Imagine there is a cylinder with a charge density of +Q per unit volume and of infinite length. Now place a spherical cavity inside it with a diameter equal to the cross-section diameter of the cylinder. Is there an electric field inside the sphere? If so, is it possible to calculate the E-field with Gauss's Law?
closed as off-topic by DilithiumMatrix, user36790, Sebastian Riese, ACuriousMind♦, Norbert Schuch Oct 5 '16 at 20:28
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Yes, you can use Gauss's law, but I will leave you to work out the details. You use the principle of superposition.
Use Gauss's law (cylindrical symmetry) to work out the E-field inside the uniform cylinder, without the spherical hole in it.
Use Gauss's law (spherical symmetry) to work out what the E- field would be due to a sphere with a negative charge density $-Q$, in the position you have shown the spherical cavity.
Your situation is equivalent to the sum of these two fields.
I do not see any symmetry which would make Gauss law useful here. There is a symmetry plane separating the upper and the lower cylinder and there is rotational symmetry around the cylinder axis. Considering these symmetries you'll probably have to use Coulomb's law for the electric fields and integrate over all charge elements in the cylinder.