(a) From Maxwell's equations, derive the conservation of energy
equation,

(∂u/∂t) + **∇∙S**
= -**E∙j**,

where u is the energy density
and **S** is the Poynting vector. Rewrite this equation in integral form and
explain why it is a statement of energy conservation.

(b) A long, cylindrical conductor of radius a and conductivity
σ carries a constant current I. Find **S** at the
surface of the cylinder and interpret your result in terms of conservation of
energy.

Consider the circuit shown.

Let V = 50 V, R_{1} = 10 Ω, R_{2} = 100 Ω, and L = 50 H.

All circuit elements are ideal and no current flows before the switch is
closed.

(a) After the switch is closed at t = 0, find the current I flowing through
the switch as a function of time.

(b) After 8 s the switch is opened again. Right after the switch is opened,
what is the voltage across R_{2} and across the switch?

A thin disc with radius R and uniform charge
density σ is centered at the origin. Its normal points along the z-axis. It is rotating
with angular velocity ω **k** about the z-axis.

(a) Find the potential and the electric field due to the disc on the z-axis.

(b) Show that the potential reduces to that of a point charge for large
distances from the origin.

(c) Find the magnetic moment of the disk.

(d) At large distances from the origin, find the magnitude and direction of the
Poynting vector **S**.

Does the rotating disk produce a radiation field? Explain.

A thin wire of radius b is used to form a circular wire loop of radius a (a
>> b) and total resistance R.

The loop is rotating about the z-axis with constant angular velocity ω**k**
in a region with constant magnetic field **B** = B_{0}**i**.

At t = 0 the loop lies in the y-z plane and the point A at the center of the
wire crosses the y-axis.

Let the (**θ**/θ) direction be tangential to the loop and be equal to the
positive z direction at point A.

Let the (**φ**/φ) direction be tangential to the wire and be equal to the
direction indicated in the figure.

(a) Find the current flowing in the loop. Neglect the
self-inductance of the loop. What is current density **J** as a
function of time?

(b) Find the thermal energy generated per unit time, averaged over one
revolution.

(c) Write down an expression for the the pointing vector
**S** on the
surface of the wire.

(d) Use **S** to find the field energy per unit time flowing into the
wire, averaged over one revolution.

A finite spherically symmetrical charge distribution disperses under
the influence of mutually repulsive forces.

Suppose that the charge
density ρ(r, t), as a function of the distance r from the center of symmetry and
of time, is known.

(a) Prove that the curl of the magnetic field is zero at any point.

(b) Use this to show that **B** is zero at any point.