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I can understand why the electric field outside a charged sphere equals kq/r².

But I remember reading somewhere that the electric field AT the surface of a charged sphere equals the electric field at a point infinitely close to the surface of the sphere divided by 2, i.e. half the field at extreme proximity.

I can't understand why that is, even though I thought a lot about it. Do you think this statement is true, and if so, can you prove it?