Jackson 1.10 Homework Problem Solution
Dr. Christopher S. Baird
University of Massachusetts Lowell
PROBLEM:
Prove the mean value theorem: For charge-free space the value of the electrostatic potential at any
point is equal to the average of the potential over the surface of any sphere centered on that point.
SOLUTION:
The potential is known on the surface, so this problem can be formulated using a Dirichlet Green's
function equation:
x= 1
40∫xGDd3x'−1
4∮
d GD
d n '
da '
where the Dirichlet Green's function must satisfy:
GD(x,x')= 1
∣
x−x'
∣
+F(x,x')
where
∇'2Fx,x'=0
and GD = 0 on the surface
In this particular case, there is no free charge,
x=0
, so that the equation simplifies to
Φ( x)=− 1
4π∮
(
Φd GD
d n '
)
da '
Because we are only measuring the potential at the center of the sphere which is centered on the origin,
x = 0 and therefore
1
∣
x−x'
∣
=1
x'
, leading to:
GD(x,x')= 1
x'+F(x,x')
In order for the green function to disappear on the surface, GD(x' = R) = 0, we must have F = -1/R. The
Green function is now:
GD(x,x')= 1
x'−1
R
Insert this into the equation:
Φ( x)=− 1
4π∮
(
Φd
d x '
[
1
x'−1
R
]
)
x'=R
da '
