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The figure below shows a hollow metal sphere with a positive point charge $Q$ sitting outside it. What is the electric field at the center of sphere ? The answer is zero (look at here at the beginning of page 4), but I do not understand why?

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Well, have you tried google? – Emilio Pisanty Nov 15 '12 at 0:58

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Suppose electric field inside the sphere is nonzero. Then since there is no charge inside the sphere and since electric lines of force do not form closed loops so we should be able to find two points A and B on the surface of sphere such that a line of force starts from A and ends at B, thus causing a potential difference between these points. But since the sphere is made of metal (which are usually good conductors) so there will be a flow of current between these two points until the potential difference between them vanishes. So in equilibrium i.e. when no current is flowing, electric field inside sphere should be zero.

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Since (1) the metallic sphere is an equipotential surface and (2) the potential inside the sphere must satisfy Laplace's equation, it follows, by the uniqueness theorem, that the potential inside the sphere is constant and thus, that the electric field inside the sphere is zero.

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So is it true that although there would not be any charge outside of the sphere, electric field inside the sphere would still be zero? – alvoutila Nov 15 '12 at 10:06
I must say that although you explained why there is not electric field inside the ball, it is still hard to understand why? I think if you could give me a picture about electric field inside a sphere(conductor), it would help me. – alvoutila Nov 15 '12 at 10:28
@alvoutila, if there is an electric field inside the hollow region, the electric potential at some points must be higher or lower than the potential of the inner surface. But, this would mean that there is a local minimum or maximum of the potential in the hollow region. But, this is mathematically impossible! At a local min or max, all the 2nd partial derivatives of the potential must have the same sign but, to satisfy Laplace's equation, they must add to zero. So, the electric field must be zero in the hollow region. – Alfred Centauri Nov 15 '12 at 14:54

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