Current Electricity_IIT Material

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CURRENT ELECTRICITY  • • • • • • • • • •

Electric current Electric current density, drift velocity Electromotive force and terminal potential difference (V) Grouping of cells Ohm’s law and resistance Temperature coefficient of resistance Kirchoff’s laws Wheatstone bridge  Meter bridge Potentiometer

Current Electricity 

RK

CURRENT ELECTRICITY 1.

Electric current : The time rate of flow of charge through any cross section is called electric  current . If charge ∆q charge passes in time ∆t. q q ∆q Average current Iav = . q q ∆t

∆q dq = . ∆t →0 ∆t dt

Instantaneous current I = Lt

Electric current is assumed to be a fundamental quantity in physics with unit ampere and dimension A. 2. The current flowing through a conductor is said to be one ampere when one coulomb of charge passes through it in one second. 18 3. If 6.25×10 electrons pass across the cross-section of a conductor in one second, the strength of the current flowing across the conductor is one ampere. 4. i) Electronic current : Here the direction of this current is taken as the direction in which the electrons are transferred. ii) Conventional current : The direction of this current is taken as opposite to that of electronic current. 5. a) Free electrons are charge carriers in metals. b) Positive and negative ions are charge carriers in liquids. c) Positive ions and electrons are charge carriers in gases. d) Holes and electrons are charge carriers in semiconductors. 6. The current in different situations is calculated as follows: a) Due to translatory motion of charge : If n particles, each of charge q passes through a nq given area in time t seconds then i = t b) Due to rotatory motion of charge : If a point charge q is moving in a circle of radius r with q e qv speed v, constant frequency f and time period T then i = = = qf = t T 2πr 7. AC and DC : a) If the magnitude and direction of current does not vary with time. It is known as direct current i DC. i t b) If a current is periodic i.e. magnitude varies t periodically and polarity reverses after each half cycle, it is known as alternating current (AC). 8. Though conventionally a direction is associated with current (opposite to the motion of electrons), it is not a vector is a) the direction merely represents the sense of charge flow and is not a true direction. b) current does not obey the law of parallelogram of forces. 9. For a given conductor current does not change with change in its cross section. 10. Conductor remains uncharged when current flows through it.  

11. Current density ( J ) : Current density at a point, is defined as the current flowing per unit area, surrounding that point, the plane of the area being normal to the direction of current fl ow. Current density is a vector quantity. The direction of current density is along that of conventional current. 1

Current Electricity 

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If ‘i' be the current flowing past a section of area ‘s’ then current density J =

i . s  

For non-uniform distribution of current over an area the average current density will be J = 

∆i . ∆s

∆i di = . ∆s→0 ∆s ds

The current density at a specific point P will be J = Lt

If the cross sectional area is not normal to the current J =

di . dscos θ   

ds

θ

i





J

di = J ⋅ ds  

i = ∫ J ⋅ ds  

  

Current is the flux of current density. Example : A potential difference applied to the ends of a wire made up of an alloy drives a current through it such that j = α + βr. where r is the distance of the point from the axis. If R be the radius of the wire, then find the total current through any cross section of the wire.

r

dr



 

di = J ⋅ ds = (α + β r)(2πrdr)cos0 R

i = ∫ di 0

=

2πR2 (3α + 2β R) 6

12. Drift velocity : a) In a conductor in the absence of electric field, the electrons or charges will be moving in random direction. This is because of thermal energy of electrons. b) Thermal energy depends on temperature of the conductor. v c) If temperature is increased, random velocities of the electrons v VA VB increases but average velocities of charges is zero. v v d) When electric field is applied to the ends of the conductor the v charges start moving in the direction of the field. e) As charge moves it collides with an atom, and it loses some momentum. f) If then again picks up velocity because of the field. g) It accelerates and gains some velocity, but within a short time, it again collides with another atom. This keeps on happening all the time because of collisions and heat is liberated in the conductor. h) Because of collision, the velocity of the charge does not increase beyond a particular value. i) The average velocity of the charge is called as Drift velocity (Vd).  j) Drift velocity is the average velocity and not the instantaneous velocity of the charge. 2

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13. Relaxation time : If τ be the average time between successive collisions, the distance drifted during this period is 1 2 1 ⎛ eE ⎞ 2  = at = ⎜ ⎟τ 2 2⎝ m ⎠ The drift speed is Vd =



1 ⎛ eE ⎞

= τ. τ 2 ⎜⎝ m ⎟⎠

It is proportional to the electric field and to the average collision time τ. 14. Relation between current and drift speed : The current ‘i' through the wire will be q i= t If n is free electron density (number of free electrons per unit volume) q = nA where A is area of cross section and  is length of the conductor. nAe i= t i = nAeVd i = neVd A J = neVd 15. Electric cell : a) It is a device which converts chemical energy into electrical energy. b) There are two types of cells i) Primary cell ii) Secondary cell c) Comparison of primary and secondary cells: Primary cell

Secondary Cell

i) Converts chemical energy into electrical energy

Electrical energy is first stored in t he form of chemical energy and then again gets converted into to electrical energy on drawing current from it.

ii) This cannot be recharged.

This can be recharged

iii) Their e.m.f is less and internal resistance Their e.m.f is more and internal resistance is more is less iv) In this the conversion of chemical energy into electrical energy is an irreversible process. Eg: Denial cell, voltaic cell, cadmium cell, dry cell, Laclanche cell etc.

In this the conversion of chemical energy in to electrical energy is a reversible process. Eg: Lead accumulators, Edison cell

16. Electromotive force (e.m.f) of a Cell : i=0

ε

a) The work done is carrying a unit positive charge once in the whole circuit including the cell, is defined as the electromotive force. b) Electromotive force is the potential difference between the terminals of a cell in open circuit. c) Electromotive force depends on –(1) nature of electrolyte (2) metal of the electrodes. 3

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d) Electromotive force does not depend on (1) area of plates (2) distance between the electrodes (3) Quantity of electrolyte (4) size of the cell. e) Electromotive force is the characteristic property of the cell. The direction of current inside the cell is always from negative to positive electrode. f) The unit of electromotive force is volt. 17. Terminal potential difference (V) : The potential difference across the cell when cell is in closed condition.

ε R+r Ri + ri = ε V = Ri V = ε − ri

V

i=

i

ε R

18. Internal resistance (r) : The internal resistance of a cell is the resistance offered by the column of the electrolyte between the positive plate and the negative plate. i) The internal resistance of a perfect cell or ideal cell is zero. ii) Internal resistance depends on a) strength of electrolyte (r ∝ strength) b) distance between plates (r ∝ d) 1⎤ ⎡ c) area of the plates ⎢r ∝ ⎥ A⎦ ⎣ 1⎤ ⎡ d) temperature of electrolyte ⎢r ∝ ⎥ t⎦ ⎣ 19. Relation between EMF and PD: 1) In case of charging of a cell a) The current flows from +ve to –ve terminal inside the cell. I b) V > E c) V = E + ir 2) In case of discharge of a cell a) The current flows from –ve to +ve terminal inside the cells I b) V < E c) V = E – ir 3) The difference between E and V is called lost volts ∴ lost volts = E – V = ir 4) A cell of emf ‘E’ and its resistance ‘r’ is connected to resistance ‘R’. I

E r R

a) i =

E R+r

b) P.D. across resistance R is given by V = iR= c) Fraction of energy useful =

ER R+r

V R = E R+r

⎛ V ⎞ ⎛  R  ⎞ d) % of fractional useful energy = ⎜ ⎟100 = ⎜ ⎟100 ⎝ E  ⎠ ⎝ R + r ⎠ d) Fraction of energy lost =

E − V ir r = = E E R+r 4

I

E

r

E

r

I

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⎛  r  ⎞ e) % of lost energy = ⎜ ⎟100 ⎝ R + r ⎠ (E − V )R V h) For single cell, the condition for maximum current is R = r. 20. Back emf : a) The copper electrode gets covered with a layer of hydrogen and this hinders flow of current. In the neighbourhood of both electrodes, the concentrations of ions get altered. This results in an emf acting in a direction opposite to t he emf of the cell. This is called back emf . b) This formation of hydrogen around the anode is called polarization. c) To reduce the back e.m.f manganese dioxide and potassium dichromite are added to electrolyte of cell. These are called depolarizers . 21. Series combination of cells : g) r =

+

E1 – r

+

E2 –

+

r

E3 – r

R

a) b) c)

d)

E = E1 + E2 + E3 + …En r = r1 + r2 + r3+ ……. rn When cells of e.m.f.’s E1, E2, E3…. and of internal resistances r1, r 2, r3….. are connected in series across an external resistance R, the current i is given by E1 + E 2 + E 3 ........ i= R + (r1 + r2 + r3 + .....) If the e.m.f s of all the n cells and their internal resistances are same, then i =

nE (R + nr )

e)

If n r >> R, then i = E/r, i.e the current obtained from n cells is equal to that obtained from a single cell. f) If n r > R then I = mE/r i = n × (current obtained from a single cell) 5

rm R

Current Electricity 

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iv) If r >R and more current is required in the circuit. v) If the e.m.f of m cells and their internal resistance are different then 1) i = i1 + i2 + i3 ……. in

⎛  E  ⎞ ⎜∑ ⎟ ⎜ r ⎟ ⎜ 1⎟ ⎜∑ ⎟ ⎝  r  ⎠

⎡ E1 E 2 E ⎤ + ....... m ⎥ ⎢ + r2 rn ⎦ ⎣ r1 2) I = = ⎡  ⎞ ⎛  1 1 1  ⎞⎤ ⎛  ⎟ ⎢1 + R⎜⎜ + + ............ ⎟⎟⎥ ⎜ 1 ⎟ ⎜ r r r ⎥ R+ 2 n  ⎠ ⎦ ⎝  1 ⎣⎢ ⎜ 1⎟ ∑ ⎟ ⎜ r  ⎠ ⎝  E 1 r 3) E total = 4) rtotal = 1 1 ∑ ∑ r r 24. If two cells of emf E1 and E2 having internal resistances r1 and r2 are connected in parallel to an external resistance R, then

∑

E1 – + r1 i2 E2 + – r2 R

i1

i) i

i

a) The effective emf, E =

E1r2 + E2r1 r1 + r2

b) The effective internal resistance, r = c) Current through the circuit, i =

r1r2 r1 + r2

E r +R

d) i = i1 + i2 E − iR E − iR e) i1 = 1 and i2 = 2 r1 r2 ii)

E1 + – r1 i2 E2 – + r2 R

i1

i

i

a) The effective emf, E =

E1r2 − E 2r1 r1 + r2

b) The effective internal resistance, r = c) Current through the circuit, i = d) i = i1 – i2 E − iR e) i1 = 1 and r1

i2 =

r1r2 r1 + r2

E r +R

E 2 + iR r2

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Current Electricity 

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25. Mixed grouping of cells : E1

E2

En

m rows

R

i) The e.m.f of cells in a row = nE. ii) Total e.m.f of the combination = nE nr iii) The total internal resistance = m iv) The total resistance of the circuit = R +

nr m

nE mnE = nr mR + nr R+ m vi) For maximum current to flow through the external circuit, the external resistance nr should be equal to the total internal resistance. or R = or, mR = nr m Power transferred to the load is maximum when load resistance is equal to internal resistance. This is known as maximum power transfer theorem. 26. Two cells if e.m.f.s E1 and E2 be connected in a circuit. Let r1 and r2 be the I E1 internal resistance of the cells. r1 E1 + E 2 a) The current through the circuit I = r1 + r2 E2 v) The current flowing through the external resistance (i) =

b) The terminal voltage across the cells V1 = E1 – Ir1 V2 = E2 – Ir2 27. Let two cells of e.m.f.s E1 and E2 be connected in parallel in a circuit. Let r1 and r2 be the internal resistance of the cells. a) The direction of the resultant current is determined by the direction of the higher e.m.f. E − E2 b) If E1 < E2, the current through the circuit is I = 1 . r1 + r2

r2

I

E1 r1

E2 r2

c)

While the cell E1 is discharging, the cell E2 is in the charging. The terminal voltage across the cells V1 = E1 – Ir1 and V2 = E2 + Ir2. 28. Ohm’s law : i

V

At constant temperature, the current (i) flowing through a conductor is directly proportional to the potential difference (V) between its ends. V α i or V = iR where R is the electrical resistance of the conductor a) Ohm’s law is not a universal law. b) Conductors which obey Ohm’s law are called ohmic (or) linear conductors. Ex. metals. 7

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c) d)

e) f)

The graph between V and I for ohmic conductor is straight line passing through the origin. Conductors which do not obey Ohm’s law are called Non ohmic (or)Non linear conductors. Ex:Carbon compounds, electrolytes, transistors, diodes, semiconductors, discharge tubes, Thermionic valves, vacuum tubes. The graph between V and i for non ohmic resistance is a curve i

i

i

V

i

V

Metal conductor

Vacuum tube

V

V

Electrolyte

Thermistor

29. Thermistor : a) It is a thermal resistor. b) It is a heat sensitive nonohmic device. c) Made of semiconductor compounds as oxides of nickel, iron, cobalt and Cu. d) It is enclosed in a capsule with an epoxy surface. e) Symbol is

or

f) One type of thermistor has high positive temperature co-efficient (PTC) of resistance. g) Another type of thermistor has high negative temperature co-efficient (NTC) of resistance. h) (i) NTC thermistor is used as resistance thermometer for measuring low temperatures of the order of 10 K. (ii) High resistance at low temperature makes it possible to measure low temperature very accurately. i) Thermistors one in the form of beads, discs or rods to which a pair of platinum wires are provided at leads.  j) A tiny bead form thermistor serves as thermometer and can measure temperature changes of –3 the order as small is 10 K. k) Thermistor used in measuring the rate of energy (power) in a mino wave beam. l) Thermistor used in radio circuits to avoid sudden and large surge of current. m) Thermistor is used as thermostat. 30. Resistance : a) The property by virtue of which a conductor opposes the flow of charge in it is known as resistance. b) It is measured as the ratio between potential difference between the ends of the conductor and current flowing in the conductor ⇒ R= V/i. c) SI unit of resistance is Ohm. 1 ohm = 1 volt / 1 amp  d) Ohm is the resistance of a conductor through which a current of 1 ampere flows when the potential difference between its ends is 1 volt. 2 –3 –2 e) Dimensions formula R is ML T I . f) For good conductors resistance is very low and for insulators or bad conductors it is high. 31. Conductance : a) The reciprocal of resistance is known as conductance ⇒ G = 1 /R. –1 b) SI unit of G is siemen (S) (or ohm or mho) c) Conductance decreases on increasing temperature 8

Current Electricity 

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32. Dependence of resistance : a) Resistance of a conductor is directly proportional to its length and inversely proportional to its area of cross section. 

R∝

R=S

⇒



or ρ



A A A Here S or ρ is known as resistance or specific resistance R= b) R =

ρ Where r is radius of cross section. πr 2

ρl A

=

ρl 2 V

=

ρV A2

=

ρm A 2d

=

ρl 2 d m

c) Resistance does not depend on current and potential difference. Through resistance of a linear conductor is independent of applied voltage, for a given body it is not unique and depends on length and area of cross section. (i.e how the potential difference is applied)If  ,b, h denote length, ρ ρh breadth and thickness of a slab, (  >b>h), Rmax = and R min = b bh 33. Specific resistance : a) It is equal to resistance of the conductor of unit length and unit area of cross section.  s RA b) R α or R= or s =  A A c) S.I. unit : Ohm – metre d) It depends only on the material of the conductor and temperature. e) It is independent of dimensions of the conductor. f) For silver and copper specific resistance is small g) For Nichrome, constantan, Maganin it is large. 34. Conductivity : (or) specific conductance ( ) :  1 a) It is reciprocal of resistivity. σ = = s RA b) S.I unit : siemen / m c) For insulators σ = 0 d) For perfect conductors, σ = infinity Example : The specific conductivity of a cylindrical conductor of length varies as σ(x) = σ0





and area of cross section A

where x is the distance along the axis of cylinder from one of its ends. x Calculate the resistance of the cylinder along cylindrical axis. Solution : dR =

=

dx

1 dx σ(x) A x

xdx Aσ0

since discs are connected in series 

R = ∫ dR = 0

2  . 3A σ0 9

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35. Vector form of Ohm’s law : i

 V

V = Ri V=ρ 1

ρ



A

i

=σ

⎛V⎞ J ⎜⎟=σ ⎝ ⎠ 



J = σE 

J = Current density, E = Electric field, σ = conductivity. Example : The space between coaxial cylinders whose radii are a and b (b > a) is filled with a conducting medium. If σ is specific conductivity of the medium, calculate the resistance between the cylinders in the radial direction. L b

a

dR = b

1 dr σ 2πrL

R = ∫ dR = a

n(b/a)

2πσL

.

36. Temperature co-efficient of resistance ( ) : a) It is defined as the change in specific resistance (or resistance) per 1°C rise of temperature to the original specific resistance (or resistance) at 0°C. y ρ − ρ0 b) α = t α = +ve c) α =

ρ0 t

↑

Rt − R0

R

α≅0 α = –ve

R0 t

t→

c) ρ t = ρ 0 (1 + α∆t ) …(1) d) Rt = R0 (1 + α∆t) …(2) ρ0 and R0 are the specific resistance and resistance at 0°C, ρt and Rt are the corresponding values at t°C. e) If R1 and R2 are resistances at t1°C and t2°C then R 2 − R1 . α= R1t 2 − R 2 t 1

10

x

Current Electricity 

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f) For small temperature variation, ρ T = ρ To [1 + α(T − To )] where ρ To and ρ T are the resistivities at temperatures To and T respectively and α is a constant for a given material and is called the temperature coefficient of resistivity. 1 dρ α= ⋅ ρ dT

EFFECTIVE RESISTANCE : 37. Series connection : i

R2

R1

R3

i) Current is the same through all the resistors ii) Total p.d.= sum of individual p.d.s across each resistor. iii) Individual p.d. is directly proportional to individual resistor. iv) Total resistance is greater than the greatest individual resistance. v) Total resistance = sum of the individual resistances. R = R1 + R2 + R3 + ……. vi) Two resistances in series : a) The total resistance RS = R1 + R2 VR1 b) V1 = R1 + R 2 c) V2 = d) i =

R1

R2

V1

V2

i V

VR 2 R1 + R 2

V1 V = 2 R1 R 2

vii) A conductor and Semi conductor are connected in series. If the resistance of the combination is same at all temperatures then R1 α1 = R2α2 where R1, R2 are resistances of conductor and semi conductor. 38. Parallel connection :

R1

i) Potential difference remains the same across each resistor.

i1

ii) Total current=sum of the individual currents.

i3

i2

R2 R3

iii) Individual currents are inversely proportional to the individual resistances. iv) Effective resistance is less than the least individual resistance. v) When a number of conductors are connected in parallel, the reciprocal value of the resultant resistance is equal to the sum of the reciprocal values of the i ndividual resistances. 1 1 1 1 = + + + ..... R R1 R 2 R 3 vi) Two resistances in parallel a) The total resistance R1R 2 RP = R1 + R 2 b) I1 =

IR 2 R1 + R 2

c) I2 =

IR1 ; R1 + R 2

I1

R1

I2

R2

V = I1R1 = I2R2 11

Current Electricity 

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39. If RS and RP be the resultant resistance of resistances R1 and R2, when connected in series and parallel then R1 =

1 ⎛   ⎞ 2 ⎜ R S + R S − 4R sRP ⎟ 2 ⎝   ⎠

R2 =

1 ⎛   ⎞ 2 ⎜ R S + R S − 4R sRP ⎟ 2 ⎝   ⎠

40. If n equal resistances each of resistance R are connected to form triangle (or) Square (or) Polygon then ⎛ n − 1 ⎞ effective resistance between any two adjacent corners is R′ = ⎜ ⎟R . ⎝  n  ⎠ 41. Effective resistance across cube : r r r r r

r

r

r r r r

r

When twelve identical resistors each of resistance R Ω are connected in the form of a skeleton cube, the effective resistance across (i) the ends of a side is (7r/12)Ω, (ii) the opposite vertices on the same face i s (3r/4)Ω and (iii) the diagonally opposite vertices is (5r/6)Ω. 42. Kirchhoff's laws : a) First law : i3

i1

i4 i2

i5

i) The algebraic sum of electric currents meeting at a junction is zero. for the junction 'P' ; i1 + i2 - i3 – i4 – i5 = 0 (or) i1 + i2 = i3 + i4 + i5 ii) Kirchhoff's first law is known as junction law or point law of kirchhoff's current law iii) Kirchhoff's first law obeys law of conservation of electric charge. b) Second Law : i1

E1

R1

R5

i5

R2 i2

R3

i3

i4

E2 R4

i) the algebraic sum of emfs or potential differences around a closed circuit is zero. For the closed circuit ABCDEA + E1 – i1 R1 – i1R2 – i3R3–E2 – i4R4 + i5R5 = 0 ii) Second law is known as loop theorem or kirchhoff's voltage law. iii) Kirchhoff's second law obeys law of conservation of energy and based on fact that the electrostatic field is conservative in nature. 12

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c) Sign convention in kirchhoff's laws: i) While going from +ve of a battery to the negative through a cell, emf is negative. ii) While going in the direction of the current through a conductor, potential difference is negative. 2A 12V 4Ω

r=2Ω y 12V

ir = 4V E=12V

8V=ir x

Potential rise and fall in a circuit. A

i path

R

A

B

B

VB – VA = + iR

VB – VA = –iR

ε

ε i path

A

i path

R

i path

A

B

B

VB – VA = –ε

VB – VA = +ε

43. Wheatstone bridge : B Q

P G

A

C S

R

D ()

i) Wheastone bridge is a circuit used to compare the ratio of nearly equal resistance. It consists of four ams, each consisting a resistor. ii) If two of the resistors of the four are known, the other two can be compared. iii) If three resistances are known the fourth one can be calculated. iv) If the current through the galvanometer in a Wheatstone bridge is made zero, then the bridge is balanced. v) Under balanced condition : P R a) = Q S b) c) d) e)

The same current passes through the P& Q. The same current passes through the R & S The P.D. across the ends of the galvanometer is zero. When galvanometer and cell are interchanged, the balance point is not effected. (P + Q)(R + S) f) The effective resistance = P +Q +R +S vi) Wheatstone’s bridge is more sensitive if P = Q = R = S vii) The number of closed circuits in bridge = 7. 13

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44. Meter Bridge : R RB

G



100 cm

Rh

i) ii)

iii)

It works on the principle of Wheatstone Bridge. It is the simplified form of Wheatstone Bridge. It is used to find a) unknown resistance of a wire b) specific resistance of the wire c) and also to compare resistances.  resistance in the left gap When the Meter bridge is balanced then = resistance in the right gap 100 −  Where

iv) v) vi)



is the balancing length from the left end.

A high resistance box is connected in series to the galvanometer to protect it from higher currents. The bridge wire (manganin wire) ha low α-value. Meter bridge is more sensitive if  1 = 50 cm

vii) The resistance of copper strip is called end resistance. viii) The resistance in two gaps (x and R) are interchanged to reduce the effect of end resistance. ix) If a conductor is connected in the left gap and it is heated then blanching point shifts towards right. x) If a semiconductor is connected in the left gap and it is heated then balancing point shifts towards left. 45. Potentiometer : Potentiometer is a device which does not draw any current from the given circuit and still measures the potential difference. Thus it is equivalent to an i deal voltmeter. i) It is a device which is used to a) compare the e.m.f.s of two cells, b) to determine the e.m.f of a cell c) determine the internal resistance of a cell d) calibrate a voltmeter and an ammeter e) determine the current in a circuit, f) determine unknown resistance, g) measure thermo emfs. Potentiometer consists of two circuits. a) Primary circuit : i

ε0 r0

RS L

A cell of ε0 and internal resistance r0 in the primary circuit maintains uniform potential gradient along the length of its wire.

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b) Secondary circuit : It consists a battery whose emf is to be determined galvanometer, high resistance (HR) and sliding jockey J. J G

H.R

Comparison of emfs using potentiometer : 2

ε0

R0

r0 J l1

ε1 H.R

G

Comparison of emfs using potentiometer : a)  1 and  2 are balancing lengths when two cells of emfs, E1 and E2 are connected in the secondary circuit. one after the other then,

E1 = E2

1 2

b) By sum and difference method, E1 + E 2 L1 E L + L2 . or 1 = 1 = E1 − E 2 L 2 E 2 L1 − L 2 46. Internal resistance of a cell r : i

ε0

r0

R0

J

l1

ε1 H.R

G

R1

i) Internal resistance of a cell

⎛  −  2  ⎞ ⎛ E − V ⎞ ⎟⎟R r =⎜ ⎟R = ⎜⎜ 1 ⎝  V  ⎠ ⎝   2  ⎠ When 2 =

1

= balancing length for the cell connected in the secondary circuit.

balancing length when a resistance R is connected in parallel to the cell.

E = emf of the cell in the secondary circuit V = Terminal voltage ii) The sensitivity of potentiometer can be increased by decreasing the potential gradient. i.e., by increasing the length of potentiometer wire for a given B. iii) The best instrument for accurate measurement of the e.m.f of cell is potentiometer because it does not draw current from cell. iv) Potentiometer acts like a voltmeter of infinite resistance. v) Eb (emf of battery in the primary circuit ) must be greater then Ec (emf of cell in the secondary circuit) otherwise e.m.f will not be balanced even over the complete length of wire. vi) + ve terminals of both battery and cell must be connected at same point otherwise Ib and Ic will be in same direction and null point is never obtained. 15

Current Electricity 

RK

47. Earthing : If some point of a circuit is earthed then its potential is taken to be zero. Example : 6V

D

E 2Ω

3V

C

B

4Ω A

F

In the above figure, VA = VB = 0 VF = VC = VD = –3V VE = –9V VB – VE = 9V VA − VF 3 = A. 4 4 V − VE 9 Current through 2Ω resistance is B = A. 2 2 48. Twelve equal resistances each of resistance 2 ohm form the edge of cube. If a 2V battery of internal resistance 0.1 atm be connected to the adjacent vertices of a cube. Find the current flowing through each side of a cube. Current through 4Ω resistance is

y R

x+2y A

y–z

K

L R

z y

z

R

R

x E

2V r

y R

x+2y C

2(y–z)

y–z B R

N y–z R R

R y

M y–z

D

Applying KVL to mesh EACE r(x + 2y) + Rx = E …(1) AKNCA : yR + zR + yR – xR = 0 …(2) BLMDB : (y – z)R + 2(y – z)R + (y – z)R – zR = 0 …(3) By solving (1), (2) and (3) 7E x= 12r + 7R 5E y= 2(12r + 7R) z=

2E . 12r + 7R

16

Current Electricity 

RK

Example : A conductor of length



has a non-uniform cross section.

The radius f cross-section varies linearly from a to b. The resistivity of the material is ρ. Assuming that b – a