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Suppose we had a hollow conducting sphere with a net charge q on it. There is no charge in the cavity; the conductor itself has a charge q. The idea is that this net charge would reside on the 'surface', since the conductor has free charges otherwise to make the net field inside the meat of the conductor zero.

  1. My question is with regard to what 'surface' means. Does 'surface' refer to the interface between the conductor and air? If so, why isn't there charge residing on the inner surface of the hollow sphere? Has this something to do with uniqueness theorems?

  2. Related question: If there is an external charge q outside an uncharged hollow conductor, why is there no induced charge on the inner 'surface'?

  3. Also related: How is it that information of charge inside a cavity is known outside, but that outside is completely unknown inside? In a sense, aren't both regions of air the same and separated by the conductor only? Better put: when there is a charge inside the cavity, the inner surface charge distribution cancels the field in the conductor due to the cavity charge, and residual charge q sits uniformly on the outer surface, thus 'revealing' the presence of q to the outside. Why does the same not happen to a charge outside?

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  • $\begingroup$ What do you mean by a hollow sphere, is it have thickness? $\endgroup$ – Mehmet Bütün Oct 5 '20 at 14:59
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    $\begingroup$ @Butane yes, it does have thickness. So think of a solid conducting sphere of radius say, 1m, and a concentric solid sphere of radius 0.5m removed from the first sphere. $\endgroup$ – PhutureFysicist Oct 6 '20 at 4:47
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Initially, the meaning of the surface for your question is the upper boundary of the sphere which is equivalent to the sphere shell with a thickness infinitesimally small (mathematically, goes to zero). When it comes to the problem of why charges are not residing at the inner surface or in between the layers is, I need to explain how the charges are defined. Charges are defined as points so if you consider the cardinality of the surface of the sphere you will see that it is capable of accomodating uncountably infinite amount of charge. If you have any suspicion about injecting charges to the initially charged surface, a sphere shell can carry all the charges on only the (mathematical) surface. Also, the charges distribute themselves in a fashion that minimizes their potential energy, so if you consider Coulomb's law, for some distribution some charges can reside in the interior of the material but this condition doesn't minimize the potential energy. Applying the same condition for your second question, you will easily see that under an electric field charges distribute themselves (for the spherical shell) symmetrically, and using the minimization of potential energy there will be no charge interior of the material. In addition, if you put a charged sphere under an electric field the distribution for the final state requires a detailed analysis and calculation to carry and I think this may help you to understand the charge distribution from a more general perspective.

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