PHY531 Problem Set 3. Due October 2, 2001.

- A complex function of a complex variable where
and are real, and

(1)

(2) - 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) 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.

- 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.

- 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 .
- 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)

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