If there is a non-conducting solid sphere with uniform volume charge density $\rho$ and inside it there is a spherical cavity located outside the center of the sphere, why is it the case that the charge density is $-\rho$?
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1 Answer
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There are a few ways of solving this. But there is a kind of elegant and simple way:
If the volumetric charge density of the large sphere is said to be $\rho$, this mean that if you enclose any volume inside the sphere, that volumetric boundary would also have a volumetric charge density of $\rho$, since the density is homogenous (constant throughout the object).
If we removed a chunk of the large sphere, the void created could, essentially, be seen as a region with the corresponding charge density but with negative sign.
Note that this solution relies on the fact that the EM fields derived from Maxwell's Equations can be superimposed: The solution for the EM fields inside the void is the same as the solution for the EM fields inside the smaller cavity superimposed with the larger sphere.
