Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

y

x

Figure 25.1 The equipotentials (dashed circles) and field lines (solid lines)

for a line charge perpendicular to thez-plane.

of solving some two-dimensional physical problems describable by a potential

satisfying∇^2 φ= 0. The general method is known as that ofcomplex potentials.

We also found that iff=u+ivis an analytic function ofzthen any curve

u= constant intersects any curvev= constant at right angles. In the context of

solutions of Laplace’s equation, this result implies that the real and imaginary

parts off(z) have an additional connection between them, for if the set of

contours on which one of them is a constant represents the equipotentials of a

system then the contours on which the other is constant, being orthogonal to

each of the first set, must represent thecorrespondingfield lines or stream lines,

depending on the context. The analytic functionfis the complex potential. It

is conventional to useφandψ(rather thanuandv) to denote the real and

imaginary parts of a complex potential, so thatf=φ+iψ.

As an example, consider the function

f(z)=

−q

2 π 0

lnz (25.1)

in connection with the physical situation of a line charge of strengthqper unit

length passing through the origin, perpendicular to thez-plane (figure 25.1). Its

real and imaginary parts are

φ=

−q

2 π 0

ln|z|,ψ=

−q

2 π 0

argz. (25.2)

The contours in thez-plane ofφ= constant are concentric circles and those of

ψ= constant are radial lines. As expected these are orthogonal sets, but in addi-

tion they are, respectively, the equipotentials and electric field lines appropriate to