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Electric field, Coulomb's and Gauss' law

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The second part of question 1 is rather complicated. I'm sure that was not intended by your Prof. However, to understand the issues here, it is necessary to consider this in full detail.

Can there be an electric field at a point where there is no charge?

Clearly, the answer is "yes". E.g., a point charge q has an electric field of E = q/(4 pi epsilon_0 r^2) in the radial direction. This field exists in the "empty space" surrounding the point charge.

Can there be a charge at a place where there is no field?

To address this question rigorously is a bit complicated because of the fact that charge is quantized, i.e. it always appears as a multiple of the elementary charge. At the exact point where the charge resides, the electric field is undefined if you use Coulomb's formula. Let's first look at this problem from a macroscopic point of view where you can pretend as if charge is a continuous quantity. Then, according to Gauss' Law, we have:

Surface integral of E dot dS = Q/epsilon_{0} (1)

Here the surface integral is over a closed surface, dS is a ...

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