Force on charge at center of spherical shell

Question

A thin metallic spherical shell contains a charge $Q$ on its surface. A point charge $q_1$ is placed at the centre of the shell and another charge $q_2$ is placed outside the shell. All the three charges are positive. Then, the force on charge $q_1$​ is:

The answer given is zero.

However, I believe that electrostatic shielding works only if there are no charges in the cavity of a conductor. In this case there is a charge in the cavity. Then, how do I prove that there is no force on the charge $q_1$?

Answer 1

Apply the integral form of Gauss's law for a spherical shell of radius R using a concentric spherical surface of r<R.

Wikipedia has a Derivation of Gauss's Law

$∯_{surface}E \cdotp dA = \frac{Q_{enclosed}}{\epsilon_0} $

Since we chose a symmetric shape for our Guassian surface (we can rotate the system however we want and it looks identical) we know that the electric field must be the same at all parts of the Gaussian surface. Therefore

$∯_{surface}E \cdotp dA = E∯_{surface}dA = Ek$

where k is the surface area of the gaussian surface.

In the absence of the test charge, for a spherical surface of $r<R, Q_{enclosed}=0$. Therefore

$E = 0$ for $r<R$

Adding a test charge to a region of zero electric field results in zero force on the test charge.