PHY531 Problem Set 4. Due October 16, 2000

- An incompressible conducting charged liquid drop is initially spherical
with a radius and has a total charge . The surface is then
slightly perturbed. The distance from the origin to the surface as
a function of the usual polar angles is

(1) - a.
- Show that the electrostatic energy and the surface area of the drop,
correct to second order in
are

(2)

- b.
- The total potential energy from a small distortion is

(3) - c.
- A raindrop can be considered to be a conductor for electrostatic
purposes. What is the potential difference, in Gaussian units, between
the drop surface and infinity for a spherical drop that is just stable if
its diameter is 1 millimeter? Convert this result to Volts. The
surface tension of water at 20C is 72.75 erg/cm from
the
*American Insitute of Physics Handbook*, third edition, Ed. D.E. Gray, (McGraw Hill, New York, 1972).

This problem was first solved by Rayleigh in 1882.

- Jackson Problem 3.23
- a.
- Show that if
is a solution of Laplace's equation,
that

(4) - b.
- Apply the result of part a to the potential of a point charge on the
axis of an infinitely long grounded conducting circular cylinder,
to show that the self capacitance of a conducting surface generated
by rotating a circle of diameter about one of its tangents is

(5) - c.
- Use the result of Jackson Problem 1.20 to give crude upper and lower
bounds to the coefficient of in the self capacitance result.
- d.
- Numerically evaluate the coefficient of in the self capacitance
to at least 3 place accuracy.
You may find
Euler's transformation, which
speeds up the convergence of well behaved alternating series,

(6)

- Jackson Problem 3.3