Capacitance

BA

TARGET IIT JEE 2007

XII (ALL)

COHTENTS KEYCONCEPTS EXERCISE-1 EXERCISE-II EXERCISE-III ANSWER KEY

KEY 1.

CAPACITANCE O F A N

CONCEPTS

ISOLATED SPHERICAL CONDUCTOR :

C = 47C G „ R in air

C = 471 e e R in a medium 0

(

This sphere is at infinite distance from all the conductors. The Capacitance C = 47T EQR exists between the surface of the sphere & earth . SPHERICAL CAPACITOR :

It consists of two concentric spherical shells as shown infigure.Here capacitance of region between the two shells is C and that outside the shell is C . We have 471 e ab C = and C = 471 e b b-a Depending on connection, it may have different combinations of C, and -C . t

2

n

2

Q

2

3.

PARALLEL PLATE CAPACITOR : (i)

UNIFORM DI-ELECTRIC M E D I U M :

If two parallel plates each of area A & separated by a distance d are charged with equal & opposite charge Q, then the system is called a parallel plate capacitor & its capacitance is given by, S)6; —A .in a medium C^ = — C= with air as medium r

U

This result is only valid when the electricfieldbetween plates of capacitor is constant, (ii)

M E D I U M PARTLY A I R :

C

=

So A

d-lt-i

When a di-electric slab of thickness t & relative permittivity e is l l l l introduced between the plates of an air capacitor, then the distance between P3 the plates is effectively reduced by irrespective ofthe position of BSSSSii® V ^rJ the di-electric slab . r

(iii)

4.

COMPOSITE M E D I U M :

c=

GA I I -rl r2 0

r3

CYLINDRICAL CAPACITOR :

It consist oftwo co-axial cylinders ofradii a& b, the outer conductor is earthed. The di-electric constant ofthe mediumfilledin the space between the cylinder is e . The capacitance per unit length is C = 2ne-ne mFarad in r

(fe^Bansal Classes

y

r

CAPACITANCE

121

CONCEPT o r VARIATION OF PARAMETERS:

e kA , ifeither ofk, A or d varies in the region between As capacitance ofa parallel plate capacitor isC = the plates, we choose a small dc in between the plates and for total capacitance of system. dx -, If all dC's are in parallel C = } dC If all dC's are in series 1 e k(x)A(x) 0

T

J

6.

COMBINATION (i)

OF

0

CAPACITORS

CAPACITORS I N

SERIES

: :

In this arrangement all the capacitors when uncharged get the same charge Q but the potential difference across each will differ (if the capacitance are unequal). 1 — 1 + 1— + —1 + + 1 (ii)

C

CAPACITORS I N

ENERGY

C

I

+

C

2

+ C

STORED IN A

C|

C2

C3

s 1 jC3,y 1 Q +v

%

+c

+

Q

% 1Cj.c,,vV

PARALLEL :

3

Q

3

When one plate of each capacitor is connected to the positive terminal of the battery & the other plate of each capacitor is connected to the negative terminals of the battery, then the capacitors are said to be in parallel connection. The capacitors have the same potential difference, V but the charge on each one is different (if the capacitors are unequal). eq.

rIMHh v, v, v, Q

CHARGED CAPACITOR

:

Capacitance C, charge Q & potential difference V; then energy stored is U = -1 CV = —1 QV = 1- Q— . This energy is stored in the electrostatic field set up in the di-electric medium between the conducting plates of the capacitor . 2

2

HEAT PRODUCED IN SWITCHING IN CAPACITIVE CIRCUIT

Due to charge flow always some amount of heat is produced when a switch is closed in a circuit which can be obtained by energy conservation as Heat = Work done by battery - Energy absorbed by capacitor. 9.

SHARING O F

CHARGES

:

When two charged conductors of capacitance C & C at potential V & V respectively are connected by a conducting wire, the charge flows from higher potential conductor to lower potential conductor, until the potential of the two condensers becomes equal. The common potential (V) after sharing of charges; C,V C V charge _ q j + q V =netnetcapacitance C, + C C+C charges after sharing qj = C,'V & q = C V. In this process energy is lost in the connecting wire C C (V,-V ) as heat. This loss of energy is U - U = ^ r ^ g s

1+

2

22

2

t

2

2

}

2

2

2

2

10

initial

2

2

eal

REMEMBER :

(i) The energy of a charged conductor resides outside the conductor in its EF, where as in a condenser it is stored within the condenser in its EF. (ii) The energy of an uncharged condenser = 0 . (iii) The capacitance of a capacitor depends only on its size & geometry & the di-electric between the conducting surface .(i.e. independent ofthe conductor, like, whether it is copper, silver, gold etc)

250 ps, I = - 0.1 i -4000(t-250)xi(r

0.04 0.015

00t

6

e

Q.18

•t(xIO^s)

a m p ;

-o.n

400 ^ - — P €

EXERCISE # III

Q.l

(i) 0.2 x 10"

Q.2

4.425 x 10~ Ampere

Q.5

F,

8

'

6

Q.9

5

5

9

QA = 90

pC,

Q B = 150

V48

Q

1.2 x lO" J ; (ii) 4.84 x 10" J ; (iii) 1.1 x 10"

2^9A

C

&

Q.3

pC, Q

C

= 210

q.4

B

pC, UJ = 4 7 . 4 MJ, U

Q.7

D

Q.10

Q0 =

F

Q.8 CVR,

Ri+R2

5

J

C K ^ /n K, (Ka-KO K, = 18 MJ

^ 2 e0

R1+R2 anda=

2 s0

Q.ll

XII (ALL)

quesjjommm.

R ifE,=E (B)C R.C,

(D) f

2

1

2

< ^ 2 1 Q.13 Aparallel plate capacitor is charged by connecting it to a battery. The battery is disconnected and the plates of the capacitor are pulled apart to make the separation between the plates twice. Again the capacitor is connected to the battery (with same polarity) then (A) Chargefromthe batteryflowsinto the capacitor after reconnection (B) Chargefromcapacitorflowsinto the battery after reconnection. (C) The potential difference between the plates increases when the plates are pulled apart. (D) After reconnection ofbattery potential difference between the plate will immediately becomes half of the initial potential difference. (Just after disconnecting the battery) Q. 14 The plates of a parallel plate capacitor with no dielectric are connected to a voltage source. Now a dielectric of dielectric constant K is inserted tofillthe whole space between the plates with voltage source remaining connected to the capacitor. (A) the energy stored in the capacitor will become K-times (B) the electricfieldinside the capacitor will decrease to K-times (C) the force of attraction between the plates will increase to K -times (D) the charge on the capacitor will increase to K-times 1

C

2

Q. 15 Four capacitors and a batteiy are connected as shown. The potential drop across the 7 pF capacitor is 6 V. Then the : J (A) potential difference across the 3 pF capacitor is 10 V (B) charge on the 3 pF capacitor is 42 pC (C) e.m.f. of the battery is 3 0 V (D) potential difference across the 12 pF capacitor is 10 V.

H 3.9(.IF

J7nF "puF

Q. 16 A circuit shown in the figure consists of a battery of emf 10 V and two capacitance C, and C of capacitances 1.0 pF and 2.0 pF respectively. The potential difference V - V is 5 V (A) charge on capacitor Cj is equal to charge on capacitor C Ao—| |—| | — | | o B (B) Voltage across capacitor Cj is 5V. c' e q, (C) Voltage across capacitor C is 10 V (D) Energy stored in capacitor C. is two times the energy stored in capacitor C . 2

A

B

2

2

2

Q.17 A capacitor C is charged to a potential difference V and batteiy is disconnected. Now if the capacitor plates are brought close slowly by some di stance: (A) some +ve work is done by external agent (B) energy of capacitor will decrease (C) energy of capacitor will increase (D) none of the above

(fe Bansal Classes

Question Bank on Capacitance [13]

Q.18 The capacitance of a parallel plate capacitor is C when the region between the plate has air. This region is nowfilledwith a dielectric slab of dielectric constant k. The capacitor is connected to a cell of emf E, and the slab is taken out (A) charge CE(k - 1 ) flows through the cell (B) energy E C(k - 1) is absorbed by the cell. (C) the energy stored in the capacitor is reduced by E C(k - 1 ) (D) the external agent has to do ^E C(k -1) amount ofwork to take the slab out. 2

2

2

Q.19 Two capacitors of capacitances 1 pF and 3 pF are charged to the same voltages 5 V. They are connected in parallel with oppositely charged plates connected together. Then: (A) Final common voltage will be 5 V (B) Final common voltage will be 2.5 V (C) Heat produced in the circuit will be zero. (D) Heat produced in the circuit will be 37.5 pJ Q. 20 The two plates X and Y of a parallel plate capacitor of capacitance C are given a charge of amount Q each. X is now joined to the positive terminal and Yto the negative terminal of a cell of emfE = Q/C. (A) Charge of amount Q willflowfromthe negative terminal to the positive terminal ofthe cell inside it (B) The total charge on the plate X will be 2Q. (C) The total charge on the plate Y will be zero. (D) The cell will supply CE amount of energy. 2

Q.21 A dielectric slab is inserted between the plates of an isolated charged capacitor. Which of the following quantities will remain the same? (A) the electricfieldin the capacitor (B) the charge on the capacitor (C) the potential difference between the plates (D) the stored energy in the capacitor. Q.22 The separation between the plates of a isolated charged parallel plate capacitor is increased. Which of the following quantities will change? (A) charge on the capacitor (B) potential difference across the capacitor (C) energy of the capacitor (D) energy density between the plates. Q.23 Each plate ofa parallel plate capacitor has a charge q on it. The capacitor is now connected to a battery. Now, (A) the facing surfaces of the capacitor have equal and opposite charges. (B) the two plates of the capacitor have equal and opposite charges. (C) the battery supplies equal and opposite charges to the two plates. (D) the outer surfaces ofthe plates have equal charges. Q. 24 Following operations can be performed on a capacitor: X - connect the capacitor to a battery of emf E. Y - disconnect the battery Z - reconnect the battery with polarity reversed. W - insert a dielectric slab in the capacitor (A) In XYZ (perform X, then Y, then Z) the stored electric energy remains unchanged and no thermal energy is developed. (B) The charge appearing on the capacitor is greater after the action XWY than after the action XYW. (C) The electric energy stored in the capacitor is greater after the action WXY than after the action XYW. (D) The electricfieldin the capacitor after the action XW is the same as that after WX. Q.25 A parallel plate capacitor is charged and then disconnectedfromthe source of potential difference. Ifthe plates of the condenser are then moved farther apart by the use of insulated handle, which one of the following is true? (A) the charge on the capacitor increases (B) the charge on the capacitor decreases (C) the capacitance of the capacitor increases (D) the potential difference across the plate increases

(fe Bansal Classes

Question Bank on Capacitance

[13]

Q.26 Aparallel plate capacitor is charged and then disconnected from the source steady E.M.F. The plates are then drawn apart farther. Again it is connected to the same source. Then: (A) the potential difference across the plate increases, while the plates are being drawn apart. (B) the charge from the capacitorflowsinto the source, when the capacitor is reconnected. (C) more charge is drawn to the capacitor from the source, during the reconnection. (D) the electric intensity between the plates remains constant during the drawing apart of plates. Q.27 When a parallel plates capacitor is connected to a source of constant potential difference, (A) all the charge drawnfromthe source is stored in the capacitor. (B) all the energy drawnfromthe source is stored in the capacitor. (C) the potential difference across the capacitor grows very rapidly initially and this rate decreases to zero eventually. (D) the capacity of the capacitor increases with the increase of the charge in the capacitor. Q.28 When two identical capacitors are charged individually to different potentials and connected parallel to each other, after disconnecting themfromthe source: (A) net charge on connected plates is less than the sum of initial individual charges. (B) net charge on connected plates equals the sum of initial charges. (C) the net potential difference across them is differentfromthe sum ofthe individual initial potential differences. (D) the net energy stored in the two capacitors is less than the sum ofthe initial individual energies. Q. 29 Aparallel plate capacitor of plate area A and plate 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 ofthe capacitor so as tofillthe space between the plates. If Q, E and W denote respectively, the magnitude of charge on each plate, the electricfieldbetween the plates (after the slab is inserted) and the work done on the system, in question, in the process of inserting the slab, then e AV s KAV V AV 1 - 1 K Q. 3 0 A parallel plate capacitor is connected to a battery. The quantities charge, voltage, electricfieldand energy associated with the capacitor are given by Q , V , E and U 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>Q (B) V > V (C) E > E (D)U