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?
Answer
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.
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