circuit shown, the potentiometer wire AB has a length = 100 cm and t resistance 10 Q. What should be the distance of the jockey from point A so the reading of the ammeter is 0,5 A? The coil resistance of the ammeter is The cell at the top has an emf = 15 Volts and internal resistance 1 O.

1—\AVv Q.22 A soap bubble of radius r is blown at the end of a capillary of length / and of internal radius R. Surface tension of soap solution is T and coefficie nt of viscosity of air is r\. The volume of air flowing per second through the c apillary is given by Q.23 soap bubble. Find the lifetime of the soap bubble. Two small balls A and B are interconnected by an inextensible string of length L. M ass of ball A= m, mass of ball B = 2m. The balls are resting on a frictionless h orizontal surface, with the distance between them = 3L/5. In this position, ball A is suddenly given a horizontal velocity v , perpendicular to the line joining the two balls. Find the speed of ball B just after the string becomes taut. Fin d the impulse of the tension in string when the string becomes taut Find the ste ady tension in string much after the string has become taut. A wooden log of mas s m with a cross-section shaped like an equilateral right-angled triangle can sl ide on a horizontal surface without friction. Two point-like bodies of masses m and 2m, tied to each other using a thread, are placed onto the log as shown in t he figure. The length of the base of the log is L-54 cm. Friction and the masses of the thread and the pulley are negligible The bodies are released at a certai n moment. What distance does the wooden log cover until the body of mass 2m reac hes its bottom? Determine the speed ofthe bodies and that ofthe wooden log when V the body of mass 2m reaches the bottom of the log. In a tennis racket, the c.m . is 12 inches from the end ofthe handle. The radius of gyration about an axis t hrough the c.m. as shown in the figure is 8 inches. If the tennis ball is hit at a distance of 20 inches from the end ofthe handle, where should the player hold his racket so as not to feel any translational force when hitting the ball? We have two liquids of different densities. A force of 1.36 N can hold the same pie ce of metal in one of them, and of 0.82 N in the other. In what volume proportio n should they be mixed so that the holding force is exactly 1 N? A cart on an in clined plane of angle 9 - 30° is balanced as shown by a weight of mass 10 kg. The cord Ais wound on a drum of diameter d, which is on the same shaft as a drum of diameter .j, d. = 3d, on which is wound cord B. What is the mass a, , M of the c art? 0 2 rn 0 5Q 8rj/ , where P is the excess pressure on (a) (b) (c) Q.24 (a) '(b) Q.25 Q.26 Q.27 J U l Q.28 Through the Looking Glass: A narrow beam oflight has entered a large thin l ass plate. Each refraction is accompanied by reflection of k = 30% of the beam's energy. What fraction ofthe light energy is transmitted through the plate 9 ^ Bansal Classes

PHYSICS [428]

Q.29 Lake Placid: A radio receiver is set up on a mast in the middle of a calm l ake to track the radio signal from a satellite orbiting the Earth. As the satell ite rises above the horizon, the intensity of the signal varies periodically, th e intensity is at a maximum when the satellite is 8j= 3° above the horizon and the n again at 9 = 6° above the horizon. What is the wavelength X of the satellite sig nal? The receiver is h = 4.0 m above the lake surface. Q.30 In the figure, water of density 1000 kg/m flows through the pipe. The cross-section area at stations 1, 2 and 3 are 1 cm , 2 cm and A cm , respectively. The thin vertical tubes tha t are connected to the pipe at these stations have water 20 cm levels as indicat ed. Find the mass flow rate of water through the pipe and v . [Take g = 10 m/s ] 2 3 2 2 2 3 2 Q.31 A metal ring having three metallic spokes of lengths r=0.2 m is in a vertic al plane and can spin around a fixed horizontal axis in a homogeneous magnetic f ield of a magnetic induction of B=0.5 T. The lines of magnetic field are perpend icular to the plane of the metal ring. Between the axis of the metal ring and it s perimeter we connect a consumer of a resistance of 0.15 with the help of two s liding contacts. We fix a thread of negligible mass to the rim of the ring and w ind it several times around the ring and to its end we fix a body of a mass of 2 0 g. At a given moment we release the body of mass m. The friction is negligible everywhere, the resistance of the ring, the spokes www and the connected wiring is also negligible. (a) What is the torque exerted on the ring with the spokes by the magnetic / B © forces when the body of m is moving with a constant velocity ? \ s® (b) What current isflowingthrough the consumer when the velocity of the bod y of mass m is 3 m/s? J •> L ® ® ® (c) What is the highest velocity of the body of mass m? ®/s \\\\\\\ / ® Q.32 Figure shows a hypothetical speed distribution for particles of a certain g as: P (v) = Cv for 0 < v < v and P(v) = 0 for v > v„. (a) Show that C = 3/vJ, dN/N =P(v) dv Find (b) the average speed of the particles, and (c) their rms speed. Q.33 A neutron moving with a kinetic energy = 65 eV collides head-on and inelast ically with a singly ionized helium atom at rest (in its ground state). Take the ionization energy of hydrogen atom =13.6 eV, Also, mass of Helium atom is four times that of a neutron. If the helium ion gets de-excited subsequently by emitt ing radiation, calculate the possible energies of the emitted photon(s) in eV. Q .34 A board of mass m is placed on a frictionless inclined plane that makes an a ngle 0 = 37° with the horizontal. A block of same mass is placed on the board and is given a quick push up the board with initial velocity v = 8 m/s. Find the dis tane d covered by the block by the time its velocity drops to v/2. The board doe s not move relative to the plane. Q.35 A 20 mH inductor is connected in series w ith a charged capacitor of capacitance 2 pF, having initial charge = 10 mC. Afte r how much minimum time will the energy in the capacitor become half of its init ial value? Leave answer in terms of n. Q.36 A uniform and slender rod of mass 2m and length L is lying on a frictionless V3horizontal surface. Two insects, of m ass m each, moving horizontally with velocities v and 2v hit the rod simultaneou sly and symmetrically and stick G3to it. 2 Q ^ Bansal Classes PHYSICS [6]

Q.37 (a) (b) Q.38 Q.39 Q.40 (a) (b) (c) Q.41 Q.42 The initial velocities of the insects are perpendicular to the rod, as shown. Th e distance of each insects's hit-point from the center of the rod is L/6. Just a fter hitting the rod, each insect starts walking along the rod, away from its ce nter, with constant speed = v relative to the rod. As the rod rotates and moves, the insects finally reach the ends. Find the total angle rotated by the rod til l this moment in radians. A thin uniform circular disc of radius R and mass m is hinged about its center point O, so that it is free to rotate about a fixed hor izontal axis through O. The plane of the disc is vertical. A small body A, of ma ss m/2, is fixed at the rim of the disc, as shown. Initially, the line OA makes an angle of 60° with the vertical. The disc is now released from rest, Find the ac celeration of point A just after release. Find the magnitude of horizontal and v ertical reaction forces: F and F on the hinge, just after the disc is released. In the figure shown, the spring constant is I6n N/m and its right end is fixed t o a vertical wall. The floor is smooth. A block of mass 1 kg is initially at a d istance of 1 m from the other 1 kg block. The left 4 m/s kg block, touching a ve rtical wall, is imparted a velocity = 4 m/s towards lm B the other block. All co llisions are elastic. Find the time period of this Hvmummm"rnrr 1kg oscillatory system. A ring of radius r = 1 m is placed on the top of an inclined plane and r eleased from rest. The inclined plane makes an angle of 30° with the horizontal. T he coefficient of friction between the ring and the incline is 0.2. Find the dis tance travelled by the centre of the ring by the time it completes one revolutio n, as it rolls down the incline. In the figure shown, a constant horizontal forc e F = mg/2 starts acting on the block of mass m, from the position shown. The sp ring is undeformed in the position shown and has a narual length L, while the bl ocks are initially stationary. The spring constant is unknown. The surface is fr ictionless. The mass of the hanging block is m/4, while [\wmmms\m the pulley is massless and frictionless. Find the initial acceleration of the block of mass. W rite the work-energy equation for the system consisting of the two blocks, and t he spring, for any general value of 9 = angle which the spring makes with the ve rtical. The maximum displacement of the bigger block is found to be LVJ . Based on this information, find the spring constant. A lift is moving up with a consta nt retardation = 2 m/s . When its upward velocity is 5 m/s, a boy in the lift to sses a coin, imparting it an upward initial velocity = 3 m/s, with respect to hi mself His fingers at the moment of toss are midway between the floor and ceiling , whose total height is 2.0 m. After how much time will the coin hit the floor o r roof of the lift? Also find the distance travelled by the coin and its displac ement in the earth frame till then. [Take g = 10 m/s ] At a distance of 20 m fro m a point isotropic source of sound, the loudness level is 30 dB. Neglecting dam ping of sound, find the loudness level at a distance of 10 m from the source and the distance where the sound is not audible by humans. hor v 2 2 2 Q.43 In the figure shown, find the relative speed of approach/ separation of the two final images formed after the light rays pass through the lens on the far r ight, at the moment when u = 30 cm. The speed of object = 4 cm/s. The two lens h alves are placed symmetrically w.r.t the moving object. f=40cm f=60cm otfcct, infer -1

V 40 cm ^ Bansal Classes PHYSICS [430]

Q.44 Figure shows the Regnault's appratus to determine the specific heat capacity of a unknown liquid. A solid sphere of known specific heat capacity s, having mass m, and initial temperature 0,, is mixed with the unknown liquid filled in a calo rimeter. Let masses of liquid and calorimeter are m and m respectively, specific heat capacities are s and s and initially they were at room temperature 0 . Whe n the hot sphere is dropped in it, the sphere looses heat and the liquid calorim eter system takes heat. This process continues till the temperature of all the e lements becomes same (say 0). Heat lost by hot sphere = mjS, (Qj—0) Heat taken by liquid & calorimeter = m s (0-0 ) + m s (0-0 ) If there were no external heat lo ss Heat given by sphere = Heat taken by liquid - calorimeter system m,Sj (0,-0) = m s (0-0 ) + m s (0-0,) mjSj(0j-0) m s Get s = m ( 0 - 0 ) m 2 3 2 3 2 2 2 2 3 3 2 2 9 2 3 3 3 3 2 HEAT CAPACITY DETERMINATION OF A LIQUID USING CALORIMETER : 2 2 2 steam Chamber "0" steam Disk D -Water (a) (b) By measuring the final (steady state) temperature of the mixture, we can estimat e s : specific heat capacity of the unknown liquid. To give initial temperature (0,) to the sphere, we keep it in steam chamber ("O"), hanged by thread. Within some time (say 15 min) it achieves a constant temperature 0,. Now the calorimete r, filled with water (part C) is taken below the steam chamber, the wooden remov able disc D is removed, and the thread is cut. The sphere drops in the water cal orimeter system and the mixing starts. If specific heat capacity of liquid (s ) were known and that of the solid ball (s^ is unknow then (m s +m s )(0-0 ) we ca n find s, = — — — — "1,(0,-9) In the exp. of finding specific heat capacity of an unknow sphere (s ) mass of the sphere and calorimeter are 1000 gm and 200 gm respectiv ely and specific heat capacity of calorimeter is equal to 1/2 cal/gm/°C. The mass of liquid (water) used is 900 gm. Initially both the water and the calorimeter w ere at room temperature 20.0°C while the sphere was at temperature 80.0°C initially. If the steady state temperature was found to be 40.0°C. estimate specific heat ca pacity of the unknow sphere (s ). (Use s = 1 cal/g/°C) Also find the maximum permi ssible error in specific heat capacity of tinkown solid. What should be final te mperature so that the error in s should ne minimum? 2 2 7 2 3 3 2 1 2 2 water Calorimeter ^Bansal Classes PHYSICS [431]

Q.45 END CORRECTIONS IN METER BRIDGE In meter bridge circuit, some extra length of wire called end corrections should be included at ends for accurate result. S uppose null point is obtained at /;, then Qi _ i Q 100-ZJ+p When known resistanc es are interchanged then balancing length is at l . R L T i ioo-/ +p The end cor rections calculated from above readings are used to modify observation If 100 fi & 200 D values of known resistance is used to give null deflection at /,= 33.0 cm & on interchanging the known resistances the null deflection is found at 67.0 cm. Find the value of end correction. INDEX ERROR IN OPTICAL BENCH In u-v metho d the distance between object or image from the pole of mirror or les is require d. Practically the position of holder when read from scale do not exactly give o bject or image distance. This mismatch is constant for every observation. To det ermine index error a needle (usually usedfor knitting) of known length is placed horizontally between the pole & object needle. The length of knitting needle gi ves actual object distance while the separation between holder index is read fro m the scale. Which becomes observed distance so index error (or excess reading) is e = Observed distance - Actual Distance For index correction the e is subtrac ted from observed reading to get correct reading. (a) When a knitting needle of length 20.0 cm is adjusted between pole and object needle, the separation betwee n the indices of object needle and mirror was observed to be 20.2 cm. Find the i ndex correction for u. (b) When the same knitting needle is adjusting between th e pole and the image needle, the separation between the indices of image needle and mirror was found to be 19.9 cm. Find the index error for v. (c) In some obse rvation, the observed object distance (Separation between indices of object need le and mirror) is 30.2 cm, and the observed image distance is 19.9 cm. Using ind ex correction from previous two equations, estimate the focal length of the conc ave-mirror. Q.47 A conducting sphere of radius a is surrounded by another spheri cal thin conducting shell of radius b The space between them isfilledwith dielec tric material of conductivity a and dielectric constant k. The charge Q. and Q a re given to the inner and outer shell at time t = 0. Find charge on outer shell at time t. Q.48 The amplitude of the electric field in an electromagnetic wave o f frequency © = 2.0 x 10 s~ changes with times as E(t) = k (1 + cos Ht), where k i s a constant and fi= 1.8 x 10 s~'. Would such a wave cause ionization of hydroge n atoms? If yes, what is the energy of the ejected electrons E ? Assume that ato ms absorb light as photons. The ionization energy of hydrogen gas is E = 13.6 eV . the Planck constant h - 1.05 J * s. Q.49 An air-filled parallel-plate capacito r with the plate area A is connected to a battery with an emf E and small intern al resistance. One of the plates vibrates so that the distance between the plate s varies as d = d + a cos ©t (a « d ). The capacitor break down when the instantaneo us current in the circuit reaches the value of I. Find the maximum possible ampl itude of vibrations a. Q.50 Two simple pendulums of length L each are attached t o the ceiling. The small balls attached to the strings have equal masses m. The weights are connected by a very light relaxed rubber band (not a spring) with th e force constant k. At a certain moment, each ball is given a light quick push a s shown, resulting in equal initial speeds. Find the period T of the ensuing mot ion. L l + a 2 2 2 2 + A = 2 2 16 x 15 e 0 Q ^Bansal Classes PHYSICS [432]

Q.51 A proton (m, e) and an alpha particle (4m, 2e) approach each other from a l arge distance. Initially, their velocities are the same (v). Find the minimum se paration r between the particles. Q.52 A wooden cube with a side of d = 0.10 m i s placed on a horizontal support. A bullet of mass m = 0.010 kg is shot vertical ly up through the support and through the cube. As the bullet passes through the cube, its speed decreases uniformly from v = 120 m/s to u = 115 m/s. Estimate t he minimum mass M of the cube that would allow it not to lose contact with the s upport. Q.53 In a strictly scientific experiment, a student athlete throws rocks out the window in all directions. All rocks have the same initial speed v. It t urns out that all rocks' landing velocities make angles 0 or greater with the ho rizontal. Find the height h of the window above the ground. Q.54 An insulated co ntainer is filled with a mixture ofwater and ice at tc = 0°C. Another container is filled with water that is continuously boiling at ^ = 100°C. In a series of exper iments, the containers are connected by various thick rods that pass through the walls of the containers (refer diagram). The rod is insulated in such a way tha t there is no heat loss to surroundings. In experiment 1, a insulation copper ro d is used and the ice melts in T, = 20 min. In experiment 2, a steel rod of the same cross section is used and the ice melts in T = 60 min. How long would it ta ke to melt the ice if the two rods are used "in series"? Q.55 How can you measur e the resistance of an unknown resistor r with an ammeter and a voltmeter if you don't know the internal resistances of these devices? A voltage source is avail able. Q.56 A dubmbell consists ofa light rod of length r and two small masses m attached to it. The dumbbell stands vertically in the corner formed by two frict ionless planes. L After the bottom end is slightly moved to the right, the dumbb ell begins to slide. Find the speed u of the bottom end at the moment the top en d loses contact with the vertical plane. Bbi_ Q.57 Find the maximum power of a h eating element that can be constructed from a piece of wire that has a resistanc e of 536 Q. The element is to be powered by a constant voltage of V = 110 V. The current through the wire cannot exceed 2 A. Q.58 A heavy block is attached to t he ceiling by a spring that has a force constant k. A conducting rod is attached to the block. The combined mass of the block and the rod is m. The rod can slid e without friction along two vertical parallel rails, which are a distance L apa rt. A capacitor of known capacitance C is attached to the rails by the wires. Th e entire system is placed in a uniform magnetic field B directed as shown. Find the period T of the vertical oscillations of the block. Neglect the electrical r esistance of the rod and all wires. 2 Q.59 An electric circuit contains a battery with emf E and internal resistance r , two coils with inductances L, and L, and a resistor R, connected as shown. On the diagram, all shown parameters are given. Initially, both switches are open. Switch S, is then closed. After a while, switch S is closed. What is the total c harge Q that passes through the resistor after S- is closed? Q.60 Figure shows t hree identical balls Mj, M and M each of radius 10 M, cm. The ball M is given a certain velocity in the direction of AB such that after collision with M , it (M ) has a head-on collision with the ball Mj. Find the distance BC (in cm) where B lies on the line joining the centres of M, and M . The balls are assumed to be perfectly elastic. Given CjC = 1 m. 2 2 3 3 2 3 2 2 2 ^ Bansal Classes PHYSICS [10]

HINTS & SOLUTIONS MATHEMATICS 1. L = Lim x->0 /n(l + x) / + n ( x + A 1 /[ + x ) t = L = Lim ^(x W l x ) - / n ( x l ) _ x-,0 . M ^ > . VT^) 2 x ^ In x + Vl + x 1+^ 2 2 2 X / n ( x + x-Zn^x + V T ^ - l ) + l)(x + Vl + x -1) 2 (x+vr+x -i) 2 note that Lim x->0 /nf(x + Vl + x -1) + 1 X + Vl + x -1 2 ->1 \ hence L= Lim x->0 /n x + Vl + x -1 vv 1 + X 2 /Y \ +1 X + Vl + X - l 2 2 x + Vl + x -1 (x + Vl + x - 1 ) 1+x 2 /—r \ A /n -1 + 1 1+x / y =i Note that Lim vv > x->0 x + Vl + x" - l ~T+x~ x + Vl + x

(V = Lim x + Vl + x - 1 - x = Lim VT+ x -1 x(l + x)(x + V1 + x -1) - x(V 1 + x + x -1) 2 z 2 x >0 2 (as Lim(l + x) = 1) x-M) L - Lim ( V ^ + l - l X V x ^ l + l ) x->0 ( V x + l + l ) - x ( V x ^ + l + x l ) 2 = x-(Vx +l+x-l V^T--(x-l)] x-2 Lim x 1 = Lim x-*o 2 (x + l ) - ( x - l ) 2x 2 2 — Lim 1 2 [(x +1)-1][VX2+T-(X-1)] 2 H 2 2 2 L + 153 (1/2)+ 153 hence — ;L = — 7 7 ^ — = 1 + 2 • 153 = 1 + 306 = 307 Ans. ] (1/4 L = J ••• — = tan9 d cos 0 2 sin (0/2) = 2 T ( cos 0 = 1 - 2 sin (0/2) ); sin (0/2) = 2 0 6. •h 100 • i •• . i i V y D' y = - j ^ - m = 2 cm Ans. ] D\fl r = R sin 0 z ° i 4tcRyJx(27cRsine)Rde = dE : h 'PAsinGde^ C O S 0 dF cos 0 = , 2hc o 1 dF cos 0 0 f

x It/2 J eff = — { s i n e COS 0 d 0 dF a(2itRt) p eff = 2c P 1 - 0 2c 2 4c 0 o2 Tt Rt = 4c P = 8 71 a Rt c 0 ^ Bansal Classes PHYSICS [437]

350x200 340 + 5 + 15 200x360 200 340 + 5 + 5 A - 200 340 + 5-10 fx 340+ 5-To 335 335 360 3 4 0 - 5 - 5 360 330 f2 = 200 x 200 x 335 340-5-15 335 x 320 360 330 3 50 200 360x330 200 360x33 y - / i = 200 x — x — - 2 0 0 x 335 335 320 -350 335 32 -3 50 335 320 / - / , « 12.68 Hz ] m m u cos 0 = — v + mv. 2 2 2 2~ i O-ucos0 v - V = u cos 0 V V 2 t u sin 8 u sin 0 Vj = u COS 0 = V, 4ucos 0 3 u cos 0 f ucosO § = tan~ 3usin0 V n 2 it $ = tan~ cot0 - tan" cot0 0 tan" cotG n -(P-0) cot 0 cot (p+0) cot 0 cot P cot 0 - 1 3 "" cot p +cot0 cot P cot 0 + cot e - 3 cot P cot e -3 i cot p 2 tan 6 + 3 tan 0 2 => cot p is mm. at tan 0 = j j hence p is max, at 0 = 30° ] 9, V. = AXp V, AX,i. k X, kX, p _ _—L p A 2 i mole of diatomic 1 A

vacuum IP 11Bansal Classes PHYSICS [15]

AW = JkXdx = Y ( X ^ - X F ) x, AQ - AU + AW = nC dT + AW= - R ( T -T,) + AW v 2 AQ = | ( R T - R T , ) + A W = | ( P V - P V ) + AW = | ( k X - k X f ) + | (X*X?) 2 2 2 1 i 2 AW 10, u ti v c o s 3 7 ° ' 0 T = t + t = 0 (k/2)(X - X ) 100 2 - 7Xf)x 1 0 0 = — % ^ (5 / 2)(kX - kXf) + (k / 2)(X 2 2 2 2 t2 ] u ev cos37° 0 0 0 u 5/8xv x4/5 Q A+iL-Ii i 2 v v v 2v sin 37° 13 2 v x 3 / 5 _ 13 v'0 ' 10 vo i 13 v = 5 J j m/s Q Q A 0 TTTTTTTTT7TTT7 4 7R 11.(a) M = p - n — 3 M,tota! 2 j 7 F = Fj - F (b) (2R y 2 If V, and V gravitational potentials, v is final speed GM removed 3R 2 f—T I 2J 4 Mremoved = P — 71 R =. 3 8 "7 2GMm ( m\ 9R 3 M 2 R 7

where V GMM 2R al G M tola! R n G M removed y v J R/2 putting the value of V, and V and solving we get, 2 v= I6GM 1 \ 21R Lj r, 12. l O2 r. R, = 3fi a-3 Lj = 20 x 10" H L = 40 x 10" H 2 3 R 2 = 6Q e = 15 sin 200 t fa Bansal Classes Z,= y'Rf-HtoLj) = V 3 7 4 = 5 2 T f PHYSICS

Z,= ,/R +(cdL ) 2 2 1 + 2 2 2 a/6 +8' 2 1 + 2 2 10 2 2 Z = ^ ( R R ) + c o ( L L ) = V 9 +12 = 15 I - ^-sin(200t- co/ 2=T I I

/ co 2 12 co/ T z= (C) C = O 3 4' 3 9 + tjl t + - V X4 6 0 N 21VQ 38/ 29v =5v -v = -^f 2 0 v co/ 2 2 y 21v, 76 ' x m x v i K = 12 29v "l6 ( + —x5mx 2i^o + —x5mx — x 2 2 12 76 ^ Bansal Classes PHYSICS [437]

1 2^ 29 +5x21 + 5x7 = 2~ n 76 2 2 m v 0 2 2 „1 2 m v 2 x 0 3291 5776 AK K„ 18. (a) 2845 xlOO =43% 5776 p,C + p C C =m p,+p 2 v 2 v J 2 'V, = 3 C = p 2 v 2 15R „ — - + 3R 5R _ 3R 2 ' 2 ~ c v 2 21R C = R + C - 2 . 1 R + R = 3.1 R y (b) S i M I L _ 31_ 31_ r 2.1R 21' ~ 21 given: P=const > T = 1200 K T, - 300 K = = V Y Pj = P 2 x 10 N/m V, =Vn 0 5 2 i =const T, = T = 300 K 4 . v=c«"t > T = 300 K p P = 2 x 10 N/'m = P P,= fjL = 4 V 2 =4V v 3 =4V 0 P ,T=300K T=1200K Po 2 3 2 s 2 0 H v 2

P.j = P = P = 2 x 10 N/m 4 0 5 2 Zo 4 -sT=3G0K. 4V mV n (c) 300K 1200K" ' Work done in entire cycle W,l->2 3P V 2-»3 0 r Kj 0 0 W = i|Bansal Classes V3 PV 8 w net l->2 2-*3 3~>l W -3PoV -l-4P V =1.6P V W =I-6P V Q,n = Q ^ = AU _ + W ^ = pC (T -T ) 3 c V 0 0 lo W = W _,J - nRT, log + W net 0 ZL W '•"Vo ' v e v4Vo J L4P V 0 0 + 0 0 0 0 nei 0 0 2

! >2 ! 2 p 2 PHYSICS 1

Qin = Ql-»2 = H - ! ( 2 - l ) 3- (H 2~ l^ l) -l ( 2 2 " l l ) Qin = Q i - > 2 = 31 (4P V -P V ) = 3P V X 3 . 1 X3 R T T = 1 RT RT = 3 P V P V 0 0 0 0 0 0 V Q 1.6P V ri % - 3P„V„x3.1 1 0 0 ° °* •o'o • n n 1 1 / 0 A W net xlOO = ——— xlOO = 9.3 x 1 OO —• 3,i 3 x 160 Tj % 9.3 = 17.2% ] 19. (a) (b) A T m 0 I ax = 4A2 I = 2A2 when A < J 2 A then detector becomes idle res 1000 1 10~ sec 3 from 1 to 2, it remains idle for phase angle of 90° => t = — = 0.5 x 10' s ] 2 idle 3 AT 20. 0.06 Q i o = 0.0125 kg/m 4.8 A- = 3.5 cm = 3.5 x lQ- m = 2 P 4.8m, 0.06 kg 02 R p, 2.56 0.0781 kg/m T = 8 0 N 2.56m, 0.2 kg f 80 V i

V 2 V 0.0781 « 32 m/s 32-80 , — 80 A = A 2~ 1 V i 2; V v r v + V + V 3.5 x 10~ 2 v 112 I 1 A* 48 x 3.5 x 10-2 = - Ja 3 x 10~2 m = 3.5 x 10l 2, 4.8 2.56 2 2 x IO" 7 m 21. ^Bansal Classes Given: I, - 0.5 A I=I I Applying Kirchoffs Law:15 - 111, -(11 - r ) x 0.5 = 0 an d IjT = I x 6 = 3 Putting I, - 3/r in eq. (1) 9.5-11 x (3/r) + 0.5r = 0 r + 19r - 66 = 0 r=3 So, length AD = 3 cm] 1+ 2 2 2 15 V •(1) .(2) I, r -WWV L.I —'VWv en 2 n in 1 •A/WV*] S w(10—r) W B vW

PHYSICS [22]

22. 4 3 v = — Ttr dv dr — = 4TO-. — dt dt TR -4 l [dt |r dr 8r)/7 j 2 [ 3 ttR x 4T — 8 T)/ r 4 TR 8rj/ t, = 1 2r)/r t,i = TR T- Ans. 3L/5 \ L VQCOSS 23. •V 0 sin6 o In CM frame velocity of B Lab frame Fig. (A) < v n cos8 ,v cos8 0 »v 0 sin8 CM frame Fig. (B) Tangential velocity remains unchanged whereas velocity change along string for B is mv sin 9 Impulse of tension 2 24.(a) Let wooden log moves distance x Displac ement of centre of mass along horizontal = 0 L L 2m 2 x , m 2 x - mx = 0 V 3L x — 8 — 20.25 cm (b) By conservation of mechanical energy : vsin9 (m + m + 2m)v = (2m - m)g(Lsin60°); 2 43.Bansal Classes PHYSICS [23]

point where a bat is held \ 25. From linear momentum linear impulse equation, we get FAt = MV ...(1) From angular impulse angular momentum equation, we get F(y - a)At = (MKL)co ...(2) (Mk co) (y-a) "MVT cm 2 For point to be stationary cm ( - ) ® V = a X Vcm k = (a - x) = ~~—~ co (y-a) mg+ 1.36 = p,vg mg + 0.82 = p vg mg + 1 = p vg Pl l+P 2 I 2 2 2 v 2 V V + V or k (y-a) 2 solving we get 1 V 27. 28. If system is balance then 2 V t — =2T* x V2> i i = T d =i (i) — 2 Mg Mg 2Tj = Mg sin 30° = , T, = ....(ii) T = 10 x 10= 100 N ,T = 100 N (iii) from (i) d = 3d, , T, x d, = T x 3d,, T, = 300 N Mxg from (ii) 300 = — , M = 120 kg M = 120 kg ] Ray 1 ca rries (1 - k) of the beam's energy; Ray 2 carries k (l - k) ofthe beam's energy; Ray 3 carries k (l - k) of the beam's energy; etc. The total fraction of transm itted energy is (1 - k) + k (l - k ) + k (I - k ) + ... = (1 - k) (l + k + k + . ..) = (1 - k) /(l - k ) = (1 - k)/(l + k) = 7/13 = 53.8% ] d T d 2 0 T d 2 2 2 2 2 2 2 2 4 2 2 2 2 2 4 2 2 2 4 2 2 2 7 Z Ans. ] ^Bansal Classes PHYSICS

29. RB = —— sin 9 RA = RB cos 2 0 h = —-r (cos 26) sin 0 path difference Ap = RB - RA h Ap = — — (1 - cos 20) = 2h sin 0 = X —0 sin X 7t 2A, 2 sin 0 = 7- => 3 x — = 2(4m) h 180 71 X = 4x—-m =0.42 ml 30 J 30. Vj - v (35-20) 2g 100 (in meter) Also, v, (1) = v (2) (by equation of continunit y) 3v.2 15 2g 100 .'. v = 1 m/s mass flow rate = a x v x ^ = 1000 x l x 2 x i o = 0.2 kg /sec 2 2 37.5-35 2.5 2~ 3 Now, 1 - 3 100 x20 3 100 2g 2 2 2 2 -4 V V V V = ] 31. Unl ^-r suppose v is the constant velocity then co = — v mg t = IAB e 1 1 n 2 x Brv 1 = R = — x — Br V— = 2R R 2 Brv IBVv t = IAB = 2R x %r B 2 R current I: when v = 3 m/sec e 1 Brv 1 1 = R 2 R — x 0.5x0.2x3 1A Ans 2 0.15 for hi ghest velocity net torque on the ring should be zero 2mgR 2x20xl0~ x 10x0.15 x = mgr = 1 B to* V-; v = B V 0.25x7tx4xl0, - 2 and t = mgr = 20 x 10~ x 10 x 0.2 4 x 1Q- N-m Ans. ] A 1 2 2 3 3 5x3.14r =0.2 m/sec Ans. 3 2 & Bansal Classes PHYSICS [25]

3 2 " ( a ) 1 dN N^ = C v 2 ' 0 3 IdN v V = ICv2dv 0 N 1 = C v q0. 1 C = - ^J v0 (b) JvdN v = fCv dv = v„ = _o N n jv dN 2 2 0.75V A (c) 33. = JCv^dv = J ^ x ^ " = v J N i mu = 4mv cosQ ...(1) mvj = 4mv sin9 ...(2) From e q. (1) & (2) u + v? = 16 v ...(3) Ej = -(13.6 eV)z = -54.4 eV o0 2 2 2 2 =0.775 v J 0 0 mvj E,i = - -(13.6z) = -6.4 eV n 1 mu Before collision v 4invo After collision AEj = E AE = E AE = E AE = E 2 3 4

2 3 4 3 - Ej = 40.8 eV - Ej = 48 eV - E , = 51 eV - E = 7.2 eV 2 2 m v 65eV= ~mvf + | ( 4 m ) v + AE = ~ f + ~(4M) 1 1 u +v* a 16 + AE ...(4) 1 65eV = -2m v .2 + 81 2+ ——- + AE -mv, 65eV 4 On substituting AE = 51 eV mv, comes out to be negative, which implies that, electron transition upto n = 3 is possib le. In subsequent de-excitation the possible energies are AEj -40.8 eV [n = 2 to n = 1] AE = 48 eV [n = 3 t o n = l ] AE = 7.2 eV [n = 3 to n = 2] ] 3 2 4 34. ^Bansal Classes Net external force = M system acm „ ( ma + 0 2mg sin 6 = 2m I 2m , => a = 2g sin 6 PHYSICS [26]

m+m . . a = m gsinG Also v - — = 2ad; 2 3v_ 8a solving we get d = 2 m/s Ans. ] 35. Let at any instant t charge is q . dq [charge decreasing] ~ dF di d^q dt dt di C dt C dt 2 ' 1 = 2 L •c '-nmp1 q = q sin .VLC t + (j) at t = 0, q = q => — 2 4 2 8 xLtan0 + W = s rT T 2 Bansal Classes PHYSICS

(c) 41. For v = 0 x = V3L V3mgL_J_ 2 4 2 k = V3mg 2L v I kL => 9 = 60° Zm JU | a = 2m/s 0 2 w.r.t. h lift to acceleration of coin, a' = g - a = 8 m/s u = 3 m/s u, maximum h eight, H' = —^ = 0.56 m < 1 m 2a' so, it cannot touch the roof of lift. Now, let t is time taken to reach the floor 0 rd 2 s 0 0 2 0 2 42. • -l=3t --~(8)t On solving, t = 1 sec w.r.t. ground Initial velocity, v = 8 m/s v Maximum height, H = — = 3.2 m Time taken for upward motion, t, = v /g = 0.8 s Time taken for downward motion, t = 1 - 0.8 sec = 0.2 Distance travelled during down ward motion = (l/2)g * (0.2) = 0.2 m Total distance travelled = 3.2 + 0.2 = 3.4 m Displacement till that time = 8(1) - (l/2)g x (l) = 3 m ] 1 Point source - sph erical wavefront => I o c ~ j I, = 41 0 2 ff 2 2 2 L = 10 log ~~ = 30 10 10 (1) 10 i0 41 Lj = 10 log ^" = 10[log 4] + L = 20 log 2 + 30 « 36 dB sound will= be inaudible if I I from (1) I= 1000I I d ~ = 1~0 0 0~ d ( 2 0 ) I d = 2007l0 m ] 2 0 G 2 2 1 W U 2 2 & Bansal Classes PHYSICS [30]

43. 40 / m = f + u 40+ (-30) m = v/u v = -120 cm v 64 cm/s 2 cm 2 cm 4 cm/s i =m v , 2 0< Vj = 1 6 x 4 = 64 cm/s Generaly equation for y-component of velocity of image A_l-± v u / 1 1 1 v - 3 0 +40 120 40 / = 40 cm / = 60 cm 1 1 40 30 v = -120 y yi 0 / / + x 0 o' x m y« f f + x Oj s dyu dt (/ +x ) 0 y0f U 2 _ x x y of =" mv.. - —f + x )- x _ ( dt " " v o dx0 u v 0 x 0 2 v Y o (1)

Applying equation (1) for first lens with V = 0 and then applying for second len s with v = - 16/5 cm y o dx v = ( / y of ) 2dt =-(16/5) cm/s +X °y o _ 0 2 Now substitute V and m = 60+ (-160) y 1 3 X / • 16zr\ 64x2x60 _ _ (60-160) 144/125 \ -> j 144 Therefore, relative velocity = 2 x 125 q 5 60 in eq. (1), to get 44.(a) To determine the specific heat capacity of unknown solid, m,s + m s t i w e use ssolid and get s = 1/2 cal/g/°C m, ~ SS J 3 2 2 0 S0l]d fds^ 1 = 2A0 e - e V s )max s s • + 2 1 1 1 2 (0.1 °C) 40.0-20.0 80.0-40.0 o , - oss y - + 1% (b) m s +m s m. y 9j-e substituting value, we get s, = 0.5 cal/g°C for finding error i n s. s. = 2 2 3 3 v y m s +m s \ (e -e)(de-de ) (0-e,)(d0,-d0) as, = mi y (G,-e) 2 2 3 3 a 2 2 r ^Bansal Classes PHYSICS [31]

ds^ _ (de - de )(6 - 9) - (9 - e ) (dO, - d6) 2 1 2 (6! - 9) + (9 - 9 ) de - de (0, - 0) - d0j (0 - 6 ^) 2 2 S, 1 2 ( 0 _ 0 ) ( 0 _ 0 ) 1 2 (0,-0)(0-02) 2 1 2 As, _ (0 -0 - 0 ) M ( H when 0 20 + 1 2 ; ) ( 2

)A0 + ( 0 - 0 ) A 0 + ( 0 - 0 ) A 0 As, _ A9[2(9,-9 )] s, ( 0 - 0 0 ) As, is minimum when (0,-9) (9 - 0 ) is maximum. 0,+0 This happens 80 => steady state temperature should be — - — = 50°C Ans. ] 2

45. 1 cm, 1 cm 46. (a) 0.2 cm; (b) -0.1 cm ; (c) 12 cm [Sol.(c) u = 30.2 - 0.2 (excess reading) = 30.2 cm v = 19.9 - (-0.1) (excess reading) = 20.0 cm. 1 1 1 7 = - + - =>f= 12 .0cm ] f v u 47, t = t Let charge on outer shell is Q => Leakage current I dQ dt at distance r from centre J = ctE I Am-' I dQ charge on innershell = Q, + Q. - Q =s r Qi + Qi-Q" 4n £n Kr 2