Suppose I have a sphere of radius $r$ with all the charge residing on the surface, distributed uniformly i.e. charge density $\sigma$ is constant. I want to find the electric field created by this sphere. All problems of this sort I have solved by enclosing the sphere in a larger imaginary sphere with radius $R$, and then using the assumption that the electric field lines are in a radial direction, in combination with the divergence theorem.
If I use that method for this question, I find that the Electric field inside the sphere is zero. I can't understand this intuitively. I would have thought that if you are in the proximity of charge, there will be an electric field. Why would the electric field lines extend outwards radially from the sphere, and not inwards too?
Answer
All of those charges on the other side of the sphere "conspire" to exactly cancel the field of the nearby charges on the surface. You're analysis is correct. This result can also be shown by integration of Coulomb's Law, but it's not an easy calculation.
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