Summary Capacitors - Electrostatics

72 Electrostatics

Capacitance.
(1) Definition : We know that charge given to a conductor increases it’s potential i.e.,
Q  V  Q  CV

Where C is a proportionality constant, called capacity or capacitance of conductor. Hence
capacitance is the ability of conductor to hold the charge.
Coulomb
(2) Unit and dimensional formula : S.I. unit is  Farad (F)
Volt
Smaller S.I. units are mF, F, nF and pF ( 1mF  10 3 F , 1 F  10 6 F , 1nF  10 9 F ,
1 pF  1 F  10 12 F )

C.G.S. unit is Stat Farad 1 F  9  10 11 Stat Farad . Dimension : [C ]  [M 1 L2 T 4 A 2 ] .

(3) Capacity of an isolated spherical conductor : When charge Q is given to a spherical
conductor of radius R, then potential at the surface of sphere is Q
+ + +
+ +
Q  1  + +
V  k. k   +
R
+
R  4  0  + +
O
+ +
+
Q 1 +
+ + +
Hence it’s capacity C   4  0 R  C  4 πε 0 R  .R
V 9  10 9
in C.G.S. CR

Note : If earth is assumed to be spherical having radius R  6400 km . It’s theortical
1
capacitance C  9
 6400  10 3  711 F . But for all practical purpose capacitance
9  10
of earth is taken infinity.
(4) Energy of a charged conductor : When a conductor is charged it’s potential increases
from 0 to V as shown in the graph; and work is done against repulsion, between charge stored
in the conductor and charge coming from the source (battery). This work is stored as
“electrostatic potential energy”
1
From graph : Work done = Area of graph  QV
2 V

1
Hence potential energy U  QV ; By using Q  CV , we can write
2

1 1 Q2 Q
U QV  CV 2 
2 2 2C
(5) Combination of drops : Suppose we have n identical drops each having – Radius – r,
Capacitance – c, Charge – q, Potential – v and Energy – u.
If these drops are combined to form a big drop of – Radius – R, Capacitance – C, Charge – Q,
Potential – V and Energy – U then –

, Electrostatics 73

(i) Charge on big drop : Q  nq
4 4
(ii) Radius of big drop : Volume of big drop = n  volume of a single drop i.e., R 3  n  r 3 ,
3 3
R  n1/ 3 r
(iii) Capacitance of big drop : C  n c
Q nq
(iv) Potential of big drop : V  V  n 2 /3 v
C n c
1 1
(v) Energy of big drop : U  CV 2  (n c) (n v ) 2 U  n 5 /3 u
2 2
(6) Sharing of charge : When two conductors joined together through a conducting wire,
charge begins to flow from one conductor to another till both have the same potential, due to
flow of charge, loss of energy also takes place in the form of heat.
Suppose there are two spherical conductors of radii r1 and r2 , having charge Q1 and Q 2 ,
potential V1 and V2 , energies U 1 and U 2 and capacitance C 1 and C 2 respectively, as shown in
figure. If these two spheres are connected through a conducting wire, then alteration of charge,
potential and energy takes place.

Q1 Q2 Q1 Q2
C1 r1 r2 C2 r1 r2
C1 C2
V1 V2 V V
U1 U2 U 1 U2
Q1=C1 V1 Q2=C2V2 Q1=C1 V1 Q2=C2V2

(A) (B)



(i) New charge : According to the conservation of charge Q1  Q 2  Q1'  Q 2'  Q (say), also
Q 1' C1 V 4  0 r1 Q1' r Q 1' r1 Q 1'  Q 2' r1  r2
  , '  1  1 1  
Q 2' C 2 V 4  0 r2 Q 2 r2 Q 2' r2 Q 2' r2

 r   r 
 Q 2'  Q  2  and similarly Q1'  Q  1 
 r1  r2   r1  r2 
Total charge Q1  Q 2 Q '  Q 2'
(ii) Common potential : Common potential (V )    1 
Total capacity C1  C 2 C1  C 2
C1 V1  C 2 V 2
V 
C1  C 2
(iii) Energy loss : As electrical energy stored in the system before and after connecting the
spheres is
2
1 1 1 1  C V  C 2 V2 
U i  C 1 V12  C 2 V22 and U f  (C 1  C 2 ) . V 2  (C 1  C 2 )  1 1 
2 2 2 2  C1  C 2 

,74 Electrostatics

C1 C 2
so energy loss ΔU  U i  U f  (V1  V2 )2
2(C 1  C 2 )

Concept

 Capacity of a conductor is a constant term, it does not depend upon the charge Q, and potential (V) and
nature of the material of the conductor.
+ + + +
+ + +
+
+
+ Examples based on sharing of charge, drops and general concept of
+ + + +
+
capacity
Example: 95 Eight drops of mercury of same radius and having same charge coalesce to form a big drop.
Capacitance of big drop relative to that of small drop will be
(a) 16 times (b) 8 times (c) 4 times (d) 2 times
Solution: (d) By using relation C  n . c  C  (8 ) . c  2 c

Example: 96 Two spheres A and B of radius 4 cm and 6 cm are given charges of 80 C
and 40 C respectively. If they are connected by a fine wire, the amount of charge flowing
from one to the other is [MP PET 1991]

(a) 20  C from A to B (b) 16  C from A to B (c) 32 C from B to A (d) 32  C from A to B
 r1 
Solution: (d) Total charge Q  80  40  120  C . By using the formula Q 1 '  Q   . New charge on sphere A
 1  r2
r 
 r 
is Q 'A  Q  A   120 
4 
  48  C . Initially it was 80  C, i.e., 32  C charge flows from A to
 rA  rB  4  6 
B.
Example: 97 Two insulated metallic spheres of 3 F and 5 F capacitances are charged to 300 V and
500 V respectively. The energy loss, when they are connected by a wire, is
(a) 0 .012 J (b) 0 .0218 J (c) 0 . 0375 J (d) 3 .75 J

C1 C 2
Solution: (c) By using ΔU  (V1  V 2 ) 2 ; ΔU  0 .375 J
2(C 1  C 2 )

Example: 98 64 small drops of mercury, each of radius r and charge q coalesce to form a big drop. The
ratio of the surface density of charge of each small drop with that of the big drop is
(a) 1 : 64 (b) 64 : 1 (c) 4 : 1 (d) 1 : 4
2
 Small q / 4 r 2  q  R 
Solution: (d)       ; since R = n1/3r and Q = nq
 Big Q / 4R 2
 Q  r 

 Small 1  Small 1
So  1/3  
 Big n  Big 4

Tricky example: 14

Two hollow spheres are charged positively. The smaller one is at 50 V and the bigger
one is at 100 V. How should they be arranged so that the charge flows from the

, Electrostatics 75

smaller to the bigger sphere when they are connected by a wire
(a) By placing them close to each other
(b) By placing them at very large distance from each other
(c) By placing the smaller sphere inside the bigger one
(d) Information is insufficient
Solution: (c) By placing the smaller sphere inside the bigger one. The potential of the smaller one
will now be 150 V. So on connecting it with the bigger one charge will flow from the
smaller one to the bigger one.



Capacitor.
(1) Definition : A capacitor is a device that stores electric energy. It is also named
condenser.
or
A capacitor is a pair of two conductors of any shape, which are close to each other and
have equal and opposite charge.
+Q –Q
+ –
+ –
+ –
+ –
+
–
+
–




(2) Symbol : The symbol of capacitor are shown below
or variable capacitor

(3) Capacitance : The capacitance of a capacitor is defined as the magnitude of the charge
Q on the positive plate divided by the magnitude of the potential difference V between the
Q
plates i.e., C 
V
(4) Charging : A capacitor get’s charged when a battery is connected across the plates. The
plate attached to the positive terminal of the battery get’s positively charged and the one joined
to the negative terminal get’s negatively charged. Once capacitor get’s fully charged, flow of
charge carriers stops in the circuit and in this condition potential difference across the plates of
capacitor is same as the potential difference across the terminals of battery (say V).
+Q –Q
C
+ –


+ – + –
V V