Physics 743 Fall 2000

Michael Fowler

Homework #1

1. A point charge Q is placed at a distance b from the center of a grounded perfectly conducting sphere of radius a, where a < b.  Suppose that the sphere is grounded by means of a long very thin conducting wire that goes straight out from the point on the sphere furthest from the charge Q.  Suppose further that this wire is so thin that the charges on the wire itself have a negligible effect on the electric field.  Now, far out on the wire (distance large compared with b) the electric field will be that of the charge Q, plus that of the (opposite sign) charge Q¢ residing on the sphere (the image charge). So, if Q¢ < Q, there will be a net field to attract in new charge from the ground.  This seems to suggest that charge will move in on the wire until Q¢ = Q.  However, this is not correct—the image charge Q¢ should equal –(a/b)Q. Explain what is wrong with this argument. Draw a picture of the field lines for the system with the correct image charge, following the diagram in Jackson page 59, but he has a large blank area where my wire would be.  Fill that in—show the field there in detail.

 

2. (Jackson problem 2.28).  A closed volume is bounded by conducting surfaces that are the n sides of a regular polyhedron (n = 4, 6, 8, 12, 20). The n surfaces are at different potentials V_i, i = 1, 2, … n.  Prove in the simplest way you can that the potential at the center of the polyhedron is the average of the potential on the n sides.

 

3. Consider the function tan z where z is a complex variable, x + iy.  Draw a contour diagram of the lines of constant modulus |tan z|.  Use the function log tan z to draw the electric field for a line of charge placed midway between two flat parallel infinite grounded conducting plates.  Find the induced charge on the plates.

 

4. If a charged soap bubble of radius r has the same air pressure inside as outside, what is the charge Q on the bubble in terms of the radius r and the surface tension?

 

5. Suppose a simple charged capacitor, consisting of two parallel circular plates of radius a separated by a distance d much smaller than a, discharges slowly when the plates are connected by a thin piece of wire.  Show on a diagram the flow of energy from the Poynting vector for two cases: first, the wire is of length d and connects the centers of the two plates across the gap; second, the wire has length several times a, and connects the centers of the plates by an outside route.

 

Next, suppose the plates discharge through a solenoid, placed outside the plates, connected to them by wires.  Assume the solenoid and connecting wires have very low resistance.  Show the Poynting vector energy flow in this system.