11.6 • Two-DIMENSIONAL ELECTROSTATICS 459Figure 11.3511.6 Two-Dimensional Electrostatics
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, as mentioned in Section ll.4. This assumption results in an
electric field E (x, y) that can be interpreted as the force acting on a unit positive
charge placed at the point (x, y). In the study of electrostatics, the vector field
E (x,y) is shown to be conservative and is derivable from a function
called the electrostatic potential, expressed as
E (x, y) =-grad</> (x, y) = -</>" (x, y) - i</>y (x, y).
If we make the additional assumption that there are no charges within the
domain D , then Gauss's law for electrostatic fields implies that the line integral
of the outward normal component of E (x, y) taken around any small rectangle
lying inside D is identically zero. A heuristic argument similar to the one we
used for steady state temperatures, with T(x,y) replaced by if>(x,y), will show
that the value of the line integral is
- (</>z:c (x, y) + i/>yy (x, y)] 6.x 6.y.
This quantity is zero, so we conclude that <f>(x,y) is a harmonic function. If we
let 1/J (x, y) be the harmonic conjugate, thenF (z) = </> (x, y) + i'l/J (x, y)
is the complex potential (not to be confused with the electrostatic potential).
The curves </> (x, y) = K 1 are called the equipotential curves, and the
curves 1/J (x, y) = K 2 a.re called the lines of flux. If a small test charge is
allowed to move under the influence of the field E (x,y), then it will travel
along a line of flux. Boundary value problems for the potential function 4> (x, y)
are mathematically the same as those for steady state heat flow, and they are
realizations of the Dirichlet problem where the harmonic function is 4> ( x, y ).