**Midterm Exam # 2, Physics
217**

**November 28, 2001, 8.30 am – 9.50
am**

**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?

**Figure
1. Problem 1.**

**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.

**Figure
2. Problem 2.**

**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.