A Proof of the Shell Theorem

1. Aug 21, 2015

justinben

I have worked through proofs of the Shell Theorem using the integral form of Gauss's Law. However, in this exercise (I'm self studying) we have not covered the integral form of Gauss's Law, so we're looking for an alternative proof. What we have is the definition of divergence and Gauss's law:
$$div(E) = \frac{\partial{E_x}}{\partial x} + \frac{\partial{E_y}}{\partial y} + \frac{\partial{E_y}}{\partial z} = 4\pi k\rho$$ The exercise picks up from a previous exercise that I was successful at where I was asked to show that a purely radial field in empty space must be $$E = \frac{a}{r^3}[x, y, z]$$ where a is some constant of integration. It then gives the following hints. First, it mentions I will need the result that I found in the previous exercise. Then, it says I should argue that far from the sphere the field must be indistinguishable from that of a particle with charge Q. Then, it suggests that I use symmetry to argue that the field at the center is 0 and use that to show that E must be 0 everywhere in the interior.

First, this is an introductory text, so I think the part where I should argue that the field far from the sphere is essentially the same as a point particle with the same charge on the sphere Q is more of an observation that at long distances the sphere and a point particle become indistinguishable and hence their fields are indistinguishable. Second, I have no problem using the symmetry argument to show E is 0 at the center, it is a simple rotational argument that must preserve the field. However, I really don't know where to go from there. I wanted to argue that inside the sphere the field cannot be changing, hence if it is 0 at the center, it is 0 everywhere, but I haven't figured out how to argue it. That's as much progress as I've been able to make. For the outside part we need to show that $$E = \frac{kQ}{r^3}[x, y, z]$$ but I'm not sure how to tie this to the observation about very long distances away from the sphere (except maybe it is as simple as that this observation provides us with the field at a point and thus determines the value of a, which would be KQ and thus the outside part of the theorem is proven).

Those are my thoughts, but help with the solution would be greatly appreciated. Thanks!

Last edited: Aug 21, 2015
2. Aug 22, 2015

haruspex

One way might be to show that at a small distance r from the centre the field magnitude is bounded by $\frac c{(R-r)^2}$ for some constant c. Combining that with your r-3 result would do it, I think.

3. Aug 22, 2015

justinben

Maybe haruspex is correct, but I think your approach sounds too complicated for an introductory book in physics. It's supposed to be a corollary of my r^-3 result, and the fact that the field is 0 on the central axis. It's supposed to follow very easily from that.

4. Aug 22, 2015

haruspex

Maybe, or maybe the only difference is that the book assumes the r-3 result applies smoothly all the way to the centre without further proof. It seemed to me that was a bit of a jump.