Roel N. Baybayon
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.
To prove this problem, we are going to use the Green’s Second Identity which is given by,
Choosing (the scalar potential), and be the integration variable, we have
Let us solve Eq.(1) term by term. For the first integral,
, since if V contains .
For the second integral,
, since .
But because there is no charge in the volume that we are integrating(Charge-free) . So the second integral becomes
For the third integral,
For the fourth integral,
But , then
Using Divergence Theorem,
the fourth integral becomes
But and (again, this is true for a charge-free volume! ), then the fourth integral would be equal to zero, that is,
Thus, Eq.(1) is simplified into
Hence, the scalar potential is then equal to
Now, we have proven the mean value theorem. Eq.(2) says that the potential at any point is equal to the average of the potential over the surface of any sphere centered on that point.