# Do charged conductors exhibit an equipotential surface even when subject to an $E$-field?

A charged conductor, in the absence of an electric field, attains an electrostatic equilibrium such that its surface has a constant potential.

When a neutral conductor is placed in a uniform electric field, the electrons will rearrange themselves so as to, again, achieve equilibrium, thereby making the surface equipotential.

But what about a charged conductor in a uniform $$E$$-field? Aren't there too many constraints on the conductor?

For example the charges above need to :

• Rearrange themselves so as to minimize their mutual repulsion.
• The new rearrangement must be one that cancels the external field inside the conductor.

From the diagram, in order to cancel the $$E$$-field inside, all the positive charges must move to the right side of the sphere to create a field $$-E$$:

The optimal configuration that minimizes the mutual repulsion is the first one shown above, where the charges are spread over the whole surface, but in the presence of the $$E$$-field, the charges will move out of this optimal position in order to satisfy the second constraint, i.e.,cancel the field inside, thus disregarding the first one, i.e., mutual repulsion, so, in some sense, is the second condition 'stronger' than the first one?