Electric Potential - in Electrostatics

In Electrostatics

The electric potential at a point r in a static electric field E is given by the line integral

where C is an arbitrary path connecting the point with zero potential to r. When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. In this case, the electric field is conservative and determined by the gradient of the potential:

Then, by Gauss's law, the potential satisfies Poisson's equation:

where ρ is the total charge density (including bound charge) and ∇· denotes the divergence.

The concept of electric potential is closely linked with potential energy. A test charge q has an electric potential energy U_E given by

The potential energy and hence also the electric potential is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential are zero.

These equations cannot be used if the curl ∇ × E ≠ 0, i.e., in the case of a nonconservative electric field (caused by a changing magnetic field; see Maxwell's equations). The generalization of electric potential to this case is described below.