Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.2 APPLICATIONS OF CONFORMAL TRANSFORMATIONS

y s s

x r r

(a)z-plane (b)w-plane (c)w-plane

Figure 25.3 The equipotential lines (broken) and field lines (solid) (a) for an

infinite charged conducting plane aty=0,wherez=x+iy, and after the

transformations (b)w=z^2 and (c)w=z^1 /^2 of the situation shown in (a).

Find the complex electrostatic potential associated with an infinite charged conducting

platey=0, and thus obtain those associated with

(i)a semi-infinite charged conducting plate(r> 0 ,s=0);

(ii)the inside of a right-angled charged conducting wedge(r> 0 ,s =0and

r=0,s>0).

Figure 25.3(a) shows the equipotentials (broken lines) and field lines (solid lines) for the
infinite charged conducting planey= 0. Suppose that we elect to make the real part of
the complex potential coincide with the conventional electrostatic potential. If the plate is
charged to a potentialVthen clearly

φ(x, y)=V−ky, (25.6)

wherekis related to the charge densityσbyk=σ/ 0 , since physically the electric fieldE
has components (0,σ/ 0 )andE=−∇φ.
Thus what is needed is an analytic function ofz,ofwhichtherealpartisV−ky.This
can be obtained by inspection, but we may proceed formally and use the Cauchy–Riemann
relations to obtain the imaginary partψ(x, y) as follows:

∂ψ

∂y

=

∂φ

∂x

=0 and

∂ψ

∂x

=−

∂φ

∂y

=k.

Henceψ=kx+cand, absorbingcintoV, the required complex potential is

f(z)=V−ky+ikx=V+ikz. (25.7)

(i) Now consider the transformation

w=g(z)=z^2. (25.8)

This satisfies the criteria for a conformal mapping (except atz= 0) and carries the upper
half of thez-plane into the entirew-plane; the equipotential planey=0goesintothe
half-planer>0,s=0.
By the general results proved,f(z), when expressed in terms ofrands, will give a
complex potential whose real part will be constant on the half-plane in question; we