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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?

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    $\begingroup$ 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. $\endgroup$ – John Rennie Apr 22 '14 at 17:08
  • $\begingroup$ 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 $\vec{E}$. $\endgroup$ – auxsvr Apr 22 '14 at 21:40
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To add to what John said. Gauss' law says that the flux of electric field through a surface is proportional to the enclosed charge. If there were a net charge on some part of the inside of the conductor, then there would be an electric field coming out of it. Since electrons are free to move, they would move towards it for a net positive charge and be repelled for a net negative charge. For a negative charge, the farthest away they can move to would be on the surface of the conductor. Once there, the negatively charged volume inside the conductor would still exert a force on the electrons at the surface, but they can't move farther away; apply Newton's third law, the electrons in the negative clump are repelled as far away as they can go; the surface. For a positive clump on the interior, the electrons of the conductor would move to neutralize the positive charge, which would leave net positive charges where they used to be. Basically, all electrons would keep moving until the positive charge was spread over the surface such that all electrons on the interior felt no net force.

The short form is as John said; if there is an electric field inside a conductor, the electrons will follow it. They can't move past the surface, so that's where the charges will end up coming to rest.

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When a conductor is placed in an electric field, the charges realign till an equilibrium is reached when the electric field developed inside the conductor becomes equal and opposite to the external one. Thus cancelling and so net electric field inside is zero.

Inside a conductor, the mobile charges are free to move anywhere. Under the influence of external electric field, they move inside the conductor until they reach the surface of the conductor and collect there because if they are inside the conductor, there can not be static concentration of charges since they can move inside. After coming to the surface, as they can not move by the boundary, they reside there. As the charges have moved to the surface and equilibrium is reached i.e., the net electric field inside is zero or the potential throughout is constant, the charges cannot move anymore thus residing on the surface.

See also this: http://en.wikipedia.org/wiki/Electrostatic_induction#Induced_charge_resides_on_the_surface

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