You can’t dig a straight tunnel through the earth and end up in China if you’re from the US: Antipode map of the world.

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If you are let’s say, 1000km deep inside the Earth, you only feel the gravity from the inner 5400km (instead of the whole 6400km when you’re on the surface, like right now I presume). The outer 1000km don’t count, their gravitational pull just compensate!

It is not difficult to prove mathematically, if you are inside a sphere, everything that’s in the outer layers balances out, only the inside counts. In your situation of a hollow Earth, if you are inside an empty shell you indeed feel no gravity.

Since the equations for electric forces are exactly the same, you end up with a similar result (check out [Gauss’ theorem](http://en.wikipedia.org/wiki/Divergence_theorem) if you’re interested or just really bored).

(all this assumes spherical symmetry)

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