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.

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