Problem 1P

Find the average potential over a spherical surface of radius R due to a point charge q located inside (same as above, in other words, only with z < R). (In this case, of course, Laplace’s equation does not hold within the sphere.) Show that, in general,

where Vcenter is the potential at the center due to all the external charges, and Qenc is the total enclosed charge.

Step 1 of 2

We have to find the average potential over a spherical surface of radius due to a point charge located inside with .

Let us calculate the average potential over a spherical surface of radius due to a single point charge located outside the sphere with . We will center the sphere at the origin and choose coordinates so that lies on the z-axis as shown in the figure below.

The potential at a point on the surface of the sphere is,

Where,

So,

Since, ,

Hence,

Therefore, the the average potential over a spherical surface of radius due to a point charge located inside with is