Capacitance

T Tooppiicc PPaaggeeNNoo.. T Thheeoorryy 0011--0022 E Exxeerrcciissee--11 0033--2211 E Exxeerrcciissee--22 2222--2299 E Exxeerrcciissee--33 3300--3333 E Exxeerrcciissee--44 3344--3366 A AnnsswweerrKKeeyy 3377--3399

Contents

Contents

Syllabus

Syllabus

CAPACITANCE

CAPACITANCE

Capacitance ; Parallel plate capacitor

Capacitance ; Parallel plate capacitor with and without dielectrics ;with and without dielectrics ; Capacitors in series

Capacitors in series and parallel ; Energy stored in and parallel ; Energy stored in a capacitor.a capacitor.

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1.

1.

C

C

APACITANCEAPACITANCE

O

O

FF

A

A

NN

II

SOLATEDSOLATED

S

S

PHERICALPHERICAL

C

C

ONDUCTORONDUCTOR

::

C =

C = 44

!! #

#

₀₀

#

#

_rrR R iin n a a mmeeddiiuumm C C = = 44

!! #

#

₀₀R in airR in air ** ThThis is spsphehere re iis as at it innfifininite te didiststaancnce fe frorom am all ll ththe ce cononduductctorors .s .

** TThhe e ccaappaacciittaanncce e C C = = 44

!! #

#

₀₀R R exists between exists between the surface of the sphere & eathe surface of the sphere & earth .rth .

2.

2.

S

S

PHERICALPHERICAL

C

C

APACITORAPACITOR

::

It consi

It consists of two consts of two concentricentric spheric spherical shelcal shells as showls as shown in figuren in figure. Here capa. Here capacitancitance of regioce of regionn bebe twtweeee n n thth ee two shells is C

two shells is C₁₁ and that outside the shell is C and that outside the shell is C₂₂. We have. We have

C C₁₁ = =

4

4

! #

! #

00

$$

ab

ab

b

b aa

and and CC22 = 4 = 4

! #

! #

00 b b

Depending on connection, it may have different combinations of C

Depending on connection, it may have different combinations of C₁₁ and C and C₂₂..

3.

3.

P

P

ARALLELARALLEL

P

P

LATELATE

C

C

APACITORAPACITOR

::

((ii)) UUNIFORMNIFORM DDII--ELECTRICELECTRIC MMEDIUMEDIUM ::

If two parallel plates each of

If two parallel plates each of area A area A & separated by a distance d are & separated by a distance d are charged withcharged with equal & opposite charge Q,

equal & opposite charge Q, then the system is called a then the system is called a parallel plate capacitor & its capacitance isparallel plate capacitor & its capacitance is given b given byy,, C = C =

#

# #

0 0

#

rrAA d d iin n a a mmeeddiiuumm ;; C C ==

#

#

₀₀AA d

d with with air as air as mediummedium This result is only valid when the electric field between plates of capacitor is

This result is only valid when the electric field between plates of capacitor is constant.constant.

((iiii)) MMEDIUMEDIUM PPARTLYARTLY AAIRIR :: C =C =

#

#

$

$ $$

% 

% 

& 

& 

''

_##

 ₎

 _)**

((

0 0AA d d tt tt rr

When a di-electric slab of thickness t & relative permittivity

When a di-electric slab of thickness t & relative permittivity

#

#

_rr isis introduced between the plate

introduced between the plates of an air capacitor, then s of an air capacitor, then the distance betweenthe distance between the plates is

the plates is effectively reduced byeffectively reduced by tt tt rr

$$

#

#

% 

% 

& 

& 

''

((

 )

 )

**

irrespective of the position ofirrespective of the position of

the di-electric slab . the di-electric slab .

((iiiiii)) CCOMPOSITEOMPOSITE MMEDIUMEDIUM :: C =C =

3 3 rr 3 3 2 2 rr 2 2 1 1 rr 1 1 tt tt tt

A

A

0 0 # # # # # #

++

++

#

#

4.

4.

C

C

YLINDRICALYLINDRICAL

C

C

APACITORAPACITOR

::

It consist of two co-axial cylinders of radii a & b, the outer conductor is earthed . The It consist of two co-axial cylinders of radii a & b, the outer conductor is earthed . The di-electric constant of the medium

di-electric constant of the medium filled in the space between the cylinder isfilled in the space between the cylinder is

#

#

_rr . The capacitance per uni . The capacitance per unit length is t length is C =C =

, , --

_aabb

n

n

2

2

_{00 rr} ! !

#

#

#

#

!!

 

 

Farad

Farad

m

m

..

5.

5.

C

C

ONCEPTONCEPT OFOF VARIATIONVARIATION OFOF PARAMETERSPARAMETERS

::

As capacitance of a parallel plate capacitor isC =

As capacitance of a parallel plate capacitor isC =

#

#

00

kA

kA

d

d

, if either of , if either of k, A k, A or d varies in the ror d varies in the region between theegion between the

plates, we choose a small dc in between the plates and for total capacitance of system. plates, we choose a small dc in between the plates and for total capacitance of system. If all dC's are in series

If all dC's are in series

1

1

//

..

dx

dx

,, IIf af alll l ddCC's 's aarre e iin n ppaarraalllleell CC ==

..

dC

dC

P

1.

1.

C

C

APACITANCEAPACITANCE

O

O

FF

A

A

NN

II

SOLATEDSOLATED

S

S

PHERICALPHERICAL

C

C

ONDUCTORONDUCTOR

::

C =

C = 44

!! #

#

₀₀

#

#

_rrR R iin n a a mmeeddiiuumm C C = = 44

!! #

#

₀₀R in airR in air ** ThThis is spsphehere re iis as at it innfifininite te didiststaancnce fe frorom am all ll ththe ce cononduductctorors .s .

** TThhe e ccaappaacciittaanncce e C C = = 44

!! #

#

₀₀R R exists between exists between the surface of the sphere & eathe surface of the sphere & earth .rth .

2.

2.

S

S

PHERICALPHERICAL

C

C

APACITORAPACITOR

::

It consi

It consists of two consts of two concentricentric spheric spherical shelcal shells as showls as shown in figuren in figure. Here capa. Here capacitancitance of regioce of regionn bebe twtweeee n n thth ee two shells is C

two shells is C₁₁ and that outside the shell is C and that outside the shell is C₂₂. We have. We have

C C₁₁ = =

4

4

! #

! #

00

$$

ab

ab

b

b aa

and and CC22 = 4 = 4

! #

! #

00 b b

Depending on connection, it may have different combinations of C

Depending on connection, it may have different combinations of C₁₁ and C and C₂₂..

3.

3.

P

P

ARALLELARALLEL

P

P

LATELATE

C

C

APACITORAPACITOR

::

((ii)) UUNIFORMNIFORM DDII--ELECTRICELECTRIC MMEDIUMEDIUM ::

If two parallel plates each of

If two parallel plates each of area A area A & separated by a distance d are & separated by a distance d are charged withcharged with equal & opposite charge Q,

equal & opposite charge Q, then the system is called a then the system is called a parallel plate capacitor & its capacitance isparallel plate capacitor & its capacitance is given b given byy,, C = C =

#

# #

0 0

#

rrAA d d iin n a a mmeeddiiuumm ;; C C ==

#

#

₀₀AA d

d with with air as air as mediummedium This result is only valid when the electric field between plates of capacitor is

This result is only valid when the electric field between plates of capacitor is constant.constant.

((iiii)) MMEDIUMEDIUM PPARTLYARTLY AAIRIR :: C =C =

#

#

$

$ $$

% 

% 

& 

& 

''

_##

 ₎

 _)**

((

0 0AA d d tt tt rr

When a di-electric slab of thickness t & relative permittivity

When a di-electric slab of thickness t & relative permittivity

#

#

_rr isis introduced between the plate

introduced between the plates of an air capacitor, then s of an air capacitor, then the distance betweenthe distance between the plates is

the plates is effectively reduced byeffectively reduced by tt tt rr

$$

#

#

% 

% 

& 

& 

''

((

 )

 )

**

irrespective of the position ofirrespective of the position of

the di-electric slab . the di-electric slab .

((iiiiii)) CCOMPOSITEOMPOSITE MMEDIUMEDIUM :: C =C =

3 3 rr 3 3 2 2 rr 2 2 1 1 rr 1 1 tt tt tt

A

A

0 0 # # # # # #

++

++

#

#

4.

4.

C

C

YLINDRICALYLINDRICAL

C

C

APACITORAPACITOR

::

It consist of two co-axial cylinders of radii a & b, the outer conductor is earthed . The It consist of two co-axial cylinders of radii a & b, the outer conductor is earthed . The di-electric constant of the medium

di-electric constant of the medium filled in the space between the cylinder isfilled in the space between the cylinder is

#

#

_rr . The capacitance per uni . The capacitance per unit length is t length is C =C =

, , --

_aabb

n

n

2

2

_{00 rr} ! !

#

#

#

#

!!

 

 

Farad

Farad

m

m

..

5.

5.

C

C

ONCEPTONCEPT OFOF VARIATIONVARIATION OFOF PARAMETERSPARAMETERS

::

As capacitance of a parallel plate capacitor isC =

As capacitance of a parallel plate capacitor isC =

#

#

00

kA

kA

d

d

, if either of , if either of k, A k, A or d varies in the ror d varies in the region between theegion between the

plates, we choose a small dc in between the plates and for total capacitance of system. plates, we choose a small dc in between the plates and for total capacitance of system. If all dC's are in series

If all dC's are in series

..

#

#

//

))

x

x

((

A

A

))

x

x

((

k 

k 

dx

dx

C

C

1

1

0 0 T T

,, IIf af alll l ddCC's 's aarre e iin n ppaarraalllleell CC_TT ==

..

dC

dC

P

6.

6.

C

C

OMBINATIONOMBINATION

O

O

FF

C

C

APACITORSAPACITORS

::

((ii)) CCAPACITORSAPACITORS IINN SSERIESERIES ::

In this arrangement all the capacitors when unchar

In this arrangement all the capacitors when unchar ged get the same charge Qged get the same charge Q but the potential difference across each will differ (if the capacitance are but the potential difference across each will differ (if the capacitance are unequal). unequal). .. eq eq

C

C

1

1

 =  = 1 1

C

C

1

1

 +  + 2 2 C C 1 1 + + 3 3

C

C

1

1

 + ... +  + ... + n n

C

C

1

1

 .  . ((iiii)) CCAPACITORSAPACITORS IINN PPARALLELARALLEL ::

When one plate of each capacitor is connected to the positive When one plate of each capacitor is connected to the positive te

termirminanal l of of ththe e babatttterery y & & ththe e ototheher r plplatate e ofof eaceach capacih capacitor istor is connected to the negative terminals of the battery, then the connected to the negative terminals of the battery, then the capacitors

capacitors are are said said to be to be in in parallel parallel connection.connection. The

The capcapaciacitortors s havhave e the the same same potpotententiaial l difdiffereferencence,, V bV but tut thehe charge on each one is different (if the capacitors are unequal). charge on each one is different (if the capacitors are unequal). C

C_eq.eq. = C = C₁₁+ C+ C₂₂+ C+ C₃₃+ ... + C+ ... + C_nn . .

7.

7.

E

E

NERGYNERGY

S

S

TOREDTORED

II

NN

A

A C

C

HARGEDHARGED

C

C

APACITORAPACITOR

::

Capacitance C, charge Q & potential difference V

Capacitance C, charge Q & potential difference V ; ; then energy then energy stored isstored is U = U = 11 2 2 CVCV 2 2 _= = 11 2 2 QV =QV = 1 1 2 2 Q Q C C 2 2  . This

 . This energy is stored in the electrostatic field set up energy is stored in the electrostatic field set up in the di-electricin the di-electric medium between the conducting plates of the

medium between the conducting plates of the capacitor .capacitor .

8.

8.

H

H

EAEATT PRODUCEDPRODUCED ININ SWITCHINGSWITCHING ININ CAPACITIVECAPACITIVE CIRCUITCIRCUIT Due to charge flow always some amount of

Due to charge flow always some amount of heat is produced when a switch is closed in a heat is produced when a switch is closed in a circuit which cancircuit which can be obtained by energy conservation as

be obtained by energy conservation as – –

Heat

Heat = = Work done Work done by by batterybattery – – Energy absorbed by capacitor. Energy absorbed by capacitor.

9.

9.

S

S

HARINGHARING

O

O

FF

C

C

HARGESHARGES

::

When two charged conductors of capacitance C

When two charged conductors of capacitance C₁₁ & C & C₂₂ at potential V at potential V₁₁ & V & V₂₂ respectively are connected respectively are connected by a conducting wire, the charge flows from

by a conducting wire, the charge flows from higher potential conductor to lower potential conductor,higher potential conductor to lower potential conductor, until the potential of the two condensers becomes equal. The common potential (V) after sharing of until the potential of the two condensers becomes equal. The common potential (V) after sharing of charges;

charges; V =

V = nneet t cch h ee n

neet t ccapapaacci i ccee arg arg tan tan  = = q q qq C C CC 1 1 22 1 1

++

22  =  = C C V V C C VV C C CC 1 1 1 1 2 2 22 1 1 22

++

++

.. charges after

charges after sharing sharing qq₁₁ =  = CC₁₁V V & & qq₂₂ =  = CC₂₂VV. In this proces. In this proces s energy is lost in the connecting s energy is lost in the connecting wire as heatwire as heat

. This l

. This loss of oss of energy energy is is UU_{initialinitial}

$$

 U U_realreal = =

,,

--C C CC C C CC 1 1 22 1 1 22 2 2

++

(V(V11

$$

 V V22)) 2 2 _. .

PART - I : OBJECTIVE QUESTIONS

PART - I : OBJECTIVE QUESTIONS

** Marked Questions are having more than one correct option.Marked Questions are having more than one correct option.

Section (A) : Definition of capacitance, Circuits with capacitor and use of KCL and KVL

Section (A) : Definition of capacitance, Circuits with capacitor and use of KCL and KVL

A-1.

A-1. In the figure initial status of capacitor and their connection is shown. Which of the followingIn the figure initial status of capacitor and their connection is shown. Which of the following is incorrect about this circuit :

is incorrect about this circuit :

(A) Final charge on

(A) Final charge on each capacitor will be zeroeach capacitor will be zero (B) Final total electrical energy of the

(B) Final total electrical energy of the capacitors will be zerocapacitors will be zero (C) Total charge flown fr

(C) Total charge flown fr om A om A to D is 30to D is 30µCµC

(D) Total charge flown fro

(D) Total charge flown fro m A m A to D isto D is – – 30µC 30µC

A-2.

A-2. One plate of a capacitor is connected with a spring as sOne plate of a capacitor is connected with a spring as shown in figure. Area of both the plates is A. In steadyhown in figure. Area of both the plates is A. In steady state separation between the plates is 0.8 d (spring was unstretched and

state separation between the plates is 0.8 d (spring was unstretched and the distance between the platesthe distance between the plates was d when the capacitor was uncharged). The

was d when the capacitor was uncharged). The force constant of force constant of the spring is approximately-the spring is

approximately-(A) (A) ₃₃ 2 2 o o d d AE AE 4 4åå (B) (B) 22 0 0 d d AE AE 5 5 .. 2 2

00

(C) (C) ₃₃ 2 2 o o Ad Ad E E 6 6åå (D) (D) ₃₃ 3 3 o o 2d 2d AE AE å å A-3.

A-3. A circuit has a section AA circuit has a section AB shown in the figure. The emB shown in the figure. The emf of the sf of the source equalsource equals

00

 = 10V = 10V, the capacitor , the capacitor capacitancescapacitances are equal to C

are equal to C₁₁ = 1.0 = 1.0

11

F and CF and C₂₂ = 2.0 = 2.0

11

FF, the , the potential differencepotential difference

22

_AA

$$ 22

_BB = 5.0V. The voltage across each = 5.0V. The voltage across each capacitor are capacitor are (A) V (A) V₁₁ == 3 3 V V 5 5 ,, V V₂₂ = = 3 3 V V 10 10 (B) V (B) V₁₁ = = 3 3 V V 10 10 , V , V₂₂ == 3 3 V V 10 10 (C) V (C) V₁₁ = = 3 3 V V 10 10 , V , V₂₂ = = 3 3 V V 5 5 (D) V (D) V₁₁ = = 3 3 V V 5 5 , V , V₂₂ = = 3 3 V V 5 5

A-4. A 1µF capacitor is connected in the circuit shown below. The e.m.f. of the cell is 3 volts and internal

resistance is 0.5 ohms. The resistors R₁ and R₂ have values 4 ohms and 1 ohm respectively. The charge on the capacitor in steady state must be :

(A) 2

1

C (B) 1

1

C (C) 1.33

1

C (D) zero

A-5. A capacitor of capacitance C is charged to a potential difference V from a cell and then disconnected from it. A charge +Q is now given to its positive plate. The potential difference across the capacitor is now :

(A) V (B) V + C Q (C) V + C 2 Q (D) V – C 2 Q

A-6. In the given arrangement of capacitors 6µC charge is added to point A, find the charge on upper capacitor:

2C A C

3C

(A) 3 µC (B) 1 µC (C) 2 µC (D) 6 µC

A-7. In the circuit shown, switch S₂ is closed first and is k ept closed for a long time. Now S₁ is closed. Just after that instant the current through S₁ is:

(A)

0

R₁  towards right (B)

0

R₁  towards left (C)zero (D) 2 1

0

R

A-8. Initially switch S is connected to position 1 for a long time. The net amount of heat generated in the circuit after it is shifted to position 2 is

A-9. A parallel plate capacitor having capacitance C is connected to a source of constant emf E. Which of the following statements is correct ?

(A) Net charge supplied by the battery to the capacitor is equal to CE. (B) Net charge supplied by the battery to the capacitor is equal to 2 CE. (C) Net charge supplied by the battery to capacitor is equal to zero. (D) None of these.

A-10.* Two capacitors of capacitances 1µF and 3µF are charged to the same voltages 5V. They are connected in

parallel with oppositely charged plates connected together. Then : (A) Final common voltage will be 5 V

(B) Final common voltage will be 2.5 V (C) Heat produced in the circuit will be zero . (D) Heat produced in the circuit will be 37.5

1

J.

A-11. A capacitor of capacitance C is charged to a potential difference V₀. The charging battery is disconnected and the capacitor is connected to a capacitor of unknown capacitance C_x. The P.D. across the combination is V. The value of C_x should be :

(A) V ) V V ( C ₀

 $

(B) C(V V )0 V

$

(C) 0 V CV (D) V CV₀

A-12. The capacitance of capacitor of plate areas A₁ and A₂ (A₁ < A₂) at a distance d is :

(A) 0A1 d

0

(B) 0A2 d

0

(C) 0(A1 A )2 2d

0

+

(D) 0 A A1 2 d

0

A-13. In the circuit shown, the energy stored in 1

1

F capacitor is

(A) 40

1

J (B)64

1

J

(C) 32

1

J (D)none

A-14. If charge on left plane of the 5

1

F capacitor in the circuit segment shown in the figure is –20

1

C, the charge on

the right plate of 3

1

F capacitor is :

(A) +8.57

1

C (B) –8.57

1

C (C) +11.42

1

C (D) –11.42

1

C

A-15. The plates S and T of an uncharged parallel plate capacitor are connected across a battery. The battery is then disconnected and the charged plates are now connected in a system as shown in the figure. The system shown is in equilibrium. All the strings are insulating and massless. The magnitude of charge on one of the capacitor plates is: [Area of plates = A]

(A)

2

mgA

#

₀ (B)

k 

mgA

4

#

₀ (C)

mgA

#

₀ (D)

k 

mgA

2

#

₀

A-16. A parallel plate capacitor has an electric field of 105_{V/m between the plates. If the charge on the capacitor}

plate is 1

1

C, then the force on each capacitor plate is

(A) 0.1Nt (B) 0.05Nt (C) 0.02Nt (D) 0.01Nt

A-17. A capacitor is connected to a battery. The force of attraction between the plates when the separation between them is halved

(A) remains the same (B) becomes eight times (C) becomes four times (D) becomes two times

Section (B) : Combination of capacitors

B-1. In the figure shown the equivalent capacitance between 'A' and 'B' is :

(A) 3.75 F (B) 2 F (C) 21 F (D) 16 F

B-2. N identical capacitors are connected in parallel to a potential difference V. These capacitors are then reconnected in series such that positively charged plate of one capacitor is connected to negatively charged plate of the other, their charges being left undisturbed. The potential difference obtained is :

(A) zero (B) (N - 1) V (C) N V (D) N2_V

B-3. 10 identical capacitors are connected as shown. The capacitance of each capacitor is 30

1

F. Find the equivalent capacitance between A and B.

A B

(A) 30

1

F (B)60

1

F (C) 120

1

F (D)

3

B-4. The equivalent capacitance between point A and B is:

B-5. Consider the circuit shown in the figure. Charge stored in capacitor of capacitance 2 C   is C  V  C  2 C  C  (A) CV  (B) C V  4 (C) V  C  2 (D) 2CV 

B-6. Fig (a) Shows two capacitors connected in series and joined to a battery. The graph in fig (b) shows the variation in potential as one moves from left to right on the branch containing the capacitors if

-(A) C₁ > C₂ (B) C₁ = C₂ (C) C₁ < C₂

(D) The information is not sufficient to decide the relation between C₁ and C₂

B-7. In the circuit shown, a potential difference of 60V is applied across AB. The potential difference between the point M and N is

(A) 10 V (B) 15 V (C) 20 V (D) 30 V

B-8. In the circuit shown in figure, the ratio of charges on 5

1

F and 4

1

F capacitor is :

(A) 4/5 (B)3/5

(C) 3/8 (D)1/2

B-9. On each side of a polygon of n  sides a capacitor of capacitanceC  is placed as shown in figure. Equivalent capacitance across A  and B is

A C  B  C  C  (A) n  C  n  1) (

$

(B) 1

$

n  nC  (C) (n  – 1)C  (D)nC 

B-10. From a supply of identical capacitors rated 8

1

F, 250 V, the minimum number of capacitors required to form a composite 16

1

F, 1000 V is :

(A)2 (B)4 (C)16 (D)32

B-11. A, B, C, D, E, F are c onducting plates each of area A and any two consecutive plates separated by a distance d. The net energy stored in the system after the switch S is closed is:

(A) 0 V2 d 2 A 3

0

(B) 0 V2 d 12 A 5

0

(C) 0 V2 d 2 A

0

(D) 0 V2 d A

0

B-12. In the circuit shown, a potential difference of 60V is applied across AB. The potential difference between the point M and N is

(A)10V (B)15V

(C)20V (D)30V

B-13. In the circuit shown in figure, the ratio of charges on 5

1

F and 4

1

F capacitor is :

(A) 4/5 (B)3/5

(C) 3/8 (D)1/2

B-14. The minimum number of capacitors each of 3

1

F required to make a circuit with an equivalent capacitance 2.25

1

F is

(A)3 (B)4 (C)5 (D)6

B-15. From a supply of identical capacitors rated 8

1

F, 250 V, the minimum number of capacitors required to form a composite 16

1

F, 1000 V is :

(A)2 (B)4 (C)16 (D)32

B-16. What is the equivalent capacitance of the system of capacitors between A & B

(A) 7

6C (B) 1.6 C (C)C (D)None

B-17. Two capacitor having capacitances 8

1

F and 16

1

F have breaking voltages 20 V and 80 V. They are combined in series. The maximum charge they can store individually in the combination is

B-18. Three plates A, B and C each of area 0.1 m2 _{are separated by 0.885 mm from each other as shown in the}

figure. A 10 V battery is used to charge the system. The energy stored in the system is

(A) 1

1

J (B)10 –1

1

_{J (C)10} –2

1

_{J (D)10} –3

1

_J

B-19. Five conducting parallel plates having area A and separation between them d, are placed as shown in the figure. Plate number 2 and 4 are connected wire and between point A and B, a cell of emf E is connected. The charge flown through the cell is

(A)

d

AE

4

3

0

₀ (B)

d

AE

3

2

0

₀ (C)

d

AE

4

0

₀ (D)

d

2

AE

0

0

B-20. Three long concentric conducting cylindrical shells have radii R, 2R and

2

2

R. Inner and outer shells are connected to each other. The capacitance across middle and inner shells per unit length is:

(A)

2

n

3

1

0 l

#

(B)

2

n

6

₀ l

#

!

(C)

2

n

0 2l

#

!

(D) None

B-21. Four metallic plates arearranged as shown in the figure. If the distance between each plate then capacitance of the given system between points A and B is (Given d << A)

(A) d A 0

0

(B) d A 2

0

₀ (C)

d

A

3

0

₀ (D)

d

A

4

0

₀

B-22. Find the equivalent capacitance across A & B

(A)

3

28

1

f (B)

2

15

1

F (C) 15

1

F (D)none

B-23. The diagram shows four capacitors with capacitances and break down voltages as mentioned. What should be the maximum value of the external emf source such that no capacitor breaks down?

[Hint: First of all find out the break down voltages of each branch. After that compare them.]

(A ) 2.5 kV (B) 10/3kV

(C)3kV (D)1kV

B-24. Three capacitors 2

1

F, 3

1

F and 5

1

F can withstand voltages to 3V, 2V and 1V respectively. Their series combination can withstand a maximum voltage equal to

Section (C) : Equation of charging and discharging

C-1. In the circuit shown the capacitor is initially uncharged. The charge passed through an imaginary circular loop parallel to the plates (also circular) and having the area equal to half

of the area of the plates, in one time constant is:

(A) 0.63

0

C (B) 0.37

0

C (C)

0

 C

2 (D) zero

C-2. An uncharged capacitor is connected in series with a resistor and a battery. The charging of the capacitor starts at t = 0. The rate at which energy in capacitor is stored :

(A) first increases then decreases (B) first decreases then increases

(C) remains constant (D) continuously decreases

C-3.* The figure shows, a graph of the current in a discharging circuit of a capacitor through a resistor of resistance 10

4

.

(A) The initial potential difference across the capacitor is 100 volt. (B) The capacitance of the capacitor is ₁₀1_!n2F..

(C) The total heat p roduced in the circuit will be

2 n 500

!  joules.

(D) The thermal power in the resistor will decrease with a time constant

2 n 2

1

!  second.

C-4. Three identical capacitors are given a charge Q each and they are then allowed to discharge through resis-tance R₁, R₂ and R₃separately. Their charges, as a function of time are shown in the graph below. The smallest of the three resistances is

C-5. A graph between current & time during charging of a capacitor by a battery in series with a resistor is shown. The graphs are drawn for two circuits. R1, R2, C1, C2 and V1V2 are the values of resistance, capacitance and

EMF of the cell in the two circuits. If only two parameters (out of resistance, capacitance, EMF) are different in the two circuits. What is /are the correct option(s)

(A) V1 = V2; R1 > R2, C1> C2 (B) V1 > V2, R1 > R2 ; C1 = C2

(C) V1 < V2, R1< R2, C1 = C2 (D) V1 < V2, C1< C2, R1 = R2

C-6.* Capacitor C₁ of capacitance 1 mircofarad and capacitor C₂ of capacitance 2 microfarad are separately charged fully by a common battery. The two capacitors are then separately allowed to discharge through equal resistors, at time t = 0 :

(A) the current in each of two discharging circuits at t = 0 are equal and non-zero. (B) The current in the two discharging circuits at t = 0 are equal

(C) The currents in the two discharging circuits at t = 0 are unequal.

(D) Capacitor C₁ loses 50% of its initial charge sooner than C₂ loses 50% of its initial charge.

C-7. A capacitor of capacitance C is charged to a potential difference V from a cell and then disconnected from it. A charge +Q is now given to its positive plate. The potential difference across the capacitor is now

(A) V (B) V +

C

Q

(C) V +

C

2

Q

(D) V –

C

Q

, if V < CV

C-8. A capacitor of capacitance C is initially charged to a potential difference of V volt. Now it is connected to a battery of 2V Volt with opposite polarity. The ratio of heat generated to the final energy stored in the capacitor will be

(A) 1.75 (B) 2.25 (C) 2.5 (D) 1/2

C-9. A conducting body 1 has some initial charge Q, and its capacitance is C. There are two other conducting bodies, 2 and 3, having capacitances : C₂ = 2C and C₃

5 3

. Bodies 2 and 3 are initially uncharged. "Body 2 is touched with body 1. Then, body 2 is removed from body 1 and touched with body 3, and then removed." This process is repeated N times. Then, the charge on body 1 at the end must be

(A) Q/3N _{(B) Q/3}N –1 _{(C) Q/N}3 _{(D) None}

C-10. A charged capacitor is allowed to discharge through a resistance 2

4

 by closing the switch S at the instant t = 0. At time t =l n 2

1

s, the reading of the ammeter falls half of its initial value. The resistance of the amm eter equal to

(A)0 (B)2

4

(C)

3

(D) 2M

4

C-11. A capacitor C = 100

1

F is connected to three resistor each of resistance 1 k

4

 and a battery of emf 9V. The switch S has been closed for long time so as to charge the capacitor. When switch S is opened, the capacitor discharges with time constant

(A)33ms (B)5ms

C-12. In the circuit shown in figure C₁=2C₂. Switch S is closed at time t=0. Let i₁ and i₂ be the currents flowing through C₁ and C₂ at any time t, then the ratio i₁ / i₂

(A) is constant

(B) increases with increase in time t (C) decreases with increase in time t (D) first increases then decreases

C-13. In the circuit shown, when the key k is pressed at time t = 0, which of the following statements about current I in the resistor AB is true

(A) I = 2mA at all t

(B) I oscillates between 1 mA and 2mA (C) I = 1 mA at all t

(D) At t = 0, I = 2mA and with time it goes to 1 mA

C-14. In the R –C circuit shown in the figure the total energy of 3.6 ×10 –3 J is dissipated in the 10

4

 resistor when

the switch S is closed. The initial charge on the capacitor is

(A) 60

1

C (B) 120

1

C (C)60

₂

1

C (D)

2

60

1

C

C-15. A charged capacitor is allowed to discharge through a resistor by closing the key at the instant t =0. At the instant t = (ln 4)

1

s, the reading of the ammeter falls half the initial value. The resistance of the ammeter is equal to

(A) 1 M

4

(B) 1

4

(C )2

4

(D) 2M

4

C-16. In the circuit shown, the cell is ideal, with emf = 15 V. Each resistance is of 3

4

. The potential difference across the capacitor is

Section (D) : Capacitor with dielectric

D-1.* A parallel plate air capacitor is conn ected to a battery. The quantities charge, voltage, electric f ield and energy associated with this capacitor are given by Q₀, V₀, E₀ and U₀ respectively. A dielectric slab is now introduced to fill the space between the plates with battery still in connection. The corresponding quantities now given by Q, V, E and U are related to the previous one as

(A) Q > Q₀ (B) V > V₀ (C) E > E₀ (D) U > U₀

D-2. A parallel plate capacitor (without dielectric) is char ged and disconnected from a battery. Now a dielectric is inserted between the plates. The electric force on a plate of the capacitor will:

(A) decrease (B) increase

(C) remain same (D) depends on the width of the dielectric.

D-3. Two parallel plate capacitors of capacitances C and 2C are connected in parallel and charged to a potential difference V by a battery. The battery is then disconnect ed and the space between the plates of capacitor C is completely filled with a material of dielectric constant K. The potential difference across the capacitors now becomes.

(A) 1 K V

+

(B) K 2 V 2

+

(C) K 2 V 3

+

(D) K 3 V 3

+

D-4. Two conductors of thickness d  are inserted inside a parallel capacitor of thickness 3d  and capacitance

C ₀. The capacitance of new arrangement is :

d d  3d  (A) C ₀ (B) 2C ₀ (C) 3C ₀ (D) 3 0 C 

D-5.* A parallel plate capacitor of plate area A and plate seperation d is charged to potential difference V and then the battery is disconnected. A slab of dielectric constant K is then inserted between the plates of the capacitor so as to fill the space between the plates. If Q, E and W denote respectively, the magnitude of charge on each plate, the electric field between the plates (after the slab is inserted) and the work done on the system, in question, in the process of inserting the slab, then

(A) Q =

d

AV

0

0

(B) Q =

d

KAV

0

0

(C) E =

_K

_d

V

(D) W = –

*

 )

 (

'

& 

%  $

0

K

1

1

d

2

AV

2 0

D-6. A parallel plate capacitor made from two square plates of side a  and separation b  (<< a ) is charged by a battery of emfV . After disconnecting the battery, a conductor of thickness slightly less than b  is inserted as shown in figure. The potential energy of the system is

b  a  x  (A) 2 3 0 ) ( 2b a  x V  a 

$

0

(B) 2 2 0 ) ( 2 a  x V  a 

$

0

(C) 2 2 0 _{( )} 2b  a  x V  a 

$

0

(D) ₂ 2 3 0 2 ) ( V  b  x  a  a 

$

0

D-7. A parallel plate capacitor without any dielectric has capacitance C_0. A dielectric slab is made up of two dielectric slabs of dielectric constants K and 2K and is of same dimensions as that of capacitor plates and both the parts are of equal dimensions arranged serially as shown. If this dielectric slab is introduced (dielectric K enters first) in between the plates at constant speed, then variation of capacitance with time will be best represented by :

(A) (B) (C) (D)

D-8. Two parallel plate air filled capacitors each of capacitance C, are joined in series to a battery of emf V. The space between the plates of one of the capacitors is then completely filled up with a uniform dielectric having dielectric constant K. The quantity of charge which flows through the battery is

-(A)

**

 )

 (

''

& 

% 

+

1 K 1  – K 2 CV (B)

**

 )

 (

''

& 

%  +

1  – K 1 K 2 CV (C) CV

**

 )

 (

''

& 

% 

+

1 K 1  – K (D) CV

**

 )

 (

''

& 

%  +

1  – K 1 K

D-9. Two identical capacitor C₁ and C₂ are connected in series with a battery. They are fully charged. Now a dielectric slab is inserted between the plates of C₂. The potential difference across C₁ will :

(A) increase (B) decrease

(C) remain same (D) depend on internal resistance of the cell

D-10. A parallel plate capacitor has two layers of dielectric as shown in figure. This capacitor is connected across a battery. The graph which shows the variation of electric field (E) and distance (x) from left plate.

D-11. In the figure shown a parallel plate capacitor has a dielectric of width d/2 and dielectric constant K = 2. The other dimensions of the dielectric are same as that of the plates. The plates P₁ and P₂ of the capacitor have area 'A' each. The energy of the capacitor is :

(A) d 3 AV2 0

#

(B) d AV 2

#

₀ 2 (C) d AV 2 3

#

0 2 (D) d 3 AV 2

#

₀ 2

D-12. In the figure a capacitor of capacitance 2µF is connected to a cell of emf 20 volt. The plates of the

capacitor are drawn apart slowly to double the distance between them. The work done by the external agent on the plates is :

(A) – 200 µJ (B) 200µJ (C) 400µJ (D) – 400 µJ

D-13. A dielectric slab of relative permittivity

0

_r and thickness t is inserted into the capacitor. Then,

(A) the capacitance of the system increases by 0

r A 1 t 1 –

0

%

(

'

₀

*

&

)

(B) free r bound r q q –1

0

/

0

(C) the fraction change in the energy stored is

0

_r –1

(D) the plates are moved apart by a relative distance t

r 1 1 –

%

(

'

*

0

D-14. A parallel plate capacitor has two layers of dielectric as shown in figure. This capacitor is connected across a battery. The graph which shows the variation of electric field (E) and distance (x) from left plate.

(A) (B) (C) (D)

D-15. A capacitor stores 60

1

C charge when connected across a battery. When the gap between the plates is filled with a dielectric , a charge of 120

1

C flows through the battery. The dielectric constant of the material inserted is :

(A)1 (B)2 (C)3 (D) none

D-16. In the adjoining figure, capacitor (1) and (2) have a capacitance‘C’ each. When the dielectric of dielectric

consatnt K is inserted between the plates of one of the capacitor, the total charge flowing through battery is (A)

1

K

KCE

+

fromBtoC (B)

K

1

KCE

+

from C to B (C)

)

1

K

(

2

CE

)

1

K

(

+

$

from B to C (D)

)

1

K

(

2

CE

)

1

K

(

+

$

from C to B

D-17. The distance between plates of a parallel plate capacitor is 5d. Let the positively charged plate is at x=0 and negatively charged plate is at x=5d. Two slabs one of conductor and other of a dielectric of equal thickness d are inserted between the plates as shown in figure. Potential versus distance graph will look like :

D-18. The distance between the plates of a charged parallel plate capacitor is 5 cm and electric field inside the plates is 200 Vcm –1_{. An uncharged metal bar of width 2 cm is fully immersed into the capacitor. The length}

of the metal bar is same as that of plate of capacitor. The voltage across capacitor after the immersion of the bar is :

(A) zero (B) 400 V (C) 600 V (D) 100 V

D-19. Condenser A has a capacity of 15

1

F

 when it is filled with a medium of dielectric constant 15. Another condenser B has a capacity 1

1

F

 with air between the plates. Both are charged separately by a battery of 100V . After charging, both are connected in parallel without the battery and the dielectric material being removed. The common potential now is

(A) 400V (B) 800V (C) 1200V (D) 1600V

D-20. Two identical capacitors 1 and 2 are connected in series to a battery as shown in figure. Capacitor 2 contains a dielectric slab of dielectric constant k as shown. Q₁ and Q₂ are the charges stored in the capacitors. Now the dielectric slab is removed and the corresponding charges are Q’₁ and Q’₂. Then

(A)

k 

1

k 

Q

Q

1 1

_/

+

6

(B)

2

1

k 

Q

Q

2 2

_/

+

6

(C) _2k  1 k  Q Q 2 2

_/

+

6

(D)

2

k 

Q

Q

1 1

_/

6

D-21. Four identical plates 1, 2, 3 and 4 are placed parallel to each other at equal distance as shown in the figure. Plates 1 and 4 are joined together and the space between 2 and 3 is filled with a dielectric of dielectric constant k = 2. The capacitance of the system between 1 and 3 & 2 and 4 are C₁ and C₂ respectively. The

ratio 2 1

C

C

 is : (A)

3

5

(B)1 (C)

5

3

(D)

7

5

PART - II : MISLLANEOUS QUESTIONS

1. COMPREHENSION :

COMPREHENSION # 1

Capacitor C₃ in the circuit is a variable capacitor (its capacitance can be varied). Graph is plotted between potential difference V₁ (across capacitor C₁) versus C₃. Electric potential V₁ approaches on asymptote of 10 V as C₃

5 3

.

C1

C2 C3

V

1. The ratio of the capacitance 1

2 C C  will be (A) 2 3 (B) 4 3 (C) 3 4 (D) 3 2 2. The value of C₃ for which potential across C₁ will become 8 V

(A) 1.5 C₁ (B) 2.5 C₁ (C) 3.5 C₁ (D) 4.5 C₁

3. The ratio of energy stored in capacitor C₁ to that of total energy when C₃

5 3

(A) Zero (B) 1

3 (C) 1 (D) Data insufficient

COMPREHENSION # 2

The potential energy of a charged conductor or a capacitor is stored in electric field. The energy per unit volume is called the energy density (u). Energy density in a dielectric media is given by u = 0KE2

2 1

₀

. This relation shows that the energy stored per unit volume depends onE2_{. If E is the electric field in a space of}

volume dV, then total stored energy in an electrostatic field is given by U = K E dV 2

1 2

0

.

0

and if E is uniform

throughout the volume, then total energy stored can be given by K E V 2

1 2

0

0

.

4. The energy density in the electric field created by a point charge falls off with distance from the point charge as. (A) r 1 (B) ₂ r 1 (C) ₃ r 1 (D) ₄ r 1

5. A charges q₁ is placed at the centre of a spherical conducting shell of radius R. Conducting shell has a total charge q₂. Electrostatic potential energy of the system is

-(A) q ₈ 2q_R1q2 2 1

!0

+

(B) q ₈ 2q_R1q2 2 2

!0

+

(C) q₄ q1_Rq2 2 1

!0

+

(D) q₄ q1_Rq2 2 2

!0

+

6. Let u_a and u_d represent the energy density in air and in a dielectric respectively, for the same field in both. Let K = dielectric constant. Then

(A) u_a = u_d (B) u_a = Ku_d (C) u_d = Ku_a (D) u_a= (K –1)u_d

7. A parallel plate capacitor is connected to a battery. The plates are pulled apart with a uniform speed. If x is the separation between the plates, then the rate of change of electrostatic energy of the capacitor is propor-tional to-(A)x (B)x2 _(C) x 1 (D) ₂ x 1

COMPREHENSION # 3

The charge across the capacitor in two different RC circuits 1 and 2 are plotted as shown in figure.

8.* Choose the correct statement(s) related to the two circuits. (A) Both the capacitors are charged to the same charge. (B) The emf's of cells in both the circuit are equal.

(C) The emf's of the cells may be different. (D) The emf E₁ is more than E₂

9. Identify the correct statement(s) related to the R₁, R₂, C₁ and C₂ of the two RC circuits. (A) R₁ > R₂ if E₁ = E₂ (B) C₁< C₂ if E₁ = E₂ (C) R₁C₁> R₂C₂ (D)

2 1

R

R

 < 1 2

C

C

COMPREHENSION # 4

In the circuit as shown in figure the switch is closed at t = 0.

10. At the instant of closing the switch

(A) the battery delivers maximum current. (B) no current flows through C

(C) Voltage drop across R₂ is zero.

(D) the current through the battery decreases with time finally becomes zero. 11.* A long time after closing the switch

(A) voltage drop across the capacitor is E. (B) current through the battery is

2 1 R R

E

+

(C) energy stores in the capacitor is

2 2 1 2 R R E R C 2 1

*

*

 )

 (

'

'

& 

% 

+

2. ASSERTION AND REASON

12. Statement-1 : The electrostatic force between the plates of a charged isolated capacitor decreases when dielectric fills whole space between plates.

Statement-2 : The electric field between the plates of a charged isolated capacitance decreases when dielectric fills whole space between plates.

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false.

(D) Statement-1 is false, statement-2 is true.

13. Statement-1 If temperature is increased, the dielectric constant of a polar dielectric decreases whereas that of a non-polar dielectric does not change significantly.

Statement-2 The magnitude of dipole moment of individual polar molecule decreases significantly with increase in temperature.

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.

14. Statement-1 : The heat produced by a resistor in any time t during the charging of a capacitor in a series circuit is half the energy stored in the capacitor by that time.

Statement-2 : Current in the circuit is equal to the rate of increase in charge on the capacitor. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false.

(D) Statement-1 is false, statement-2 is true.

3. MATCH THE COLUMN

15. Consider the situation shown. The switch S is open for a long tim e and then closed. Then:

E S

C C

Column I Column II

(A) Charge flown through battery when S is closed (p)

2

2

CE 

(B) Work done by battery. (q)

2

CE 

(C) Change in energy stored in capacitor. (r)

2

4

CE 

(D) Heat developed in the system.

(t) 8 CE2 (s) 4 CE

16. In the figure shown, area of each plate is A. Match the following : 2d V d 1 2 3 4 5 6 Column-I Column-II

(A) Charge on plate 3 (p) zero

(B) Charge on plate 5 (q) V

(C) Potential difference between plates 2 and 3 (r) 0

A

V

2d

0

(D) Potential difference between plates 2 and 5 (s) 0

A

V

d

0

(t) none of these

17. Arrangements of Parallel Plates Capacitance

(A) (p) 3 A

0

_2d0 (B) (q) 3 A

0

_d0 (C) (r) 2 A0 3d

0

(D) (s) 2 A0 d

0

(t) none of these

4. TRUE & FALSE :

18. If the charge on capacitor is constant, on increasing the separation (still keeping it very less change) between its plates the force between the plates does not change.

19. If the potential difference between two plates of a capacitor is constant, on increasing the plate's separation the electric field remains constant.

5. FILL IN THE BLANKS :

20. Two parallel plate capacitors of capacitances C and 2C are connected in parallel and charged to a potential difference V. The battery is then disconnected and the region between the plates of capacitor C is completely filled with a material of dielectric constant K. T he potential difference across the capacitors now becomes ...

21. The capacity of a conductor ... when an earth connected uncharged conductor is brought near it. 22. Capacity of parallel plate capacitor ... by decreasing the separation between two plates.

MIXED OBJECTIVE

* Marked Questions are having more than one correct option.

1. The plates of a parallel plate capacitor are charged with surface densities

7

₁  and

7

₂respectively. The electric field at points :

(A) inside the region between the plates will be zero

(B) above the upper plate & below the lower plate will be zero (C) every where in the space will be zero

(D) inside the region between the plates will be uniform & non-zero

2. Two large conducting plates A and B have charges Q₁ and Q₂ on them. The charges on the sides 1, 2, 3, and 4 respectively are : (A) q₁ = q₄ = 2 Q Q₁

 +

₂  and q₂= – q₃ = 2 Q Q₁

 $

₂ (B) q₁ = q₃ = 2 Q Q₁

 +

₂  and q₂ = q₄ = 2 Q Q₁

 $

₂ (C) q₂ = q₃ = 2 Q Q₁

 +

₂  and q₁ = q₄ = 2 Q Q₁

 $

₂ (D) q₁ = q₂ = q₃= q₄= 2 Q Q₁

 +

₂

3. A parallel plate capacitor is connected to a battery. The plates are pulled apart with a uniform speed. If X is the separation between the plates, then the rate of change of the electrostatic energy of the capacitor is proportional to :

(A) X2 _{(B)X (C)1/X (D)1/X}2

4. Two metal spheres of radii a and b are connected by a thin wire. Their separation is large compared with their dimensions. The capacitance of this system is :

(A) 4

!#

₀ab (B) 2

!#

₀(a + b) (C) 4

!#

₀(a + b) (D) 4

!#

₀(a2 _{+ b}2_)/2

5.* An uncharged capacitor having capacitance C is connected across a battery of emf V. Now the capacitor is disconnected and then reconnected across the same battery but with reversed polarity. Then :

(A) after reconnection, thermal energy produced in the circuit will be equal to 2CV2_.

(B) after reconnection, thermal energy produced in the circuit will be equal to two-third of the total energy supplied by the battery.

(C) after reconnection, no energy is supplied by the battery.

6.  A parallel plate capacitor is filled with a uniform dielectric. Maximum charge that can be given to it, does not depend upon :

(A) dielectric constant of the dielectric. (B) dielectric strength of the dielectric.

(C) separation between the plates. (D) area of the plates.

7. In the given figure, a capacitor of non-parallel plates is shown. The plates of capacitor are connected by a cell of emf V₀. If

7

  denotes surface charge density and E denotes electric field. Then :

A

B D F V₀

(A)

7

_A >

7

_B (B) E_F > E_D (C) E_F = E_D (D)

7

_A =

7

_B

8. In the circuit, capacitor is initially uncharged. The equivalent resistance will be (in steady – state) :

(A) 1

4

(B) 3

4

(C) 4

4

(D) 5

4

9. In the circuit shown the cells are ideal & of equal e.m.f. , the capacitance of the capacitor is C & the resistance of the resistor is R . X is first joined to Y and then to Z . After a long time the total heat produced in the resistor will be :

(A) equal to the energy finally stored in the capacitor (B) half of the energy finally stored in the capacitor (C) twice the energy finally stored in the capacitor (D) 4 times the energy finally stored in the capacitor .

10.* In the circuit shown, all the capacitors are initially uncharged. When s witch S is closed, a total charge of 12

1

C passes through point A and a charge of 8

1

C passes through point B.

9V S

C1 C2=3 F1

A B

C3 C4=4 F1

(A) Value of capacitance of C₁ is 2

1

F (B) Value of capacitance of C₁ is 4

1

F (C) Value of capacitance of C₃ is 2

1

F (D) Value of capacitance of C₃ is 6

1

F

11. An ideal cell is connected across a capacitor as shown in figure. T he initial separation between the plates of a parallel plate ca pacitor is d. The lower plate is p ulled down with a uniform velocity v. Neglect the resistance of the circuit. Then the variation of charge on capacitor with time is given by

v C E (A) q t (B) q t (C) q t (D) q t

12.* A parallel plate capacitor of capacitance 'C' has charges on its plates initially as shown in the figure. Now at t = 0, the switch 'S' is closed. Select the correct alternative(s) for this circuit diagram.

t=0 -2 c0 0c

0

S A B

(A) In steady state the charges on the outer sur faces of plates 'A' and 'B' will be same in magnitude and sign.

(B) In steady state the charges on the outer sur faces of plates 'A' and 'B' will be same in magnitude and opposite in sign.

(C) In steady state the charges on the inner s urfaces of the plates 'A' and 'B' will be same in magnitude and opposite in sign.

(D) The work done by the cell by the time steady state is reached is 2

C 5

0

2

.

13. An isolated parallel plate capacitor of capacitance C has four surfaces with charges Q₁, Q₂, Q₃ and Q₄ as shown in figure. T he potential difference between the plates is

Q1 Q3

Q2 Q4

(A) Q1

+

Q2₂

+

_CQ3

+

Q4 (B) Q2₂

 +

_CQ3

14. Two metallic spheres of radii a and b are separated by a distance d as shown in figure. The capacity of the system is :

(A) 4

!#

₀ /(1/a + 1/b – 2/d) (B) 2

!#

₀ /(1/a – 1/b + 1/d)

(C) 4

!#

₀ /(1/a + 1/b – 1/d) (D) 4

!#

₀(a + b)

15. A capacitor of capacity C is charged to a steady potential difference V and connected in series with an open key and a pure resistor 'R'. At time t = 0, the key is closed. If I = current at time t, a plot of logI against 't' is as shown in (1) in the graph. Later one of the parameters i.e. V, R or C is changed keeping the other two constant, and the graph (2) is recorded. Then

(A) C is reduced (B) C is increased (C) R is reduced (D) R is increased

16. The distance between plates of a parallel plate capacitor is 5d. Let the positively charged plate is at x=0 and negatively charged plate is at x=5d. Two slabs one of conductor and other of a dielectric of equal thickness d are inserted between the plates as shown in figure. Potential versus distance graph will look like :

(A) (B) (C) (D)

17. A capacitor is charged fully using a cell. With the cell connected, the capacitor plates are slowly pulled apart so that new capacitance becomes half of the original capacitance. Let the work done by pulling agent be w (A) Energy absorbed by the cell will be less than w

(B) Energy absorbed by the cell will be more than w (C) Energy stored in the capacitor will increase by w (D) There will be heat loss in this process.

18.* A parallel plate capacitor of area A and separation d is charged to potential difference V and removed from the charging source. A dielectric slab of constant K = 2, thickness d and area

2 A

 is inserted, as shown in the figure. Let

7

₁ be free charge density at the conductor-dielectric surface and

7

₂ be the charge density at the conductor-vacuum surface.

d A K 72  –72  –71 71

(A) The electric field have the sam e value inside the dielectric as in the free space between the plates. (B) The ratio 2 1

7

7

 is equal to 1 2 . (C) The new capacitance is

d 2

A 3

#

₀

(D) The new potential difference is

3 2

V

19. A and C are concentric conducting spherical shells of radius a and c respectively. A is surrounded by a concentric dielectric medium of inner radius a, outer radius b and dielectric constant k. If sphere A is given a charges Q, the potential at the outer surface of the dielectric is.

(A) ₄ Q_kb 0

!0

(B) 4 ₀ Q

!0

 ₎**  ( '' &  %  $ + ) a b ( k 1 a 1 (C) ₄Q _b 0

!0

(D) None of these

20. If n drops, each of capacitance C and charged to a potential V, coalesce to form a big drop, th e ratio of the energy stored in the big drop to that in each small drop will be

(A) n : 1 (B) n4/3 _{:1 (C) n}5/3 _{: 1 (D) n}2 _{: 1}

21. Figure shows a part of network of a capacitor and resistors. The potential indicated at A, B and C are with respect to the ground. The charge on the capacitor in steady state is

B A C 24 44 84 44 +8V +4V +6V 1 F1 10V (A) 4

1

C (B)6

1

C (C)10

1

C (D)16

1

C

22. A student charges a capacitor in such a manner that it stores energy of 1 J. Now he wants to increase the potential energy to 4 J. He should :

(A) quadruple the potential difference across the capacitor without changing the carge (B) double the potential difference across the capacitor without changing the charge (C) double both the potential difference and charge

(D) double the charge without changing the potential difference

23. The capacitance (C) for an isolated conducting sphere of radius (a) is given by 4

!0

₀a. If the sphere is enclosed with an earthed concentric sphere. The ratio of the radii of the spheres being

)

1

n

(

n

$

 then the

capacitance of such a sphere will be increased by a factor

(A)n (B)

)

1

n

(

n

$

(C)

n

)

1

n

(

$

(D) a . n

24. A parallel plate capacitor is connected to a battery. The quantities charge, voltage, electric field and energy associated with the capacitor are given by Q₀, V₀, E₀ and U₀ respectively. A dielectric slab is introduced between plates of capacitor but battery is still in connection. The corresponding quantities now given by Q, V, E and U related to previous ones are

(A) Q > Q₀ (B) V > V₀ (C) E > E₀ (D) U < U₀

25. In the transient circuit shown the time constant of the circuit is : (A)

3

5

RC (B)

2

5

RC (C)

RC

4

7

(D)

RC

3

7

26. Find heat produced on closing the switch S

(A) 0.0002 J (B) 0.0005 J

(C)0.00075 (D)zero

Multiple Choice Questions :

27. Two capacitors of 2

1

F and 3

1

F are charged to 150 volt and 120 volt respectively. The plates of capacitor are connected as shown in the figure. A discharged capacitor of capacity 1.5

1

F falls to the free ends of the wire. Then

(A) charge on the 1.5

1

F capacitors is 180

1

C (B) charge on the 2

1

F capacitor is 120

1

C

(C) positive charge flows through A from right to left. (D) positive charge flows through A from left to right.

28. In the circuit shown, each capacitor has a capacitance C. The emf of the cell is E. If the switch S is closed (A) positive charge will flow out of the positive terminal of the cell

(B) positive charge will enter the positive terminal of the cell (C) the amount of charge flowing through the cell will be CE. (D) the amount of charge flowing through the cell will be 4/3 CE.

29. In the circuit shown initially C₁, C₂ are uncharged. After closing the switch (A) The charge on C₂ is greater that on C₁

(B) The charge on C₁ and C₂ are the same

(C) The potential drops across C₁ and C₂ are the same

(D) The potential drops across C₂ is greater than that across C₁

30. A circuit shown in the figure consists of a battery of emf 10 V and two capacitance C₁ and C₂ of capacitances 1.0

1

F and 2.0

1

F respectively. The potential difference V_A – V_B is 5V

(A) charge on capacitor C₁ is equal to charge on capacitor C₂ (B) Voltage across capacitor C₁ is 5V.

(C) Voltage across capacitor C₂ is 10 V

(D) Energy stored in capacitor C₁ is two times the energy stored in capacitor C₂.

31. If Q is the charge on the plates of a capacitor of capacitance C, V the potential difference between the plates, A the area of each plate and d the distance between the plates, the force of attraction between the plates is

(A) _**  )  ( ' ' &  %  0 A Q 2 1 0 2 (B) _**  )  ( ' ' &  %  d CV 2 1 2 (C) _**  )  ( ' ' &  %  0₀ 2 A CV 2 1 (D) _**  )  ( ' ' &  %  0 ! 2 0 2 d Q 4 1

32. A capacitor C is charged to a potential difference V and battery is disconnected. Now if the capacitor plates are brought close slowly by some distance :

(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

33. Four capacitors and a battery are connected as shown. The potential drop across the 7

1

F capacitor is 6 V. Then the: (A) potential difference across the 3

1

F capacitor is 10 V

(B) charge on the 3

1

F capacitor is 42

1

C (C) e.m.f. of the battery is 30 V

(D) potential difference across the 12

1

F capacitor is 10 V.

34. The capacitance of a parallel plate capacitor is C when the region between the plate has air. This region is now filled with 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 E2_{C(k – 1) is absorbed by the cell.}

(C) the energy stored in the capacitor is reduced by E2_C(k – 1)

(D) the external agent has to do 1

2E

2_{C(k – 1) amount of work to take the s lab out.}

35. A parallel plate air-core capacitor is connected across a source of constant potential difference. When a dielectric plate is introduced between the two plates then :

(A) some charge from the capacitor will flow back into the source. (B) some extra charge from the source will flow back into the capacitor. (C) the electric field intensity between the two plate does not change. (D) the electric field intensity between the two plates will decrease.

36. A parallel plate capacitor of plate area A and plate seperation d is charged to potential difference V and then the battery is disconnected. A slab of dielectric constant K is then inserted between the plates of the capacitor so as to fill the space between the plates. If Q, E and W denote respectively, the magnitude of charge on each plate, the electric field between the plates (after the slab is inserted) and the work done on the system, in question, in the process of inserting the slab, then

(A) Q =

d

AV

0

0

(B) Q = d KAV 0

0

(C) E = _KV_d (D) W = –

*

 )

 (

'

& 

%  $

0

K 1 1 d 2 AV2 0

37. A parallel plate capacitor has a parallel slab of copper inserted between and parallel to the two plates, without touching the plates. The capacity of the capacitor after the introduction of the copper sheet is :

(A) minimum when the copper slab touches one of the plates. (B) maximum when the copper slab touches one of the plates. (C) invariant for all positions of the slab between the plates.