Electromagnetics - potential inside a spherical surface

[zeeva]

Homework Statement

Find the average potential over a spherical surface of radius R due to a point charge q located inside.

Homework Equations

V(ave)=V(centre) + Q(enc)/(4*pi*epsi*R)
Where V(centre) is the potential at the centre due to all the external charges, and Q(enc) is the total of the enclosed charge.

The Attempt at a Solution

I think what they are asking is, what is the potential where the charge is located, I thought that because at the centre, r=0, then the V is proportional to 1/r, therefore it's not possible. If you can see the question differently, let me know because it's really doing my head in.
Thanks

 

[Mindscrape]

Hmm, yeah, I agree the wording is ambiguous. You may have to use Laplace's equation. If you haven't learned that yet, then maybe we need a clearer wording.

 

[gabbagabbahey]

I think the woording is perfectly clear: whenever you are asked for the average of some quantity over a surface, you simply integrate that quantity over the surface and then divide by the surface area.

V_{\text{ave}}=\frac{\int_{\mathcal{S}}Vda}{\int_{\mathcal{S}}da}

You are essentially averaging the potential over the surface by taking the value of the potential on each small patch of the surface, multiplying it by the area of that patch, adding up that product for every piece of the surface and then dividing by the total area of the surface.

Similarly, if you are ever asked to average a quantity over a 3d region, you would integrate the quantity over that region and then divide by its volume. And, if you are asked to average a quantity over time, you would integrate over some time interval and then divide by the length of that time interval.