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How can we show that charges should go to the surface of a conductor, assuming that system should minimize its energy? (With no additional assumptions and maybe using variation method!)

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    $\begingroup$ Possible duplicate of Does the induced charge on a conductor stay at the surface? $\endgroup$ – Semoi Jun 18 '19 at 21:46
  • $\begingroup$ the questions are the same and I know the reasons people say there but I want to look different to this problem and use variation method and minimizing the energy of system to proof charges should go to the surface! $\endgroup$ – a.p Jun 18 '19 at 21:53
  • $\begingroup$ Systems don't move until their energy is minimized. They move until forces are balanced and at equilibrium $\endgroup$ – Andrew Jun 18 '19 at 23:15
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How can we show that charges should go to the surface of a conductor, assuming that system should minimize its energy? (With no additional assumptions and maybe using variation method!)

See the diagrams below.

The top diagram shows an isolated conductor not in the presence of an electric field. The distribution of charges in the conductor are randomly since there is no electric force field acting upon them.

The bottom diagram shows the same conductor in a uniform electric field. By convention, the direction of the electric field (direction of the arrows) is the direction of the force that a positive charge would experience if placed in the field.

Now the electrons in a conductor are highly mobile and free to move in response to an external force. In the bottom diagram the electrons in the conductor are attracted to the surface of the left side of the conductor by the force of the field leaving the right surface a deficit of electrons and thus being positively charged.

The electric field produced within the conductor due to the movement of charge to the surfaces is equal and opposite to the external electric field, and therefore the net electric field within the conductor is zero.

Hope this helps.

enter image description here

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