For those who have griffiths its problem 3.41 (its not exactly the same but that's what this question is modelled after) Alright so I was going through how to solve a problem in my textbook when I got stuck midway through. The approach I had taken wasn't working and I wasn't quite sure where to proceed from where I was. Here's the question. "The average of the field E inside a ball Vr of radius R due to all the charges within the ball, is Eave = 1/4piepsilon p/R^3 (*) we will demonstrate this in the following way " a.) show that the average field due to a single charge q at a point r inside the ball is the same as the field at r due to a uniformly charged ball with pdensity = - 3q/(4piR^3) Alright so I began the question by finding the average field using the formula (and I now know I probably couldn't do this) then by comparing this to the electric field of a uniformly charged sphere. I found the dipole moment (using the charge distribution) and substituted into the eqn. for the average electric field inside the sphere. What I got was E = 1/4(pi)epsilon (-3qr)/R^3 Then I did the same thing for the sphere (solved for Qenc and applied Gauss' Law) and got the same answer. (note: is that what the field will be at r in the ball given the charge density?) then part b says B.) Express the Answer to A in terms of the dipole moment. Now this question threw me off track as there isn't really an "answer" to A. So I thought it must mean what I found the electric field to be. Substituting that in thought simply takes me to the thing I'm trying to prove (*). C.) Then prove the (*) for an arbitrary charge distribution applying the superposition principle to B. This is where I have no clue how to proceed. I imagine that what we want to do is use the fact that we have the average field for a point charge is equal the field in a uniformly charged sphere and apply the principle of superposition to the point charges. However I don't really have an idea of what I should be doing. I think my biggest problem with the question is that I don't really know how to take the average of the electric field. Its equal to the electric field divided by the volume integral over the region (from what I can remember from calculus) but I'm lost and need an idea of where I should be going. Any help with this would be greatly appreciated (I've been trying all sorts of things for over 4 hours and haven't gotten anywhere- I think its probably something pretty obvious but it doesn't feel that way right now.)
Look up dipole moments. The solution is, however, greatly simplified if you are at the center of the sphere.