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If you have a sphere of radius $R$ with a uniformly distributed charge $Q$ on the surface and a point of charge $q$ with a distance to the center of the sphere on $d$, what would be the force experienced by the point charge? Would it be more correct to treat the sphere as a single point, using Coloumbs Law, or would it be more correct to use integrals to calculate the force from each 'point' on the sphere?

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For a general charge distribution the correct way is to use integrals to calculate the force from each point.

But a sphere uniformly charged is a special case, you can check with Gauss's law that it behaves just like a point charge in the center, so you can in this case use both ways: Coulomb's law (only in this special case) or integrals (always correct).

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    $\begingroup$ It doesn't even have to be uniformly charged; all that is required is that the charge distribution be spherically symmetric, so it could vary with the distance from the center. $\endgroup$ – Javier Jan 21 '17 at 17:57

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