Jan 22, 2001

Notes for Lecture Notes for Lecture #3

The ``mean value theorem'' for solutions to the Laplace equation

Consider an electrostatic field F(r) in a charge-free region so that it satisfies the Laplace equation:
Ñ2 F(r) = 0.
(1)
The ``mean value theorem'' value theorem states that the value of F(r) at the arbitrary (charge-free) point r is equal to the average of F(r¢) over the surface of any sphere centered on the point r (see Jackson problem #1.10). One way to prove this theorem is the following. Consider a point r¢ = r + u, where u will describe a sphere of radius R about the fixed point r. We can make a Taylor series expansion of the electrostatic potential F(r¢) about the fixed point r:
F(r + u ) = F(r) + u·Ñ F(r) + 1
2!
(u·Ñ)2 F(r) + 1
3!
(u·Ñ)3 F(r) + 1
4!
(u·Ñ)4 F(r) + ¼.
(2)
According to the premise of the theorem, we want to integrate both sides of the equation 2 over a sphere of radius R in the variable u:
ó
õ


sphere 
dSu = R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu).
(3)
We note that
R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu) 1 = 4 pR2,
(4)
R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu) u ·Ñ = 0,
(5)
R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu) (u ·Ñ)2 = 4 pR4
3
Ñ2,
(6)
R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu) (u ·Ñ)3 = 0,
(7)
and
R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu) (u ·Ñ)4 = 4 pR4
5
Ñ4.
(8)
Since Ñ2 F(r) = 0, the only non-zero term of the average it thus the first term:
R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu) F(r + u) = 4 pR2 F(r),
(9)
or
F(r) = 1
4 pR2
    R2 ó
õ
2p

0 
dfu ó
õ
+1

-1 
dcos(qu) F(r + u).
(10)
Since this result is independent of the radius R, we see that we have proven the theorem.


File translated from TEX by TTH, version 2.20.
On 22 Jan 2001, 08:56.