# Does the induced charge on a conductor stay at the surface?

My textbook says that when a conductor is placed in an electric field, the electrons in it realign so that the net electric field inside the conductor is zero. There isn't a proof for this. It merely states that the electrons drift opposite to the applied electric field and create an opposite electric field inside the conductor canceling the external one.

I imagine Gauss's Law would provide a simple proof. If we try enclose a point inside the conductor in a gaussian surface, and note that the enclosed charge is zero, so the flux through the surface is zero and so is the electric field at the surface.

When I looked back at my reasoning, I realized that I had assumed that the charges induced on the conductor always stay at the surface, which I had no way of proving. How do I prove this formally?

• The electrons in a conductor are free to move. Therefore if there is any potential gradient in a conductor the electrons will move along it. At equilibrium there cannot be any potential gradient (i.e. electric field) inside the conductor or the electrons would move and it wouldn't be in equilibrium. – John Rennie Apr 22 '14 at 17:08
• This is true in the static case, not in general. In the general case you need to use the continuity equation and Ohm's law, which yield $\rho(t) = \rho(0)e^{-\frac{\sigma}{\epsilon}t}$; since $\vec{J}$ vanishes asymptotically (static case), so does $\vec{E}$. – auxsvr Apr 22 '14 at 21:40