Phys 312 - Assignment 1 - Due 22 Jan 98

1.

A parallel plate capacitor has a small hemispherical boss of radius a on one of its plates. To be definite, you can assume that A is the area of each plate, d the separation between them, with ; take the plates to be perpendicular to the z axis and put the boss at the center of the lower plate. However, the results you will obtain hold more generally, as long as a is much smaller than the distance of the boss from the closest edge. The problem is to find the electric field.

You can imagine that the lower plate is a flat plain, the upper plate is the bottom of a large cloud, and the hemispherical boss represents a hill in the middle of the plain, or maybe a tree in the middle of a field. The problem then tells you something about the chances of being struck by lightning in hilly country, or why isolated trees are at risk of being struck.

(a) If is the charge density on the lower plate far away from the boss, what is the electric field far away from the boss, ? You can (approximately) relate to the area, A, and to the total charge on the plate, Q, but it is not necessary to do so.

(b) Find the field and the charge density everywhere on the lower plate. Hints for doing this will be given in class.

(c) Where on the lower plate does the maximum electric field occur, and how much larger than is it?

(d) Where on the lower plate does the minimum electric field occur, and how much smaller than is it?

(e) Using a computer program, draw a set of equipotentials, starting from the lower plate. The graph should illustrate how the equipotentials become smooth for .