PHY531 Problem Set 3. Due October 2, 2001.

1. A complex function of a complex variable where and are real, and
 (1)

is said to be analytic in a region if the derivative defined by the limit
 (2)

is independent of how the complex variable is taken to zero.

a.
Show that is analytic in a region if and only if
 (3)

b.
Show that if is analytic in a region, both and satisfy the two dimensional Laplace's equation there.
c.
Show that if both and are analytic functions in the appropriate regions, that is also analytic.
d.
Show that the function
 (4)

is analytic except at the origin and along the negative axis. Here is the quadrant correct arctangent that goes from to and has a discontinuity along the negative axis.

What simple physical situations lead to electric or magnetic fields given by the negative gradient of the functions and corresponding to the complex function ?

Many two-dimensional electrostatic problems can be solved by conformal mapping which uses the properties proved above to transform the boundaries of a known solution to other solutions.

2. A region of space in circular cylindrical coordinates is defined by
 (5)

where . The potential on all surfaces is zero except the surface which is held at a fixed potential . Calculate an expansion of the potential everywhere inside the region in terms of modified Bessel functions.

Show that near , the charge density on the walls near has the form of Jackson Eq. 2.75 as expected.

3. A long conducting circular cylinder of radius has its axis along . At the cylinder has a small slit cut into it so the portion along is insulated from the portion along . A battery is connected across the gap so that the potential of the upper half of the cylinder differs from the lower half by . Calculate a series expansion of the potential everywhere. Calculate the electric field along the axis and make an accurate plot of versus .

4. Repeat problem 3 for the case where the cylinder has a square cross section of side .

A conformal mapping of the inside of a square to a circle leads to an elliptic function transformation. One result is that away from the corners, the square of side behaves approximately like a circle of radius,

 (6)

where is a complete elliptic integral of the first kind,
 (7)

Plot the versus for and on the same graph as for problem 3 for comparison.