Home Work Set # 11, Physics 217, Due: December 12, 2001

Problem 1

A coaxial cable consists of two very long cylindrical tubes, separated by linear insulating material of magnetic susceptibility . A current I flows down the inner conductor and returns along the outer one; in each case the current distributes itself uniformly over the surface (see Figure 1). Find the magnetic field in the region between the tubes. As a check, calculate the magnetization and bound currents and confirm that (together of course with the free currents) they generate the correct field.

Figure 1. Problem 1.

Problem 2

a) A current I flows down a straight wire of radius R. If the wire is made of linear material (copper or aluminum) with susceptibility , and the current is distributed uniformly, what is the magnetic field a distance r from the center?
b) Find all the bound currents.
c) What is the net bound current flowing down the wire?

Problem 3

Notice the following parallel:

Thus, the transcription , , , and turns an electrostatic problem into an analogous magnetostatic one. Use this observation, together with your knowledge of the electrostatic results, to calculate
a) The magnetic field inside a uniformly magnetized sphere.
b) The magnetic field inside a sphere of linear magnetic material in an otherwise uniform magnetic field.
c) The average magnetic field over a sphere, due to steady currents within the sphere.

Problem 4

A familiar toy consists of donut-shaped permanent magnets (magnetization parallel to the axis), which slide frictionless on a vertical rod (see Figure 2). Treat the magnets as dipoles, with mass M and dipole moment .

Figure 2. Problem 4.
a) If you put two back-to-back magnets on the rod, the upper one will "float" - the magnetic force upward balancing the gravitational force downward. At what height z does it float?
b) If you now add a third magnet (parallel to the bottom one), what is the ratio of the two heights? (Determine the actual number, to three significant digits).

Problem 5

At the interface between one linear magnetic material and another magnetic material the magnetic field lines bend (see Figure 3). Show that , assuming there is no free current at the boundary.

Figure 3. Problem 5.