Electric Field Inside of a Conductor

Question

I have seen many explanations for why the net electric field inside of a conductor is zero (assuming only electrical forces are acting on the particles inside of it). These explanations typically say that the charges in the conductor will move the outside and, to ensure that they remain stationary, there must be a net electric field of zero everywhere inside of the conductor. There seems to me to be a very large disconnect in this logic. If all of the charges are on the surface, then yes, I understand that the electric field near the surface of the conductor must be zero, but why does it have to be zero at, say, the center of the conductor (where there are presumably no charges).

Answer 1

I think that part of the problem stems from the (somewhat misleading) sentence "The charges inside a conductor move to its surface.". That is true only of the excess charge inside the conductor.

Using an example to clarify things, if a conductor is composed of, say, 10000 protons and 15000 electrons, we can expect some 5000 electrons to migrate to the surface of the conductor. However, the remaining 20000 particles will stay inside the conductor. Now, these particles are still charged, and they are more or less free to move (in metals, it's mostly the electrons that can do so). Thus, if an electric field existed inside the conductor, the charges would quickly start moving around (even if, on average, the interior of the conductor remained neutral). That would generate a current, magnetic fields, and so on, contradicting the assumption that we are dealing with a static conductor.

So, if we insist on a static situation, we must also insist on the fact that inside the conductor the electric field must be zero. If there is an excess of charge in the conductor, the $\mathbf{E} = 0$ condition will be achieved by a "smart" distribution of this excess on the boundary.

Feynman's lectures 5 and 6 on electromagnetism (particularly sections 5.9, 5.10 and 6.6) offer some more insight into this subject.

Answer 2

Well it goes with the assumption that it's a charge distribution in equilibrium and there isn't any current. It's important to note that electric field is not zero in a conductor carrying current. But here we are assuming no current.

If there was an electric field in the center of a conductor then, unless the conductivity is zero, there will be some current resulting from the electric field, violating our assumptions. Current density and electric field are related by

$$\mathbf J = \sigma \mathbf E$$

If you allow non-equilibrium conditions then there is little restriction on where you can place the charges and what the resulting electric field inside the conductor will be. But the charges won't necessarily stay there.

Discussing the equilibrium conditions is a very useful constraint.