Physics 4B Lecture Notes
261
Chapter 26  Capacitance
Problem Set #5  due:
Ch 26  2, 3, 5, 7, 9, 15, 22, 26, 29, 61, 63, 64
The ideas of energy storage in Efields can be carried a step further by understanding the concept of
"Capacitance."
Lecture Outline
1. The Definition of Capacitance
2. Capacitors in Circuits
3. Energy Storage in Capacitors and Electric Fields
4. Dielectrics in Capacitors
1. The Definition of Capacitance
Consider a sphere with a total charge, Q, and a radius, R.
From previous
problems we know that the potential at the surface is, V
=
k
Q
R
.
Putting more
charge on the sphere stores more energy, but the ratio of energy or potential to
charge depends only on R, not on Q or V.
That is,
Q
V
=
R
k
.
It's true for all
charged objects that the ratio of potential to voltage depends only on the shape,
so this ratio is defined as the capacitance.
C
≡
Q
V
The Defintion of Capacitance
The units of capacitance are
1 Coulomb
1 Volt
≡
1 Farad
≡
1 F.
Example
1:
Calculate the capacitance of two equal but oppositely charged plates of area, A, and
separation, d.
Neglect any edge effects.
The potential difference between the plates is,
∆
V
= 
r
E
•
d
r
s
∫
.
The field between the plates is just the sum of the fields due to the
individual plates (see Ch 23  example 9),
r
E
=
r
E
+
+
r
E

=
σ
2
ε
o
ˆ
k
+
σ
2
ε
o
ˆ
k
=
σ
ε
o
ˆ
k
=
q
ε
o
A
ˆ
k .
Using
d
r
s
=
dz
ˆ
k , the voltage on the capacitor can be written as,
∆
V
= 
q
ε
o
A
ˆ
k
•
dz
ˆ
k
= 
∫
q
ε
o
A
dz
= 
∫
q
ε
o
A
∆
z
⇒
V
=
qd
ε
o
A
.
Applying the definition of capacitance,
C
≡
Q
V
=
q
qd
ε
o
A
⇒
C
= ε
o
A
d
.
Note that the capacitance only depends on the shape.
R
Q
+q
d
q
z
+
E

E
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Physics 4B Lecture Notes
262
Example
2:
Find the capacitance of two concentric cylindrical conductors of radii a and b with a
length,
l
.
Show that the result is consistent with example 1.
Assume the cylinders have equal and opposite charges,
Q.
Then the potential difference between them is,
∆
V
= 
r
E
•
d
r
s
∫
where
r
E
=
2k
λ
r
ˆ
r
=
2kQ
r
l
ˆ
r from example 7 of chapter 24.
Using
d
r
s
=
dr
ˆ
r , the voltage on the capacitor can be
written as,
∆
V
= 
2kQ
r
l
ˆ
r
•
dr
ˆ
r
= 
∫
2kQ
l
1
r
dr
a
b
∫
= 
2kQ
l
ln
b
a
⇒
V
=
2kQ
l
ln
b
a
.
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 Spring '10
 Basu
 Capacitance, Energy, Electric charge, Dielectrics

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