9. Current Electricity

CURRENT ELECTRICITY This topic is taken from our Book: ISBN : 9789386320094 Product Name : Current Electricity for JEE Main & Advanced (Study Package for Physics) Product Description : Disha's Physics series by North India's popular faculty for IIT-JEE, Er. D. C. Gupta, have achieved a lot of acclaim by the IIT-JEE teachers and students for its quality and indepth coverage. To make it more accessible for the students Disha now re-launches its complete series in 12 books based on chapters/ units/ themes. These books would provide opportunity to students to pick a particular book in a particular topic. Current Electricity for JEE Main & Advanced (Study Package for Physics) is the 9th book of the 12 book set. • The chapters provide detailed theory which is followed by Important Formulae, Strategy to solve problems and Solved Examples.

• Each chapter covers 5 categories of New Pattern practice exercises for JEE - MCQ 1 correct, MCQ more than 1 correct, Assertion & Reason, Passage & Matching and Integer Answer & Subjective Questions. • The book provides Previous years’ questions of JEE (Main and Advanced). Past years KVPY questions are also incorporated at their appropriate places. • The present format of the book would be useful for the students preparing for Boards and various competitive exams.

Contents 3. DC and DC Circuits

Contents 217-306

4. Thermal & Chemical Effects of Current 307-338



3.1 Introduction

218





3.2 Ohm’s law

221





3.3 Electricity from chemicals : cell 226



3.4 Electromotive force (emf)

227



3.5 Circuit analysis

229



3.6 Electrical instruments

250



Review of formulae and



important points

256



Exercise 3.1- 3.6

258-286



Hints & solutions

287-306

4.1 Thermal effect of current : joule’s law

308



4.2 Electrical appliances

310



4.3 Seebeck effect

315



4.4 Peltier effect

317



4.5 Thomson effect

318



4.6 Chemical effect of current

318



4.7 Faraday’s law of electrolysis

319



Review of formulae and



important points



Exercise 4.1-4.6

322-332



Hints & solutions

333-338

320

Chapter 3

DC and DC Circuits

221

3.2 OHM'S LAW If the physical conditions of conductor (temperature, length etc.) do not change, then the current in the conductor is directly proportional to the potential difference across the conductor.

Resistivity ( r ) It can be defined as the ratio of electric field applied to current density.Thus r

=

E . J

Resistivity is the property of material which does not depend on size and shape of the conductor. SI unit of resistivity is Ohm-meter ( W -m). A perfect conductor would have zero resistivity and a perfect insulator have infinite resistivity. Metals and alloys have the lowest resistivities and hence they are the best conductors. It is a familiar fact that good electrical conductors, such as metals, are good conductors of heat, while poor electrical conductors such as plastics are also poor thermal conductor.

Variation of resistivity with temperature The resistivity of metallic conductors increases with increasing temperature as rt

= r0 [ 1 + at + bt 2 ]

where r0 = resistivity of conductor at 0o C rt = resistivity of conductor at t o C

t = the temperature difference between t and 0o C, and

a & b are temperature coefficients of resistance

a > b , and positive for metals and negative for non-metals.

(a)

Note: The resistivity of the alloy manganin is practically independent of the temperature.

Resistance

uur uur ur ur In the relation E = rJ , it is difficult to measure E and J directly. It is, therefore very convenient to put this relation in macro parameters like current and potential difference. To do this let us consider a conductor of length l and cross-sectional area A as shown in fig. 3.11. ur uur We can write = rJ E As \

E V l

=

V i and J = l A

= r

i A

(b) Fig. 3.10

Fig. 3.11

222

ELECTRICITY & MAGNETISM or

V

æ rl ö = iç ÷ = iR è Aø

This relation is often referred to as Ohm's law. where R is called resistance of the specimen and is equal to R =

rl . A

. The SI unit of R is ohm ( W ). Its symbol is The material obeying Ohm's law is called ohmic conductor or a linear conductor. If ohm's law is not obeyed, then the conductor is non-ohmic or non linear. Fig 3.12 (a) is a curve for metalic conductor. The straight line shows that the resistance of conductor is the same no matter what applied voltage is used to measure it. fig.3.12 (b) is not straight line and the resistance depends on voltage applied, the resistance is called dynamic resistance and the conductor is called non-ohmic conductor.

Thermistor Thermistor is the heat sensitive resistor. It is usually made by semiconducting material for which V – i plot is not linear. The temperature coefficient of thermistor is negative and Fig. 3.12

usually large. It is of the order of -0.04 / o C. Thermistors are used for making resistance thermometers which can measure very low temperature.

Note: 1.

Fig. 3.13

2.

Resistance depends on length of wire not on the shape of the wire. All the wires have same resistance between A & B. l is the straight length of wire between its ends. V = i R is the general relation between i and V which is applicable to both type of conductors, ohmic as well as non-ohmic.

Variation of resistance with temperature As the resistance of any specimen is proportional to its resistivity, which varies with temperature, and also resistance varies with temperature. Therefore we have Fig. 3.14

for small t,

2

Rt

= R0 (1+ at + bt )

Rt

; R0 (1+ at )

Temperature coefficient of resistance ( a ) If R1 and R2 are the values of resistance at temperature t1 and t2 respectively, then R 1 = R0 ( 1 + at1 ) and R2 = R0 ( 1 + at2 ) \

R1 R2

or

a

1 + at1 = 1 + at 2

=

R2 - R1 . R1t2 - R2 t1

Note: We have assumed a to be constant for all temperature. But actually it varies with temp. If Rt is the resistance at any temperature, then Rt = R0 ( 1 + at ), differentiating above equation w.r.t. temp, we get a

=

1 R0

æ dRt ö çè dt ÷ø .

DC AND DC CIRCUITS

223

Resistance of conductor of non-uniform cross-section Consider a conductor of length l and radius at its ends are r1 and r2. The resistance of element under consideration

where, \

dR

=

r

=

dR

=

R

=

r dx pr 2 ær -r ö r1 + ç 2 1 ÷ x è l ø

Fig. 3.15

r dx é æ r -r ö ù p ê r1 + ç 2 1 ÷ x ú è l ø û ë

2

Resistance of whole conductor l

ò dR = 0

or

R

=

l

ò

0

r dx é ær -r ö p ê r1 + ç 2 1 ÷ è l ø ë

ù xú û

2

rl . p r1r2

Why conductor offers resistance? Resistance means the hinderance offered to the flow of charge. Electrons in their motion collide with the positive ions and themselves, due to which resistance in motion occurs. The resistance mainly occurs due to collisions of electrons with the positive ions.

Super conductors Kamerlingh found that mercury offers zero resistance at 4.2 K. This phenomenon is called super conductivity and the metal is called superconductor. Certain alloys become superconductors at rather high temperature. The resistance of material in the superconducting state is zero and the currents once established in closed superconducting circuits persist for weeks, even though there is no battery in the circuit.

Stretching a wire Consider a wire of length l, radius of cross-section r and is of resistance R. It is stretched to length l'. Let its resistance becomes R '. Assuming volume of material remains constant after stretching or compression, and so

or

pr 2 l

=

l' l

=

Fig. 3.16 Resistance-temperature graph

pr ' 2 l '

for superconductor

r2 r '2 rl

and R ' =

We have,

R

=

\

R'

=

R

l ' r2 ´ l r '2

=

R

l' l' ´ l l

pr

2

rl '

pr '2

224

ELECTRICITY & MAGNETISM or

R'

Also

R'

or

R'

=

æ l 'ö Rç ÷ è lø

=

R

=

ærö Rç ÷ è r 'ø

r2 r '2

2

´

r2 r '2

4

% change in resistance, if there is small change in length DR ´ 100 R

=

2

Dl ´ 100 l

Resistors in series and parallel

(a)

(b) Fig. 3.17

Series : The resistors are said to be in series, if they provide a single path between the points. The current is same in each resistor in series, but p.d. across resistor is proportional to its resistance. Let us consider two resistances R1 and R2 connected in series. For series connection V = V1 + V2 Since i is same in both the resistors, iR = iR1 + iR2 \ or R = R1 + R2 The above equation is true for any number of resistors in series. Thus for n-resistors R = R1 + R2 + ......+Rn Parallel : The resistors are said to be in parallel between the points, if each provides an alternative path between the points. The potential difference is the same across each resistor, but current divide in inverse ratio of their resistances. For parallel connection of resistors i = i1 + i2

or

Fig. 3.18

For n-resistors in parallel

V R

=

V V + R1 R2

1 R

=

1 1 + R1 R2

1 R

=

1 1 1 + + ..... + Rn R1 R2

Effective value of a (i) Fig. 3.19

In series : Suppose a1 and a 2 are the temperature coefficient of resistance of the resistors R1 and R2 respectively. Let a be their effective value. =

At 0°C R01 At t°C R01 (1 + a1t )

R02

R02 (1 + a 2t )

R0 = R01+ R02 R0 (1 + at )

Their equivalent resistance at any temperature in series is R t = R1t + R2t

DC AND DC CIRCUITS or

R0 ( 1 + at )

or

(R01 + R02) ( 1 + at )

or (R01 + R02) + (R01 + R02) at After solving, we get (ii)

a

= R01 ( 1 + a1t ) + R02 ( 1 + a 2t ) = (R01 + R02) + ( R01 a1 + R02 a 2 ) t = (R01 + R02) + (R01 a1 + R02 a ) t 2 =

æ R01 a1 + R02 a 2 ö çè ÷ø R01 + R02

In parallel :

= At any temperature t o C

or

or

or

Fig. 3.20 1 Rt

=

1 1 + R1t R2t

1 R0 (1 + at )

=

1 1 + R01 (1 + a1t ) R02 (1 + a 2 t )

=

1 1 + R01 (1 + a1t ) R02 (1 + a 2 t )

=

1 1 + R01 (1 + a1t ) R02 (1 + a 2 t )

=

1 1 -1 -1 1 + a1t ) + 1 + a 2t ) ( ( R01 R02

1 æ R01 R02 ö çè R + R ÷ø (1 + at ) 01 02 R01 + R02 R01 R02 (1 + at )

( R01 + R02 ) (1 + at )-1 R01 R02

Solving by Binomial theorem, we have 1 1 (1 - at ) + R (1 - at ) R01 02

=

1 1 (1 - a1t ) + R (1 - a2t ) R01 02

or

æ 1 1 ö at ç + ÷ R R è 01 02 ø

=

æ a1 a 2 ö çè R + R ÷ø t 02 01

or

a

=

æ a1 R02 + a 2 R01 ö çè R + R ÷ø 01

02

225