How Do You Solve Griffiths' Problem 3.41 on Electric Fields Inside a Sphere?

[utorstudent]

For those who have griffiths its problem 3.41 (its not exactly the same but that's what this question is modeled 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.)

 

[Chronos]

Look up dipole moments. The solution is, however, greatly simplified if you are at the center of the sphere.