Electric potential

In electrostatics the electric field is $E=-\nabla \Phi$, where $\Phi$ is the electric potential.

An image charge is a fictitious point charge placed outside the physical solution region so that its potential, combined with the real sources, enforces the required conductor boundary values. The Uniqueness of the Dirichlet problem then identifies the constructed potential with the physical one. The image need not describe the actual induced surface-charge distribution.

A real charge of height $z$ has a reflected image charge at $-z$. Its acceleration is $\ddot{z}=-\kappa /z^2$, with $\kappa =q^2/(16\pi {\epsilon }_{0}m)$. Released from rest at height $d$, its ideal-model arrival time isThe image follows the instantaneous reflected position. The induced-charge energy carries one-half the full real-image pair interaction factor, consistent with this force.

For a charge $q$ at $\mathbf{p}$, $R=|\mathbf{p}|>a$, outside a grounded sphere of radius $a$, the displayed image gives zero potential on the sphere and decay at infinity. On $|\mathbf{x}|=a$, one has $|\mathbf{x}-{\mathbf{p}}^{\prime }|=(a/R)|\mathbf{x}-\mathbf{p}|$, proving boundary cancellation. The induced force is inward, of magnitude $q^{2}aR/[4\pi {\epsilon }_0(R^2-a^2{)}^2]$, from Coulomb's law applied to the image field. The self-field is excluded.

An electric potential difference compares the electric potentials at two points. In an electrical circuit it is the voltage across the terminals of a component. A vertex potential in an electrical network is specified relative to a reference vertex; terminal differences are independent of an additive change of reference.

A point charge idealizes electric charge concentrated at one point. In vacuum, a charge $Q$ at the origin has potential $Q/(4\pi {ϵ}_{0}r)$.

An equipotential is a level set of the electric potential. The electric field is normal to every smooth equipotential.