Say you have a hollow sphere with a uniformly distributed charge on the surface. Why is the electric field everywhere inside the sphere zero?
For the centre, its easy to add the vectors from the surface charges and show they sum to zero because of symmetry.
But can you show me how the field cancels out to zero for points other than the centre using vector addition?
It is very simple, but strictly applies to a conducting sphere.
From ANY point inside a sphere, draw a double cone that is zero dimension at that point.
The cross-section of the cone, can be any shape.
The two ends of the cone intersect the sphere in two similar shaped curved surfaces. Any line from a point inside one of those areas, through the tip of the cone to the opposite surface has two sections with a ratio of A:B in length.
The areas of the two end cap surfaces, are also in the ratio of A^2:B^2 and so are the charges on those two areas.
Since the inverse distances squared are also A^2:B^2, the force on a charge at the cone tip is net zero. This is true for any point in the sphere, and any cone angle or cross-section shape.
QED
The record that Virginia.edu explanation of the direction of gravity, is simply awful, although it gets the correct answer. The direction is obvious, once it is understood that the gravity inside a shell is zero, so only the smaller sphere inside produces a gravity field, which can only point to the cg of the sphere.
Any surface that is perfectly NON-CONDUCTING can not have electric charges moving on its surface.
Therefore it is impossible to put a uniform distribution of charge, on a non-conducting sphere.
My proof is ONLY valid for a conducting surface.
Because any Gauss surface made inside the hollow sphere encloses no charge so the net flux is zero. And if the Gaussian surface is shrunk to zero dimension, the electric field is zero at that point.
