Chapter 25
Capacitance
In this chapter we will cover the following topics:
-Capacitance C of a system of two isolated conductors.
-Calculation of the capacitance for some simple geometries.
-Methods of connecting capacitors (in series, in parallel).
-Equivalent capacitance.
-Energy stored in a capacitor.
-Behavior of an insulator (a.k.a. dielectric) when placed in the
electric field created in the space between the plates of a capacitor.
-Gauss’ law in the presence of dielectrics.
(25-1)
)
Capacitance
A system of two isolated conductors, one with a charge
V q and the other - q, separated by an insulator (this
can be vacuum or air) is known as a "capacitor." The
symbol used to indicate a capacitor is two parallel lines.
We refer to the conductors as "plates." We refer to the
"charge" of the capacitor as the absolute value of the
charge on either plate.
As shown in the figure, the charges on the capacitor plates create
V+
C
q
q
V
q = CV
an electric field in the surrounding space. The electric potential
of the positive and negative plate are V and V , respectively.
V
O
We use the symbol V for the potential difference V  V between
the plates (V would be more appropriate).
If we plot the charge q as a function of V we get the straight line shown in the figure.
The capacitance C is defined as the ratio q / V .
SI Unit : Farad (symbol F) We define a capacitor of C  1 F as one that
acquires a charge q = 1 C if we apply a voltage difference V  1 V between its plates.
(25-2)
Checkpoint 1:
Does the capacitance C of a capacitor increase, decrease, or remain
the same
a) When the charge q on it is doubled and
b) When the potential difference V across it is tripled?
q
C
V
The capacitance of the capacitor depends only on the geometry of
the plates and not on their charge or potential difference.
Parallel Plate Capacitor
A parallel plate capacitor is defined as a capacitor made up from two parallel
plane plates of area A separated by a distance d . The electric field between the
plates and away from the plate edges is uniform. Close to the plates' edges the
electric field (known as "fringing field") becomes nonuniform.
V
+
_
(25-3)
Batteries
A battery is a device that maintains a constant potential difference V
between its two terminals. These are indicated in the battery symbol
using two parallel lines unequal in length. The longer line indicates
the terminal at higher potential while the shorter line denotes
the lower-potential terminal.
-q
+q
+q
(25-4)
-q
Charging a Capacitor
One method to charge a capacitor is shown in
the figure. When the switch S is closed, the electric
field of the battery drives electrons from the battery
negative terminal to the capacitor plate connected to
it (labeled " " for low). The battery positive
terminal removes an equal number of electrons from
the plate connected to it (labeled "h" for high).
Initially the potential difference V between the
capacitor plates is zero. The charge on the plates
as well as the potential difference between the plates
increase, and the charge movement from the battery
terminals to and from the plates decreases.
All charge movement stops when the potential
difference between the plates becomes equal
to the potential difference between the
battery terminals.
V+
V-
Calculating the Capacitance
The capacitance depends on the geometry of the plates
(shape, size, and relative position of one with respect
to the other). Below we give a procedure for
calculating C.
Recipe :
1. Assume that the plates have charges  q and - q.
2. Use Gauss' law to determine the electric field


E between the plates ε0  E  dA  q enc .
3. Determine the potential difference V between
the plates using the equation

V   E  ds along any path that connects the

(25-5)
negative with the positive plate.
4. The capacitance C is given by the equation
q
C .
V
P
S
n̂
Capacitance of a Parallel Plate Capacitor
The plates in the figure have area A and are
separated by a distance d . The upper plate
has a charge  q and the lower plate a charge - q.
N
We apply Gauss' law using the Gaussian surface S shown in the figure.
The electric flux   EA cos 0  EA.
q
q
q
From Gauss' law we have:    EA   E 
.
0
0
A 0
The potential difference V between the positive and the negative plate is




given by: V   Eds cos 0 E  ds Ed 
The capacitance C 
qd
.
A 0
A
q
q

 0.
V qd / A 0
d
A 0
C
d
(25-6)
Cylindrical Capacitor
n̂
S
It consists of two cylinders of radii a and b
with a common axis. The two cylinders have
a height L. We choose a Gaussian surface S
that is also a cylinder with radius r and height L.
The flux of the electric field through S is
  2 rLE cos 0  2 rLE.
q
Using Gauss' law we have:   .
0
If we combine these equations we have:
q
E
.
 0 2 rL
The potential difference V between the positive and the negative plate is

q
dr
q
q
a
b
given by: V   Edr cos180  


ln
r

ln


 .

b
 0 2 L b r
 0 2 L
 0 2 L  a 

The capacitance C 
a
2 L 0
q
q


.
V  q / 2 L 0  ln  b / a  ln  b / a 
(25-7)
Spherical Capacitor
It consists of two concentric spheres of radii a and b.
We choose a Gaussian surface S that is also a
sphere with radius r.
The flux of the electric field through S is:
n̂
S
  4 r 2 E cos 0  4 r 2 E.
Using Gauss' law we have:  
q
0
.
If we combine these equations we have:
q
E
2 .
4 0 r
The potential difference V between the positive and the negative plate is

a
dr
q 1
q 1 1
given by: V   Edr cos180  


  .
2



4 0 b r
4 0  r  b 4 0  a b 

The capacitance C 
q
q

V
a
4 0
q
 ab 

 4 0 
.

q 1 1 1 1
ba



 

4 0  a b   a b 
(25-8)
Checkpoint 2:
For capacitors charged by the same battery, does the charge stored by the
capacitor increase, decrease, or remain the same in each of the following
situations?
a) The plate separation of a parallel-plate capacitor is increased.
b) The radius of the inner cylinder of a cylindrical capacitor is increased
c) The radius of the outer spherical shell of a spherical capacitor is increased
Equivalent Capacitor
Consider the combination of capacitors
shown in the figure to the left and to the
right (upper part). We will substitute
these combinations of capacitor with a
single capacitor Ceq that is
"electrically equivalent"
to the capacitor group it substitutes.
This means that if we apply the same voltage V across the
capacitors in fig. a and fig. b (either right or left) by connecting
to a battery, the same charge q is provided by the battery. Alternatively,
if we place the same charge q on plates of the capacitors in fig. a
and fig. b (either right or left), the voltage V across them is identical.
This can be stated in the following manner: If we place the capacitor
combination and the equivalent capacitor in separate black boxes,
by doing electrical measurements we cannot distinguish between the two.
(25-9)
Ceq  C1  C2  C3
Capacitors in Parallel
In fig. a we show three capacitors
connected in parallel. This means that
the plate of each capacitor is connected
to the terminals of a battery of voltage V .
We will substitute the parallel combination
of fig. a with a single equivalent capacitor
shown in fig. b, which is also connected
to an identical battery.
The three capacitors have the same potential difference V across their plates.
The charge on C1 is q1  C1V . The charge on C2 is q2  C2V .
The charge on C3 is q3  C3V . The net charge q  q1  q2  q3   C1  C2  C3  V .
q  C1  C2  C3 V

 C1  C2  C3 .
V
V
For a parallel combination of n capacitors it is given by the expression:
The equivalent capacitance Ceq 
n
Ceq  C1  C2  ...  Cn   C j
j 1
(25-10)
Capacitors in Series
In fig. a we show three capacitors connected in series.
This means that one capacitor is connected after the other.
The combination is connected to the terminals of a battery
of voltage V . We will substitute the series combination
of fig. a with a single equivalent capacitor shown in fig. b,
which is also connected to an identical battery.
The three capacitors have the same charge q on their plates.
The voltage across C1 is V1  q / C1.
The voltage across C2 is V2  q / C2 .
The voltage across C3 is V3  q / C3 .
The net voltage across the combination V  V1  V2  V3 .
(25-11)
 1
1
1 
Thus we have: V  q  
 .
 C1 C2 C3 
q
The equivalent capacitance Ceq  
V
1
1
1
1
 

Ceq C1 C2 C3
q

 1
1
1 
q 
 
 C1 C2 C3 
More Complex Capacitor Systems
In general, a capacitor system
may consist of smaller capacitor
groups that can be identified as
connected "in parallel" or
"in series."
For example, C1 and C2 in fig. a are connected in parallel.
They can be substituted by the equivalent capacitor C12  C1  C2 as
shown in fig. b. Capacitors C12 and C3 in fig. b are connected in series.
They can be substituted by a single capacitor C123 as shown in fig. c.
C123 is given by the equation
(25-12)
1
1
1


.
C123
C12 C3
Checkpoint 3:
A battery of potential V stores charge q on a combination of two identical
capacitors. What are the potential difference across and the charge on either
capacitor if the capacitors are
a) In parallel
b) In series ?
-q'
-
q'
+
+
+
+
+
dq'
V'
Charge
A
q
q'
B Voltage
O
V' V
q
1  q 2 
q2
W   
C  2  0 2C
Energy Stored in an Electric Field
Consider a capacitor C that has a charge q.
We can calculate the work W required to charge
the capacitor by assuming that we transfer a charge
dq from the negative plate to the positive plate.
We assume that the capacitor charge is q and the
corresponding voltage V . The work dW required
q
for the charge transfer is given by: dW  V dq  dq.
C
We continue this process till the capacitor charge is
q
1
equal to q. The total work W   V dq   qdq.
C0
CV 2
If we substitute q  CV we get: W 
2
or W 
qV
.
2
Work W can also be calculated by determining the area A of triangle OAB, which is
Vq
equal to  V dq : Area  W  .
2
(25-13)
E
-q
-A
-
q
+
+
A +
+
+
d
q 2 CV 2
U

2C
2
u
0E2
2
Potential Energy Stored in a Capacitor
The work W spent to charge a capacitor is stored in the form of
potential energy U  W that can be retrieved when the capacitor is
q 2 CV 2 qV
discharged. Thus U 


.
2C
2
2
Energy Density
We can ask the question: Where is the potential energy of a charged
capacitor stored? The answer is counterintuitive. The energy is
stored in the space between the capacitor plates where a uniform
electric field E  V / d is generated by the capacitor charges.
In other words, the electric field can store energy in empty space!
We define energy density (symbol u ) as the potential energy per unit volume: u 
The volume V between the plates is V  Ad , where A is the plate area.
0  V  0E2
U
CV 2
V2
Thus the energy density u 


   
Ad 2 Ad 2 Ad 2  d 
2
This result, derived for the parallel plate capacitor, holds in general.
2
(25-14)
U
.
V
C   Cair
Capacitor with a Dielectric
In 1837 Michael Faraday investigated what happens to the
q
V
-q
capacitance C of a capacitor when the gap between the plates
is completely filled with an insulator (a.k.a. dielectric).
q'
Faraday discovered that the new capacitance is given by
C   Cair . Here Cair is the capacitance before the insertion
V
-q'
q
V
-q
q
-q
of the dielectric between the plates. The factor  is known
as the dielectric constant of the material.
Faraday's experiment can be carried out in two ways:
1. With the voltage V across the plates remaining constant.
In this case a battery remains connected to the plates.
This is shown in fig. a.
V'
2. With the charge q of the plates remaining constant.
In this case the plates are isolated from the battery.
This is shown in fig. b.
(25-15)
C   Cair
q
V
-q
q'
V
-q'
q
V
-q
After the dielectric is inserted between the capacitor plates
the plate charge changes from q to q   q.
q κq
q
The new capacitance C  
 κ  κCair .
V
V
V
Fig. b : Capacitor charge q remains constant
This is because the plates are isolated.
After the dielectric is inserted between the capacitor plates
-q
q
Fig. a : Capacitor voltage V remains constant
This is because the battery remains connected to the plates.
V'
the plate voltage changes from V to V  
The new capacitance C 
V

.
q
q
q

    Cair .
V V /
V
(25-16)
In a region completely filled with an insulator of
dielectric constant  , all electrostatic equations
containing the constant  0 are to be modified by
q
replacing  0 with  0 .
Example 1 : Electric field of a point charge inside
a dielectric is: E 
conductor
dielectric
1
q
.
2
4 0 r
Example 2 :
The electric field outside an isolated conductor
immersed in a dielectric becomes:
E

.
 0
(25-17)
Example 1:
Example 2:
Example 3:
Example 4:
Example 5:
Example 6:
Example 7:
Example 8:
Example 9: