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The sphere is grounded but has a uniform charge density $\rho$ inside. If I were to draw a picture of this, could I label the surface as $\sigma =0$?

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No, a grounded sphere can have a charge on the surface, e.g. induced charges by a nearby charge. If you have a uniform charge ρ inside a grounded sphere, this means that the sphere is not a conductor inside because conductors can only have surface charges due to the conductivity. You cannot draw a picture labeling the surface of this grounded sphere with σ=0 because this is , in general, not the case.

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  • $\begingroup$ Then what does the term grounded mean? $\endgroup$ – whatwhatwhat Oct 4 '16 at 22:20
  • $\begingroup$ As far as I understand it, the sphere is a hollow metal shell, which is grounded. Inside you have an insulator (or vacuum) which has a constant charge density ρ. $\endgroup$ – freecharly Oct 4 '16 at 22:34
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Grounding a conductor means that its potential is essentially fixed. Generally this fixed value is taken as zero. if a body is grounded, then it means that any amount of finite charge on addition(or removal) from the grounded conductor does not change its potential(which does happen for a non-grounded conductor).

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