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I am wondering about the answer to the PSE question here: Electric Field inside a hollow ball, excentred of a homogeneous charged ball

I understand the solution, but I don't understand how it corresponds to the Gauss's law. The charge inside the hollow is equal to zero, so why isn't the electric field?

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  • $\begingroup$ Where on that page do you see that the field is nonzero inside the hollow? $\endgroup$ Commented Nov 13, 2017 at 22:44
  • $\begingroup$ It is not written there, but it is certainly nonzero ;) $\endgroup$ Commented Nov 13, 2017 at 23:08
  • $\begingroup$ Oh, the hollow is displaced from the center. $\endgroup$ Commented Nov 13, 2017 at 23:10
  • $\begingroup$ -1. On this site it is considered a bad practice to post "link only" questions or answers. If the linked material is deleted then your question becomes incomprehensible. You should always summarise the essential information from the link in your own question. $\endgroup$ Commented Nov 14, 2017 at 0:47
  • $\begingroup$ @sammygerbil, the link is to a question on this site (if it makes any difference). $\endgroup$ Commented Nov 14, 2017 at 0:49

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The superposition principle tells us that the field due to the two objects (a fully charged sphere, and a smaller sphere with opposite charge density) can be added together. It is entirely possible to have a non-zero electric field in a region without any charge in it (think for example about the space between the plates of a parallel plate capacitor). You just need a field whose divergence is zero, that is $\nabla \cdot E = 0$. This is true, for example, for a uniform field, or a field of the form $e\propto \frac{1}{r}$ .

The absence of a charge does not imply the absence of field - just the absence of divergence.

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but I don't understand how it corresponds to the Gauss's law.

Gauss' law tells you that the electric flux through the surface that defines the hollow volume is zero since the charge enclosed is zero.

But that doesn't imply that the electric field is zero within the hollow volume only that the electric field is special in the sense that all of the field lines entering the volume leave the volume since there is no electric charge there on which electric field lines originate and/or terminate.

That is, as Floris points out, the electric field within the hollow volume has zero divergence.

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