Introduction

A two-dimensional electrostatic field is produced by a system of charged wires, plates, and cylindrical conductors that are perpendicular to the z plane. The wires, plates, and cylinders are assumed to be so long that the effects at the ends can be neglected. This sets up an electric field that can be interpreted as the force acing on a unit positive charge placed at the point (x,y). In the study of electrostatics the vector field is shown to be conservative and is derivable from a function , called the electrostatic potential, as expressed by the equation

If we make the additional assumption that there are no charges within the domain D, then Gauss' law for electrostatic fields implies that the line integral of the outward normal component of taken around any small rectangle lying inside D is identically zero. A heuristic argument similar to the one for steady state temperatures will show that the value of the line integral is

Since this quantity is zero, we conclude that is a harmonic function. We let denote the harmonic conjugate, and

is the complex potential (not to be confused with the electrostatic potential).

The curves are called the equipotential curves, and the curves are called the lines of flux. If a small test charge is allowed to move under the influence of the field , then it will travel along a line of flux. Boundary value problems for the potential function are mathematically the same as those for steady state heat flow, and they are realizations of the Dirichlet problem where the harmonic function is .

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(c) John Mathews, 1998, 2006