Apparent contradiction while calculating potential inside shell due to off center charge

Question

Consider the following scenerio$-$

A point charge $Q$ is placed at an off center point $B$ in a spherical shell made out of a conducting material. We are required to find the potential at the center.

I understand how to calculate potentials due to induced charge on the inner surface of the shell and outer surface charge of the shell.

Now, since the conductor is equipotential, the work done in bringing a test charge from infinity to anywhere inside the conductor material is same. Let us bring our test charge, first through the path $C\rightarrow A$ and then through the path $E\rightarrow A$. Since electric field is conservative, potential due to the point charge must be constant at a point, however upon integrating along the two paths, we get clearly different results ( as travelling in the first path gives 0 work done by the field due to $Q$, but along the second path, the work done is definitely non zero).

What am I missing here?

Answer 1

travelling in the first path gives 0 work done by the field due to Q

Here is the mistake. $Q$ is not the only charge you need to consider. You must also consider the charge induced on the inner surface of the conductor. It is not spherically symmetric, so the field is not zero. The work done by that field is also nonzero on the path you describe (neglecting the problem at $Q$ itself).