 Show that the retarded potentials
$$\begin{array}{llll}\hfill V(\overrightarrow{r},t)& =\frac{1}{4\pi {\epsilon}_{0}}\int \frac{\rho ({\overrightarrow{r}}^{\prime},{t}_{r})}{R}\phantom{\rule{3.26212pt}{0ex}}{d}^{3}{\overrightarrow{r}}^{\prime}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \overrightarrow{A}(\overrightarrow{r},t)& =\frac{{\mu}_{0}}{4\pi}\int \frac{\overrightarrow{j}({\overrightarrow{r}}^{\prime},{t}_{r})}{R}\phantom{\rule{3.26212pt}{0ex}}{d}^{3}{\overrightarrow{r}}^{\prime},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
where $\overrightarrow{R}=\overrightarrow{r}{\overrightarrow{r}}^{\prime}$
and ${t}_{r}=tR\u2215c$,
satisfy the Lorenz gauge condition:
$$\nabla \cdot \overrightarrow{A}+\frac{1}{{c}^{2}}\frac{\partial V}{\partial t}=0.$$
(Note: ${d}^{3}{\overrightarrow{r}}^{\prime}$
denotes volume integration with respect to the vector
${\overrightarrow{r}}^{\prime}$.)

 Show that in a static electric field a point charge cannot have a
stable equilibrium point in vacuum!
 Show that in a static magnetic field a tiny magnet cannot have a
stable equilibrium point in vacuum. First solve the problem in the
simpler case when the spatial orientation of the magnet is fixed
(i.e. its magnetic moment $\overrightarrow{m}=\mathrm{\text{const.}}$.
Then solve the problem in the general case when the magnet can
rotate freely.
 A star is visible at altitude ${\phi}_{v}$.
The index of refraction of air is $n\left(h\right)$
as a function of height above the ground. What is the real altitude
(${\phi}_{r}$)
of the star (correcting for atmospheric refraction) if it is high enough so that
the curvature of the earth can be ignored? (Consider the Earth flat.)
 A plasma is an ionized gas, containing positively charged ions and
negatively charges electrons.
An electromagnetic wave is travelling through a plasma, which contains
$n$
electrons per unit volume. The electrons are brought into motion by the
electric field, but the ions, which have a much lower specific charge, can be
safely considered stationary.
Calculate the relative permittivity
${\epsilon}_{r}$
of the plasma as a function of the angular frequency
$\omega $
of the electromagnetic wave.
 What is the electrostatic energy per atom of a linear ionic crystal?
Hint: The Taylor expansion of the logarithm function will prove
helpful.
 A ball is falling through water with steady speed. Calculate the
velocity field of water (the velocity of flow at every point in space:
$\overrightarrow{v}\left(\overrightarrow{r}\right)$)!
(See fig. 2.)
Assume that water is incompressible (its density is constant), and that the flow is
irrotational ($\nabla \times \overrightarrow{v}=0$).
These assumptions are rarely correct for real water, however, if the ball is
falling neither “too fast” (there is no turbulence), nor “too slow” (the viscous
forces are negligible), then we can get an approximation about how the
water is flowing around the ball.
Hint: There is an electrostatic analogue for the equations!