How does symmetry affect the shell theorem?

[Bipolarity]

Gauss' shell theorem states that if given a spherical shell of charge such that the charge is uniformly distributed on the surface, the net electric field anywhere inside the sphere is zero.

But I'm wondering (and turns out this was on a past exam), what happens if the charge is not uniformly distributed?

Will the field anywhere and everywhere inside the shell STILL be 0?

My hypothesis is that it cannot be 0 everywhere, since even though Gauss' law can be used to prove that the flux is zero, spherical symmetry cannot be used to take the E vector out of the loop integral since there is no uniform charge distribution.

My other explanation was that 0 field everywhere inside would necessarily imply a uniform charge distribution which is against the framework of the question...

Thoughts?

 

[obafgkmrns]

It's non-zero. You can prove it by superposition: Suppose the charge is distributed uniformly everywhere except at a point. Then the field is the sum of a uniform field (i.e., zero inside the sphere) and the excess field at the point (non-zero everywhere).