Why is it necessarily true that all charges occupy themselves only on the surface of a conducting sphere, and not anywhere inside the sphere? One argument is that if a charge were to be inside a conducting sphere, then it would exert forces on other particles inside the sphere and there would be internal currents.
Now, my question is – do we have experimental evidence in every case that there are no currents or magnetic fields generated when a charged conducting sphere is held stationary with respect to another body? Or is our reason for believing that there are no charges inside the sphere of a more mathematical and theoretical nature?
I was discussing this with someone and they brought up Gauss' Law, but it seemed to me that the law is predicated on there not being any $E$ field inside the conductor for there to be no charges, which seemed like a somewhat circular argument. A counterargument was that the charges inside a conductor may exert forces, but it may not always end up producing a steady current flow.
I admit my question is of a very qualitative nature, but what are some strong reasons for why we posit that there can be no charges inside a conducting sphere?
Best Answer
The Gauss's law argument is as follows:
1) We know that there cannot be an E field inside the conductor, because if there was a net E field inside the conductor, then it would move charges, and the staticity assumption would break.
2) Now, assume that, in some region of the conductor, we have a net charge accumulated in some region.
3) Then, we can enclose that net charge in a Gaussian surface, and necessarily, it will have to obey $\oint {\vec E}\cdot {\vec dA} = q/\epsilon_{0}$. Since the RHS is nonzero, the LHS has to be nonzero, therefore, we have a net E field. We have arrived at a contradiction, so therefore, our assumptiont hat we can accumulate a net charge in the interior of the conductor must be false.