[Physics] Is electric field inside a cavity necessarily zero

Question

Would the field in a cavity inside an arbitrary conductor with some charge, which is NOT SYMMETRIC be zero?

My book only says that field inside a cavity is always zero when there is no charge in cavity. But they prove this by taking a sphere with a symmetrical cavity and using Gauss law. But how can I prove this in cases where I can't take $E$ (field) out of:

$$\oint\pmb{E}\cdot d\pmb{A}=\frac{Q_{enclosed}}{\varepsilon_0}.$$

Answer 1

In a static situation, there can be no field inside a conductor. If there were, charges would move until there was no field. This means that every point within a conductor (including points on the surface of an empty cavity) is at the same potential. This in tern means there can be no field inside the cavity.