Problem 1 (35 points)
A certain coaxial cable consists of a copper wire, of radius a, surrounded by an infinitesimal thin concentric copper tube of radius c (see Figure 1). The charge on the wire is λ C/m and the charge on the tube is -λ C/m. The space between the wire and the tube is partially filled (from b to c) with a linear dielectric of susceptibility χ_e.
a) What is the magnitude and direction of the electric displacement in the three regions a < r < b, b < r < c, and c < r?
b) What is the magnitude and direction of the electric field in the three regions a < r < b, b < r < c, and c < r?
c) What is the capacitance per unit length of this cable?

Problem 2 (35 points)
Consider a circular current loop of radius R, lying in the xy plane, and carrying a current I in the direction indicated (see Figure 2).
a) Find the exact magnetic field (magnitude and direction) a distance z above the center of the current loop.
b) What is the magnetic dipole moment of the current loop?
c) Verify that for z » R the exact magnetic field calculated in a) is consistent with the field of a magnetic dipole.

Problem 3 (35 points)
A uniform line charge λ is placed on an infinite straight wire, a distance d above a grounded conducting plane. The wire runs parallel to the x axis and directly above it. The conducting plane is the xy plane.
a) Find the potential in the region above the grounded plane.
b) Find the charge density σ induced on the conducting plane.
Note: The potential generated by a uniform line charge λ on an infinite straight wire is equal to

where r is the distance from the line charge.