Capacitor | Physics Optional Notes for UPSC PDF Download

 Page 1


 
CAPACITOR 
Capacitance of a conductor: 
The charge given to a conductor is directly proportional to its potential (assuming zero 
potential at infinity). That means. 
Q ? V ? Q = CV
C =
Q
V
 
It means the capacitance of a conductor is the amount of charge required to raise the 
potential of a conductor by unit amount. 
A single conductor can also act as a capacitor, here we will find the capacitance of a 
single isolated sphere. For this, let a charge q be given to a spherical conductor of radius 
R, then the potential on it is 
V =
1
4?? ?? 0
q
R
 
The other conductor is supposed to be at infinity, whose potential will be taken as zero. 
So the potential difference between he sphere and the conductor at infinity becomesV-
0 = V. Then capacitance is 
 
C =
q
V
= 4?? ?? 0
R 
SI units of capacitance of 1farad 
Example. Find capacitance of earth. 
Solution: V =
kQ
R
?
Q
V
=
R
k
 
C =
Q
V
=
R
k
=
6400× 10
3
9 × 10
9
= 7.1× 10
-4
 F 
Page 2


 
CAPACITOR 
Capacitance of a conductor: 
The charge given to a conductor is directly proportional to its potential (assuming zero 
potential at infinity). That means. 
Q ? V ? Q = CV
C =
Q
V
 
It means the capacitance of a conductor is the amount of charge required to raise the 
potential of a conductor by unit amount. 
A single conductor can also act as a capacitor, here we will find the capacitance of a 
single isolated sphere. For this, let a charge q be given to a spherical conductor of radius 
R, then the potential on it is 
V =
1
4?? ?? 0
q
R
 
The other conductor is supposed to be at infinity, whose potential will be taken as zero. 
So the potential difference between he sphere and the conductor at infinity becomesV-
0 = V. Then capacitance is 
 
C =
q
V
= 4?? ?? 0
R 
SI units of capacitance of 1farad 
Example. Find capacitance of earth. 
Solution: V =
kQ
R
?
Q
V
=
R
k
 
C =
Q
V
=
R
k
=
6400× 10
3
9 × 10
9
= 7.1× 10
-4
 F 
 
Farad unit is very large for practical purposes. 
Therefore practical purposes we used ?? F,pF. 
Capacitor: 
It is a combination of two conductors kept close to each other (w.r.t. their dimensions) 
and having equal and opposite charges. 
? These conductors are known as plates of the capacitor. 
? Charge on the capacitor means charge on its positive plate although net charge on the 
capacitor is always zero. 
Capacitance of capacitors: 
Charge on a capacitor is directly proportional to the potential difference across the plates 
of the capacitor. 
Q ? ?V ? Q = C?V ? C =
Q
?V
 
It means the capacitance of a capacitor is the amount of charge required to maintain unit 
potential difference between its plates. 
Conductor 
?? ? Charge on conductor 
V ? Potential of conductor 
Capacitor 
Charge on plate 
The potential difference between plates 
Capacitance of a capacitor depends on:- 
(1) Shape and size of the plate 
Page 3


 
CAPACITOR 
Capacitance of a conductor: 
The charge given to a conductor is directly proportional to its potential (assuming zero 
potential at infinity). That means. 
Q ? V ? Q = CV
C =
Q
V
 
It means the capacitance of a conductor is the amount of charge required to raise the 
potential of a conductor by unit amount. 
A single conductor can also act as a capacitor, here we will find the capacitance of a 
single isolated sphere. For this, let a charge q be given to a spherical conductor of radius 
R, then the potential on it is 
V =
1
4?? ?? 0
q
R
 
The other conductor is supposed to be at infinity, whose potential will be taken as zero. 
So the potential difference between he sphere and the conductor at infinity becomesV-
0 = V. Then capacitance is 
 
C =
q
V
= 4?? ?? 0
R 
SI units of capacitance of 1farad 
Example. Find capacitance of earth. 
Solution: V =
kQ
R
?
Q
V
=
R
k
 
C =
Q
V
=
R
k
=
6400× 10
3
9 × 10
9
= 7.1× 10
-4
 F 
 
Farad unit is very large for practical purposes. 
Therefore practical purposes we used ?? F,pF. 
Capacitor: 
It is a combination of two conductors kept close to each other (w.r.t. their dimensions) 
and having equal and opposite charges. 
? These conductors are known as plates of the capacitor. 
? Charge on the capacitor means charge on its positive plate although net charge on the 
capacitor is always zero. 
Capacitance of capacitors: 
Charge on a capacitor is directly proportional to the potential difference across the plates 
of the capacitor. 
Q ? ?V ? Q = C?V ? C =
Q
?V
 
It means the capacitance of a capacitor is the amount of charge required to maintain unit 
potential difference between its plates. 
Conductor 
?? ? Charge on conductor 
V ? Potential of conductor 
Capacitor 
Charge on plate 
The potential difference between plates 
Capacitance of a capacitor depends on:- 
(1) Shape and size of the plate 
(2) Distance between plates 
(3) Medium between the plates 
CALCULATION OF CAPACITANCE OF CAPACITOR 
(i) Parallel plate capacitor : 
? ?? ? ??? ??? = ???? =
?? ?? ?? 0
?? ? ?? = (
?? 0
?? ?? )??? ..
 
 
Comparing (i) and (ii) 
C =
?? 0
 A
 d
 
(ii) Spherical Capacitor: 
C =
4?? ?? 0
ab
b- a
 
 
In a spherical capacitor with two concentric conducting shells, one inside the other, the 
outer shell is connected to the ground (earthed). If we place a positive charge +Q on the 
inner shell, it will distribute on the outer surface of that shell. This induces a negative 
charge -Q on the inner surface of the outer shell. As a result, a positive charge +Q will 
flow from the outer shell to the ground (earth). 
Consider a Gaussian spherical surface of radius ?? such that ?? < ?? < ?? . From Gauss's law, 
the electric field at distance ?? > ?? is 
E =
Q
4?? ?? 0
r
2
 
Page 4


 
CAPACITOR 
Capacitance of a conductor: 
The charge given to a conductor is directly proportional to its potential (assuming zero 
potential at infinity). That means. 
Q ? V ? Q = CV
C =
Q
V
 
It means the capacitance of a conductor is the amount of charge required to raise the 
potential of a conductor by unit amount. 
A single conductor can also act as a capacitor, here we will find the capacitance of a 
single isolated sphere. For this, let a charge q be given to a spherical conductor of radius 
R, then the potential on it is 
V =
1
4?? ?? 0
q
R
 
The other conductor is supposed to be at infinity, whose potential will be taken as zero. 
So the potential difference between he sphere and the conductor at infinity becomesV-
0 = V. Then capacitance is 
 
C =
q
V
= 4?? ?? 0
R 
SI units of capacitance of 1farad 
Example. Find capacitance of earth. 
Solution: V =
kQ
R
?
Q
V
=
R
k
 
C =
Q
V
=
R
k
=
6400× 10
3
9 × 10
9
= 7.1× 10
-4
 F 
 
Farad unit is very large for practical purposes. 
Therefore practical purposes we used ?? F,pF. 
Capacitor: 
It is a combination of two conductors kept close to each other (w.r.t. their dimensions) 
and having equal and opposite charges. 
? These conductors are known as plates of the capacitor. 
? Charge on the capacitor means charge on its positive plate although net charge on the 
capacitor is always zero. 
Capacitance of capacitors: 
Charge on a capacitor is directly proportional to the potential difference across the plates 
of the capacitor. 
Q ? ?V ? Q = C?V ? C =
Q
?V
 
It means the capacitance of a capacitor is the amount of charge required to maintain unit 
potential difference between its plates. 
Conductor 
?? ? Charge on conductor 
V ? Potential of conductor 
Capacitor 
Charge on plate 
The potential difference between plates 
Capacitance of a capacitor depends on:- 
(1) Shape and size of the plate 
(2) Distance between plates 
(3) Medium between the plates 
CALCULATION OF CAPACITANCE OF CAPACITOR 
(i) Parallel plate capacitor : 
? ?? ? ??? ??? = ???? =
?? ?? ?? 0
?? ? ?? = (
?? 0
?? ?? )??? ..
 
 
Comparing (i) and (ii) 
C =
?? 0
 A
 d
 
(ii) Spherical Capacitor: 
C =
4?? ?? 0
ab
b- a
 
 
In a spherical capacitor with two concentric conducting shells, one inside the other, the 
outer shell is connected to the ground (earthed). If we place a positive charge +Q on the 
inner shell, it will distribute on the outer surface of that shell. This induces a negative 
charge -Q on the inner surface of the outer shell. As a result, a positive charge +Q will 
flow from the outer shell to the ground (earth). 
Consider a Gaussian spherical surface of radius ?? such that ?? < ?? < ?? . From Gauss's law, 
the electric field at distance ?? > ?? is 
E =
Q
4?? ?? 0
r
2
 
The potential difference is 
?? ?? - ?? ?? = -? ?
?? ?? ?? ? 
· ????
???? 
= -? ?
?? ?? ?? 4?? ?? 0
?? 2
???? 
Since ?? ?? = 0, we have 
?? ?? = ? ?
?? ?? ?? 4?? ?? 0
?? 2
???? =
?? 4?? ?? 0
(
1
?? -
1
?? ) =
?? (?? - ?? )
4?? ?? 0
????
 
Therefore, capacitance is 
 
C =
Q
V
a
- V
b
=
Q
V
a
=
4?? ?? 0
ab
b- a
 
(iii) Cylindrical capacitor 
C =
2?? ?? 0
l
ln(
R
2
R
1
)
 
 
 
A cylindrical capacitor is made of two coaxial cylinders with radii R1 and R2, where R2 
is larger than R1. The outer cylinder is connected to the ground (earthed). Assuming the 
cylinders are long enough to ignore the electric field fringing at the ends, the electric field 
between them is radial. Its strength depends on the distance from the central axis. 
If we consider a Gaussian surface between the cylinders with length y and radius r such 
that R1<r<R2, the flux through the plane surface is zero because the electric field and the 
area vector are perpendicular to each other. 
Page 5


 
CAPACITOR 
Capacitance of a conductor: 
The charge given to a conductor is directly proportional to its potential (assuming zero 
potential at infinity). That means. 
Q ? V ? Q = CV
C =
Q
V
 
It means the capacitance of a conductor is the amount of charge required to raise the 
potential of a conductor by unit amount. 
A single conductor can also act as a capacitor, here we will find the capacitance of a 
single isolated sphere. For this, let a charge q be given to a spherical conductor of radius 
R, then the potential on it is 
V =
1
4?? ?? 0
q
R
 
The other conductor is supposed to be at infinity, whose potential will be taken as zero. 
So the potential difference between he sphere and the conductor at infinity becomesV-
0 = V. Then capacitance is 
 
C =
q
V
= 4?? ?? 0
R 
SI units of capacitance of 1farad 
Example. Find capacitance of earth. 
Solution: V =
kQ
R
?
Q
V
=
R
k
 
C =
Q
V
=
R
k
=
6400× 10
3
9 × 10
9
= 7.1× 10
-4
 F 
 
Farad unit is very large for practical purposes. 
Therefore practical purposes we used ?? F,pF. 
Capacitor: 
It is a combination of two conductors kept close to each other (w.r.t. their dimensions) 
and having equal and opposite charges. 
? These conductors are known as plates of the capacitor. 
? Charge on the capacitor means charge on its positive plate although net charge on the 
capacitor is always zero. 
Capacitance of capacitors: 
Charge on a capacitor is directly proportional to the potential difference across the plates 
of the capacitor. 
Q ? ?V ? Q = C?V ? C =
Q
?V
 
It means the capacitance of a capacitor is the amount of charge required to maintain unit 
potential difference between its plates. 
Conductor 
?? ? Charge on conductor 
V ? Potential of conductor 
Capacitor 
Charge on plate 
The potential difference between plates 
Capacitance of a capacitor depends on:- 
(1) Shape and size of the plate 
(2) Distance between plates 
(3) Medium between the plates 
CALCULATION OF CAPACITANCE OF CAPACITOR 
(i) Parallel plate capacitor : 
? ?? ? ??? ??? = ???? =
?? ?? ?? 0
?? ? ?? = (
?? 0
?? ?? )??? ..
 
 
Comparing (i) and (ii) 
C =
?? 0
 A
 d
 
(ii) Spherical Capacitor: 
C =
4?? ?? 0
ab
b- a
 
 
In a spherical capacitor with two concentric conducting shells, one inside the other, the 
outer shell is connected to the ground (earthed). If we place a positive charge +Q on the 
inner shell, it will distribute on the outer surface of that shell. This induces a negative 
charge -Q on the inner surface of the outer shell. As a result, a positive charge +Q will 
flow from the outer shell to the ground (earth). 
Consider a Gaussian spherical surface of radius ?? such that ?? < ?? < ?? . From Gauss's law, 
the electric field at distance ?? > ?? is 
E =
Q
4?? ?? 0
r
2
 
The potential difference is 
?? ?? - ?? ?? = -? ?
?? ?? ?? ? 
· ????
???? 
= -? ?
?? ?? ?? 4?? ?? 0
?? 2
???? 
Since ?? ?? = 0, we have 
?? ?? = ? ?
?? ?? ?? 4?? ?? 0
?? 2
???? =
?? 4?? ?? 0
(
1
?? -
1
?? ) =
?? (?? - ?? )
4?? ?? 0
????
 
Therefore, capacitance is 
 
C =
Q
V
a
- V
b
=
Q
V
a
=
4?? ?? 0
ab
b- a
 
(iii) Cylindrical capacitor 
C =
2?? ?? 0
l
ln(
R
2
R
1
)
 
 
 
A cylindrical capacitor is made of two coaxial cylinders with radii R1 and R2, where R2 
is larger than R1. The outer cylinder is connected to the ground (earthed). Assuming the 
cylinders are long enough to ignore the electric field fringing at the ends, the electric field 
between them is radial. Its strength depends on the distance from the central axis. 
If we consider a Gaussian surface between the cylinders with length y and radius r such 
that R1<r<R2, the flux through the plane surface is zero because the electric field and the 
area vector are perpendicular to each other. 
 
In the long cylindrical capacitor, the linear charge density ?? is assumed to be positive in 
this figure. The magnitude of the charge in a length of either cylinder is ?? L 
For the curved part 
?? = ? E
? ? 
· ds
???? 
= ? Eds= E? ds= E2?? ry 
Charge inside the Gaussian surface is 
?? =
????
?? 
From Gauss's law 
?? = E2?? ry =
Qy
L?? 0
 or E =
Q
2?? ?? 0
Lr
 
Potential difference is 
?? ?? 2
- ?? ?? 1
= -? ?
?? 2
?? 1
?? ? 
· ????
???? 
= -? ?
?? 2
?? 1
?? 2?? ?? 0
????
???? = -
?? 2?? ?? 0
?? ? ?
?? 2
?? 1
1
?? ???? 
or  ?? ?? 1
=
?? 2?? ?? 0
?? ln 
?? 2
?? 1
 
Capacitance is C =
Q
V
R
2
-V
R
1
=
Q
V
R
1
=
2?? ?? 0
 L
ln (
R
2
R
1
)
 
Symbolic representation and sign conversion: 
 
CHARGING OF CAPACITOR 
At t = 0