Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

A conducting circular cylinder of radiusais placed with its centre line passing through

the origin and perpendicular to a uniform electric fieldEin thex-direction. Find the charge

per unit length induced on the half of thecylinder that lies in the regionx< 0.

As mentioned immediately following the previous example, the appropriate complex
potential for this problem isf(z)=−E(z−a^2 /z). Writingz=rexpiθthis becomes

f(z)=−E

[

rexpiθ−

a^2

r

exp(−iθ)

]

=−E

(

r−

a^2

r

)

cosθ−iE

(

r+

a^2

r

)

sinθ,

so that onr=athe imaginary part offis given by

ψ=− 2 Easinθ.

Therefore the induced chargeqper unit length on the left half of the cylinder, between
θ=π/2andθ=3π/2, is given by

q=2 0 Ea[sin(3π/2)−sin(π/2)] =− 4  0 Ea.

25.2 Applications of conformal transformations

In section 24.7 of the previous chapter it was shown that, under a conformal

transformationw=g(z)fromz=x+iyto a new variablew=r+is,ifasolution

of Laplace’s equation in some regionRof thexy-plane can be found as the real

or imaginary part of an analytic function§ofz, then the same expression put in

terms ofrandswill be a solution of Laplace’s equation in the corresponding

regionR′of thew-plane, and vice versa. In addition, if the solution is constant

over the boundaryCof the regionRin thexy-plane, then the solution in the

w-plane will take the same constant value over the corresponding curveC′that

boundsR′.

Thus, from any two-dimensional solution of Laplace’s equation for a particular

geometry, typified by those discussed in the previous section, further solutions for

other geometries can be obtained by making conformal transformations. From

the physical point of view the given geometry is usually complicated, and so

the solution is sought by transforming to a simpler one. However, working from

simpler to more complicated situations can provide useful experience and make

it more likely that the reverse procedure can be tackled successfully.

§In fact, the original solution in thexy-plane need not be given explicitly as the real or imaginary

part of an analytic function. Any solution of∇^2 φ= 0 in thexy-plane is carried over into another

solution of∇^2 φ= 0 in the new variables by a conformal transformation, and vice versa.