Jackson 1.8 Homework Problem Solution
Dr. Christopher S. Baird
University of Massachusetts Lowell
PROBLEM:
(a) For the three capacitor geometries in Problem 1.6 calculate the total electrostatic energy and express
it alternatively in terms of the equal and opposite charges Q and -Q placed on the conductors and the
potential difference between them.
(b) Sketch the energy density of the electrostatic field in each case as a function of the appropriate
linear coordinate.
SOLUTION:
(a) For a simple capacitor, the total energy is given by W = ½QV. In problem 1.6, we found the
following results.
Parallel plates capacitor:
V=Q d
Aϵ0
and
E=Q
Aϵ0
Concentric spheres capacitor:
V=Q
4π ϵ0
(
1
a−1
b
)
and
E=1
4πϵ0
Q
r2
Concentric cylinders capacitor:
V=Q
2πϵ0Lln
(
b
a
)
and
E=Q
2πrϵ0L
It is straight-forward to substitute these equations into the energy equation and find the following:
Parallel plates capacitor:
W=Q2d
2Aϵ0
and
W=V2Aϵ0
2d
Concentric spheres capacitor:
W=Q2
8πϵ0
(
1
a−1
b
)
and
W=V22πϵ0a b
b−a
Concentric cylinders capacitor:
W=Q2
4π ϵ0Lln
(
b
a
)
and
W=V2π ϵ0L
ln
(
b/a
)
