Transcript Quantum ElectroDynamics

TOPIC 5
Capacitors and
Dielectrics
1
Capacitors
Capacitors are a means of storing electric charge
(and electric energy)
• It takes energy to bring charge together
• A capacitor allows more charge to be stored for a
given energy
• It does this by reducing the potential at which the
charge is stored
• It can do this by bringing an opposite charge into
close proximity, to reduce the overall repulsion
++++++
– – – – – –
2
Capacitance
Capacitance (C) is charge per unit potential
difference
C = Q/V
Unit is Farad (F): 1 F = 1 Coulomb/Volt
Typical capacitances measured in F (10–6 F) or
pF (10–12 F)
3
Parallel Plate Capacitor
A
+
+
+
+Q
+
+
+
+
V
E
–
–
–
–
–
–
–
–Q
Two plates, area A, separation d, carrying charge Q.
Gauss’s Law (using dotted Gaussian surface shown)
E A = Q/0  E = Q/0 A

V   E  dl  E d

Q 0 A
C 
V
d
Qd
V
0 A
4
Example 1 – Parallel Plate Capacitor
A parallel plate capacitor has plates with dimensions
3 cm by 4 cm, separated by 2 mm. The plates are
connected across a 60 V battery. Find:
(a) the capacitance;
(b) the magnitude of charge on each plate;
(c) the energy stored in the capacitor – see later!
5
Example 2 – Cylindrical Capacitor
What is the capacitance of a long cylindrical
(coaxial) cable of inner radius a, outer radius b and
length L as shown?
a
b
6
Example 3 – Spherical Capacitor
What is the capacitance of two concentric spherical
conducting shells of inner radius a and outer radius
b?
7
Capacitors in Parallel
V
C1
C2
0V
Capacitors connected as shown, with terminals
connected together, are said to be in parallel.
They behave as a single capacitor with effective
capacitance C.
Total charge Q = Q1 + Q2 = C1V + C2V
Therefore
C = Q/V = C1 + C2
8
Capacitors in Series
Capacitors connected together as
shown, sharing one common terminal, +Q
are said to be in series.
–Q
They behave as a single capacitor with
+Q
effective capacitance C.
–Q
The external charge stored is Q.
The voltages across the capacitors
Vi = Q/Ci must add up to V.
Therefore
V
C1
C2
0V
V = Q/C1 + Q/C2 = Q/C
1 1
1


C C1 C2
9
Example 4 –Capacitor Network
If each of the individual capacitors in the network
below has a capacitance C, what is the overall
effective capacitance?
10
Energy stored in a Capacitor
Adding an increment of charge dq to a capacitor
requires work dW = V dq = q/C dq
This is obviously the increase in (potential) energy
stored of the capacitor U
The total energy required to charge a capacitor from
zero charge to Q is therefore
Q
q
Q2
U   dq 
C
2C
0
Since Q = C V, we can express this in other ways:
1 Q2 1
1
U
 QV  CV 2
2 C 2
2
11
Example 1 – Parallel Plate Capacitor
A parallel plate capacitor has plates with dimensions
3 cm by 4 cm, separated by 2 mm. The plates are
connected across a 60 V battery. Find:
(a) the capacitance;
(b) the magnitude of charge on each plate;
(c) the energy stored in the capacitor – see later!
Previously:
C = 5.3 pF
Q = 3.210–10 C
12
Energy stored in a Capacitor (2)
The energy stored in the capacitor can also be considered as
the energy stored in its electric field.
We have
U  CV
1
2
2
For the parallel plate capacitor we also have
So

A
0
V = E d and
C
d
1 0 A
1
2
2
U
 Ed   0 AdE
2 d
2
But A d is the volume where the electric field exists, so the
energy density is u  1  E 2
2
0
This is a general result for the energy density in a field.
13
Dielectrics
A conductor contains free charges that
can move through the material.
+
–
A dielectric contains bound charges,
which cannot move freely but will
displace through small distances
when affected by an electric field.
This leaves excess bound
charges on the surface of the
material.
This reduces the electric field
within the bulk of the material.
+
–
+
–
+
–
+
–
+
–
+
–
+
–
+
+
+
–
–
–
14
E
Dielectrics and Capacitors
The factor by which the electric field is reduced is
known as the dielectric constant k (or r).
If the gap between the plates of a capacitor is filled
with dielectric material, the voltage between the
plates for a given charge will also be reduced by
the factor k.
Since C = Q / V, this means that C is increased by k.
For the parallel plate capacitor, we therefore have
k 0 A
C
d
15
Example 5
Demonstrate that the energy stored in a spherical
capacitor is consistent with an energy density stored
in the field of
u  12 0 E 2
16