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There are two conducting spheres of different charge and a conducting wire. After they are connected by the wire, charge flows between the spheres. The charge distributes itself so that the spheres are at the same potential, but I have not been able to find an explanation for this. Why wouldn't the charge be distributed so that the electric field is the same, since that is what is moving the charges in the wire?

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    $\begingroup$ In the given link and more precisely in Electrically connected charged balls you could find a full explanation and solution. The instance the two balls are connected there exists potential difference $\:-\boldsymbol{\nabla}\phi\:$ and potential difference means non-zero electric field $\:\mathbf{E}\:$. $\endgroup$ – Frobenius Feb 6 '17 at 18:48
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    $\begingroup$ This field is responsible for the motion of charge from the ball with the higher potential to the other ball until the whole system (the two balls and the wire) turns to be an equipotential region. Then in this region $\:\mathbf{E}=\boldsymbol{0}\:$ so there is no charge motion. The system is balanced. $\endgroup$ – Frobenius Feb 6 '17 at 18:49
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Voltage is electrical pressure. Think about two tanks of water, one big and other small both connected by a pipe. Now if you connect the two, water will flow between the two till the level is same in the two tanks. Similarly, charge will flow till the metal sphere are at same potential.

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Once short circuited potential difference drops to zero. Only when there is a difference of charge field lines and equipotential lines can exist.

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