Electric field inside a cavity within a sphere

[Amith2006]

Homework Statement

This question has already been asked before. The electric field inside a spherical cavity carved out of a larger sphere is uniform which I have derived. What I don't really understand is that if you construct a gaussian surface inside the cavity, it wouldn't enclose any charge. So, there shouldn't any electric field which is contradictory to my previous results. I think I'm missing a subtle point but can't figure out. Thanks in advance.

Homework Equations

The Attempt at a Solution

 

[LaissezDairy]

If the cavity is concentric with the larger sphere, it is uniform - zero.

If the cavity is not concentric, then the thing about the gaussian surface is the fact that you have no enclosed charge means that the NET flux is zero on the surface - not that the flux is zero everywhere. It is not constant over the surface because your problem doesn't have the spherically symmetry necessary.

Here's an intuitive example - two parallel, infinite charged planes. Obviously there's a uniform E-field perpendicular to the planes between them. Construct a gaussian spherical surface between the two planes - no enclosed charge!

That means the net flux through that gaussian sphere is zero - because the flux entering it on one side is the same as the flux exiting it on the other. That does NOT mean the E-field is zero, because again it's not spherically symmetrical so you can't assume the flux is constant over the surface.

 

[Amith2006]

I get it now. Thanks.