PH 106
Chapter 26:
Dr. T. Mewes
Capacitance and dielectrics
Capacitance
A capacitor is a device that stores electric charge. One example is a parallel plate
capacitor, with charges +Q and -Q stored on the two conductors with area A at a
distance d.
-Q
+Q
+
-
Area: A
d
The capacitance C of a capacitor is defined as:
Q
C=
(1)
ΔV
where Q is the magnitude of the charge on either conductor and ΔV is the magnitude
of the potential difference between the conductors.
As capacitance is defined as the ratio of the magnitude of the stored charge and the
magnitude of the potential difference it is always a positive quantity.
C
1F = 1
Capacitance has the unit of farad (F):
V
Capacitance of a parallel plate capacitor
In chapter 25 we found that the potential difference is:
ΔV = Ed
(2)
The surface charge density on either plate is
Q
(3)
σ=
A
The electric field between the two plates is:
Q
σ
E=
=
(4)
ε0 ε0 A
Use (4) in (2):
ε A
Q
Q
Q
ΔV =
d now use this in the definition of the capacitance: C =
=
= 0
Q
ε0 A
ΔV
d
d
ε0 A
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PH 106
Dr. T. Mewes
ε A
C= 0
d
(5)
That is the capacitance C of a parallel plate capacitor is proportional to the area A of its
plates and inversely proportional to the distance d of the plates.
b
a
Capacitance of a cylindrical capacitor:
C=
l
2ke ln(b a )
(6)
l
where a is the inner radius, b the outer radius and l the length of the
capacitor.
Combinations of capacitors
Instead of drawing elaborate sketches of the elements in an electric circuit we use
simplified circuit symbols and call the result a circuit diagram.
is replaced by:
or
Parallel combination
C1
C2
+
-
The two capacitors carry a charge Q1 and Q2:
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PH 106
Dr. T. Mewes
Q1 = C1ΔV and Q2 = C2 ΔV
Thus we find for the total charge stored by a parallel combination of capacitors:
Q = Q1 + Q2 = C1ΔV + C2 ΔV
(7)
We can rewrite this by using an equivalent capacitor with capacitance Ceq:
Q = Ceq ΔV = (C1 + C2 )ΔV (8)
that is for a parallel combination of capacitors the equivalent capacitance is simply the
sum of the individual capacitances.
Ceq
+
-
For N parallel capacitors we thus get:
N
Ceq = C1 + C2 + C3 + ... + C N = ∑ Ci
(9)
i =1
Series combination
Another possible way to combine two capacitors is to connect them in series:
C1
C2
+
-
Here we know that the charges on the individual capacitors have to be the same:
Q1 = Q2 = Q
(10)
the total voltage drop of the battery ΔV is split between the two capacitors:
ΔV = ΔV1 + ΔV2
(11)
Q
Q
With: ΔV1 =
and ΔV2 =
(12)
C1
C2
If we want to replace the two capacitors with one equivalent capacitor we have:
Q
ΔV =
(13)
Ceq
Use (13) and (12) in (11):
Q
Q
Q
=
+
or by canceling Q:
Ceq C1 C2
1
1
1
=
+
Ceq C1 C2
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(14)
PH 106
Dr. T. Mewes
Therefore for a series combination of N capacitors we get for the equivalent
capacitance:
N
1
1
1
1
1
=
+
+ ... +
=∑
(15)
Ceq C1 C2
C N i =1 Ci
Energy stored in a capacitor
Suppose at an instant in time we have a charge q stored on a capacitor and the potential
difference at that time is ΔV = q / C . Now we want to add a small amount dq of
additional charge. The work required to achieve this is:
q
dW = ΔV dq = dq
(16)
C
with this we can now calculate the total work required to charge a capacitor from q=0 to
q=Q:
W =
Q
Q
0
0
q
Q2
1
dq
qdq
=
=
(17)
∫C
C∫
2C
The work done in charging the capacitor is the electric potential energy U stored by the
capacitor:
Q2
(18)
2C
By using the definition of the capacitance C (equation (1)) we get:
U=
U=
Q 2 (C ⋅ ΔV ) 2 1
=
= C ⋅ ΔV 2
2C
2C
2
(19)
For a parallel plate capacitor we can also express the energy stored in a capacitor in
ε A
terms of the electric field, by using ΔV = Ed and C = 0 (equation (5)):
d
1
1 ε0 A 2 2 1
U = C ⋅ ΔV 2 =
E d = ε 0 AdE 2 (20)
2
2 d
2
From this we can calculate the energy density uE by dividing by the volume Ad:
U
1
uE =
= ε0E 2
(21)
Ad 2
although (21) was derived for a parallel plate capacitor this result is generally valid:
The energy density of any electric field is proportional to the square of the magnitude
of the electric field.
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PH 106
Dr. T. Mewes
Dielectrics
Electric dipole in an electric field
An electric dipole consists of a negative charge -q and a positive charge +q separated
by a distance 2a.
+q
2a
-q
r
We define the electric dipole moment p as:
r
p = 2aq ⋅ rˆ
(22)
where r̂ is a unit vector pointing from the negative charge to the positive charge.
r
r
In a uniform electric field E the force F on the positive charge has the same magnitude
but the opposite direction as the force on the negative charge:
+q
2a
θ
-q
r
Thus there is no net force acting on the electric dipole, however there is a torque τ
acting on it that tends to rotate the dipole:
r r r
τ = p×E
(23)
Thus electric dipoles will try to align themselves parallel to the electric field. The potential
energy U of a dipole in an electric field is:
r r
U = −p⋅E
(24)
or
U = − p ⋅ E cos θ
(24’)
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PH 106
Dr. T. Mewes
One can see from equation (24’) that the energy is minimal for θ = 0 - so a dipole in an
electric field will tend to rotate until θ = 0 .
Capacitors with dielectrics
A dielectric is a nonconducting material, as for example rubber, glass or paper.
Suppose a dielectric has electric dipoles, what will happen as one inserts it into a
capacitor?
+
-
+
-
-
+
+
-
-
+
-
+
+
+
-
-
-
+
+
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
r
In the electric field E 0 of the capacitor the electric dipoles of the dielectric will align
themselves. This gives rise to an induced surface charge σ ind on the dielectric. Which in
r
r
r
turn creates an electric field Eind that opposes E 0 . Therefore the net electric field E in a
capacitor with a dielectric has a magnitude of:
E = E0 − Eind
(25)
One can characterize a dielectric by its dielectric constant κ that describes how much
the electric field is reduced:
r
r E0
E=
(26)
κ
Vacuum has a dielectric constant of κ=1, whereas strontium titanate for example has
κ=230.
For a capacitor the capacitance increases when a dielectric is inserted between the two
plates of the capacitor:
C = κ ⋅ C0
(27)
where C0 is the capacitance without dielectric.
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