Physics 743 Fall 2000

Michael Fowler

Homework # 2

Meissner Effect. On cooling a (type I) superconductor below a certain temperature, it becomes a perfect diamagnet, at least in weak magnetic fields. That is to say, if a warm sphere of superconducting material is placed in a constant magnetic field, then cooled down, when it goes superconducting surface supercurrents are generated, and no magnetic field remains in the bulk of the superconductor. For an external magnetic field of strength H, find the current density of the supercurrents, and find the new magnetic field outside the superconductor. Draw a picture of the field lines, and explain how you found them. If the superconductor is at a temperature T below T_c (which is where it goes superconducting in zero magnetic field) it will expel an external magnetic field provided that field is weaker than a value H(T). Discuss how that field value is related to the free energy difference between the electron gas in the normal metal and that in the superconducting state.

Poynting Vectors and Energy Flow (again). Heald (Am. Journ. Phys.52, 522 (1984)) gives a nice discussion of the electric field for a circular, or rather cylindrical, circuit. He takes a unit circle centered at the origin, with voltage V₀q /2p, so there is a battery at the point –1 supplying voltage V₀. He expands the voltage in Fourier series. The point of this question is to see how to derive the same field using complex variable techniques. First, we need to find a function of z = x + iy so that its imaginary part, say, equals q on the unit circle. The first thought is log z, but that won’t do here, because it has a singularity at the origin, and our potential is smooth except at the battery point, z = -1. But note that if you draw a line from a point q on the circle to the point –1, the angle that line makes with the real axis is just q /2. So ½log(z + 1) has the right properties. Draw the lines of constant imaginary part of this function, then draw the lines of constant real part. What is the physical significance of these sets of lines? Outside the cylinder, this function is no good, because the potential must go down at infinity, not diverge. However, the function we use outside can have singularities inside the cylinder. Show that a suitable combination of log (z + 1) and log z gives the field outside the cylinder. Draw the field lines and also the equipotentials.

Suppose a hollow cylinder has a current J flowing parallel to its axis, the current being uniformly distributed on the surface of the cylinder. There is also an insulated wire parallel to the axis of the cylinder, glued to the surface, and carrying a current –J. Sketch the lines of force of the magnetic field generated by this pair of currents, both inside and outside the cylinder.

(Problem 3.11 of Griffiths) Two long, straight copper pipes, each of radius R, are held a distance 2d apart. One is at potential V₀, the other at potential –V₀. Find the potential everywhere. Griffiths suggests using the results of his problem 2.47, which was to find the potential from two infinite parallel lines of equal and opposite charge, and show that the equipotentials in that problem were cylinders. I would like to see both these problems done using—of course—complex variable methods.