**39. **Show that the mapping
is conformal except at a finite set of points.

A parallel plate capacitor has plates that extend from to . Find an appropriate scaling that allows you to place the plates at Show that the given transformation maps the plates to the lines Solve for the potential between the plates in the plane, map to the plane and hence find the equipotential surfaces at the ends of the capacitor. Sketch the field lines. This is the so-called fringing field.

** **Choose
where
is a coordinate measured perpendicular to the plates, and
is the plate separation. The function
is analytic everywhere, and the derivative is

is non-zero except at the points

or, equivalently,

The mapping takes the form:

Then for
ranges from
to
i.e. we get the whole real axis in the
plane.
The line
maps to
ranges from
at
to
at
This is the top plate of the capacitor. Similarly
maps to the lower plate.

The mapping has a branch point at each of the points Each 2wide strip of the plane maps to the whole plane For each branch there are two points in the plane at which the mapping is not conformal.

In the
plane
we can write the potential as
giving a complex potential
with the complex part being the physical potential. Equipotentials correspond
to
const
The corresponding curves in the
plane
are:

Thus

and

The equipotenials are shown in the figure.