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Let us now investigate another trick for solving Poisson's equation (actually
it only solves Laplace's equation).
Unfortunately, this method can only be applied in two dimensions.
The complex variable is conventionally written

(740) 
( should not be confused with a coordinate: this is a strictly twodimensional problem). We can write functions of the complex variable just like
we would write functions of a real variable. For instance,
For a given function, , we can substitute
and write

(743) 
where and are two real twodimensional functions. Thus, if

(744) 
then

(745) 
giving
We can define the derivative of a complex function in just the same manner as
we would define the derivative of a real function. Thus,

(748) 
However, we now have a slight problem. If is a ``welldefined''
function (we shall leave it to the mathematicians to specify exactly what
being welldefined entails: suffice to say that most functions we can think
of are welldefined) then it should not matter from which direction in the complex
plane we approach when taking the limit in Eq. (748).
There are, of course, many
different directions we could approach from, but if we look at a regular complex
function, (say), then

(749) 
is perfectly welldefined, and is, therefore, completely independent of the details of
how the limit is taken in Eq. (748).
The fact that Eq. (748)
has to give the same result, no matter which path we approach
from, means that there are some restrictions on the functions and in
Eq. (743).
Suppose that we approach along the real axis, so that
.
Then,
Suppose that we now approach along the imaginary axis, so that
. Then,
If is a welldefined function then its derivative must also be
welldefined,
which implies that the above two expressions are equivalent. This
requires that
These are called the CauchyRiemann relations, and are, in fact, sufficient to ensure
that all possible ways of taking the limit (748) give the same answer.
So far, we have found that a general complex function can be written

(754) 
where
. If is welldefined then and automatically satisfy the CauchyRiemann relations.
But, what has all of this got to do with electrostatics? Well, we can combine the
two CauchyRiemann relations. We get

(755) 
and

(756) 
which reduce to
Thus, both and automatically satisfy Laplace's equation in
two dimensions; i.e., both and are possible twodimensional scalar potentials
in free space.
Consider the twodimensional gradients of and :
Now

(761) 
It follows from the CauchyRiemann relations that

(762) 
Thus, the contours of are everywhere perpendicular to the contours of .
It follows that if maps out the contours of some free space scalar potential
then indicates the directions of the associated electric fieldlines,
and vice versa.
Figure 45:

For every welldefined complex function we can think of, we get two sets
of free space potentials, and the associated electric fieldlines. For example,
consider the function , for which
These are, in fact, the equations of two sets of orthogonal hyperboloids.
So, (the solid lines in Fig. 45)
might represent the contours of some scalar potential and
(the dashed lines in Fig. 45)
the associated electric field lines, or vice versa. But, how could we
actually generate a hyperboloidal potential? This is easy. Consider the contours
of at level . These could represent the surfaces of four hyperboloid
conductors maintained at potentials . The scalar potential in the
region between these conductors is given by
, and the associated
electric fieldlines follow the contours of .
Note that

(765) 
Thus, the component of the electric
field is directly proportional to the distance
from the axis. Likewise, for component of the field is directly proportional
to the distance from the axis. This property
can be exploited to make devices (called quadrupole electrostatic lenses) which
are useful for focusing particle beams.
As a second example, consider the complex function

(766) 
where is real and positive. Writing
, we find that

(767) 
Far from the origin,
, which is the potential of a
uniform electric field, of unit amplitude, pointing in the direction. The locus
of is , and

(768) 
which corresponds to a circle of radius centered on the origin. Hence,
we conclude that the potential

(769) 
corresponds to that outside a grounded, infinitely long, conducting cylinder of radius , running parallel
to the axis, placed in a uniform directed electric field of
magnitude . Of course, the potential inside the cylinder (i.e.,
) is zero. The induced charge density on the surface of
the cylinder is simply

(770) 
where , and
. Note that zero
net charge is induced on the surface.
We can think of the set of all possible welldefined complex functions as a
reference library of solutions to Laplace's equation in two
dimensions. We have only considered a couple of examples, but there are, of course,
very many complex functions which generate interesting potentials.
For instance,
generates the potential around a semiinfinite,
thin, grounded, conducting plate placed
in an external field, whereas
yields the potential outside a
grounded, rectangular, conducting corner under similar circumstances.
Next: Separation of variables
Up: Electrostatics
Previous: The method of images
Richard Fitzpatrick
20060202