Potentials and Charged Conductors

When a conductor becomes charged, that charge distributes across its surface until electrostatic equilibrium is reached. Its surface is equipotential.

All points within a charged conductor experience an electric field of 0. This is because field lines from charges on the surface of the conductor oppose one another equally . However, having the electric field equal to zero at all points within a conductor, the electric potential within a conductor is not necessarily  equal to zero for all points within that same conductor. This can be proven by relating electric field and potential.

Electrical Charge at a Sharp Point of a Conductor

Repulsive forces towards the more sharply curved surface on the right aim more outward than along the surface of the conductor.

Given that work is the difference in final and initial potential energies (∆U), we can relate this difference to the dot product of force at every infinitesimal distance l along the path between the points within the conductor:

$\Delta U = - \int_i^f \! \vec F \cdot \mathrm{d} \vec l$

This is the equation for work, with ∆U substituted in place of W. Rewriting U as the product of charge (q) and potential difference (V), and force as the product of charge and electric field (E), we can assert:

$\Delta (qV) = - \int_i^f \! (q\vec E) \cdot \mathrm{d} \vec l$

Dividing both sides by the common term of q, we simplify the equation to:

$\Delta V = - \int_i^f \! \vec E \cdot \mathrm{d} \vec l$

Finally we derive the equation :

$dV = - \vec E \cdot \mathrm{d} \vec l = 0$

Thus we can conclude that, given that the electric field is constantly 0 for any location within the charged conductor, the potential difference in that same volume needs to be constant and equal to 0.

On the other hand, for points outside a conductor, potential is nonzero and can be defined by the very same equation, according to field and distance from the conductor.