Closed form analytical expressions for the electric field interior and exterior to a metal plate, due to alternating current injected at its surface, are derived. Assuming that the current is injected and extracted by wires oriented perpendicular to the surface of the plate, the problem is decomposed into two cylindrically symmetric systems in which a single current-carrying wire lies on the coordinate axis. This simplified problem is formulated in terms of a single magnetic potential and the solution obtained by use of the Hankel transform. The resulting expression for the electric field in the plate takes the form of an analytic series. In exterior regions, the electric field is expressed in terms of Hankel transforms. The result for the physical system with two current-carrying wires is obtained by superposition.