Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.1 COMPLEX POTENTIALS

the field produced by the line charge. The minus sign is needed in (25.1) because

the value ofφmust decrease with increasing distance from the origin.

Suppose we make the choice that the real partφof the analytic functionf

gives the conventional potential function;ψcould equally well be selected. Then

we may consider how the direction and magnitude of the field are related tof.

Show that for any complex (electrostatic) potentialf(z)the strength of the electric field

is given byE=|f′(z)|and that its direction makes an angle ofπ−arg[f′(z)]with the

x-axis.

Becauseφ= constant is an equipotential, the field has components

Ex=−

∂φ

∂x

and Ey=−

∂φ

∂y

. (25.3)

Sincefis analytic, (i) we may use the Cauchy–Riemann relations (24.5) to change the
second of these, obtaining

Ex=−

∂φ

∂x

and Ey=

∂ψ

∂x

; (25.4)

(ii) the direction of differentiation at a point is immaterial and so

df

dz

=

∂f

∂x

=

∂φ

∂x

+i

∂ψ

∂x

=−Ex+iEy. (25.5)

From these it can be seen that the field at a point is given in magnitude byE=|f′(z)|
and that it makes an angle with thex-axis given byπ−arg[f′(z)].

It will be apparent from the above that much of physical interest can be

calculated by working directly in terms offandz. In particular, the electric field

vectorEmay be represented, using (25.5) above, by the quantity

E=Ex+iEy=−[f′(z)]∗.

Complex potentials can be used in two-dimensional fluid mechanics problems

in a similar way. If the flow is stationary (i.e. the velocity of the fluid does not

depend on time) and irrotational, and the fluid is both incompressible and non-

viscous, then the velocity of the fluid can be described byV=∇φ,whereφis the

velocity potential and satisfies∇^2 φ= 0. If, for a complex potentialf=φ+iψ,

the real partφis taken to represent the velocity potential then the curvesψ=

constant will be the streamlines of the flow. In a direct parallel with the electric

field, the velocity may be represented in terms of the complex potential by

V=Vx+iVy=[f′(z)]∗,

the difference of a minus sign reflecting the same difference between the definitions

ofEandV. The speed of the flow is equal to|f′(z)|. Points wheref′(z) = 0, and

thus the velocity is zero, are calledstagnation pointsof the flow.

Analogously to the electrostatic case, a linesourceof fluid atz=z 0 , perpendic-

ular to thez-plane (i.e. a point from which fluid is emerging at a constant rate),