Questions

1)

A spherical void of radius $R$ is in an otherwise homogeneous dielectric material of dielectric constant $\epsilon$. At the centre of the void is an electric dipole $p$. Assume that the size of the dipole is infinitesimally small.

(a) Suppose that $R\rightarrow \infty$. Then very close to the centre of the sphere, what is the approximate form of the electric potential as a function of the distance $r$ from the centre? (10 points)

(b) With a finite $R$, write down (i) the equation for the electric potential inside and outside of the void, and (ii) the boundary conditions on the potential at the surface of the void. (10 points)

(c) Solve the equation and the boundary conditions to find the electric potential both inside and outside the void. (20 points)

2)

In a medium with a frequency-dependent permittivity, the electric displacement (in the frequency domain) can be written as ${\bf D} ({\bf r}, \omega) = \epsilon (\omega) {\bf E}({\bf r}, \omega)$.

(a) Assuming a medium with no free charge or free current with a frequency-dependent dielectric constant $\epsilon (\omega)$, rewrite the four Maxwell's equations in the frequency and wave-vector domain. Hint: Fourier transform the Maxwell equations. (10 points)

(b) Now we consider a conductor where the charge-currents are induced by the electric field ( ${\bf J} =\sigma {\bf E}$ where $\sigma$ is the constant conductivity). Modify the Maxwell's equations obtained in (a) and derive the electromagnetic wave equations in the $(\omega , {\bf k})$ domain. (10 points)

(c) Conductors also contain bound electrons which give a dielectric constant

Using the wave equations from (b), obtain the complex wave number in terms of the frequency. Assuming $\chi \ll 1$, obtain the leading term in the absorption coefficient. Would you be able to observe the anomalous dispersion in such a conductor? Explain your answer. (20 points)

3)

A very long solenoid of radius $a$, with $n$ turns per unit length, carries a current $I_s$. Coaxial with the solenoid, at a radius $b\gg a$, is a circular ring of wire, with resistance $R$. When the current in the solenoid is (gradually) decreased, a current $I_r$ is induced in the ring.

(a) Calculate $I_r$, in terms of $dI_s/dt$. (10 points)

(b) The power ($I_r^2 R$) delivered to the ring must have come from the solenoid. Confirm this by calculating the Poynting vector just outside the solenoid (the electric field is due to the changing flux in the solenoid; the magnetic field is due to the current in the ring.) Ignore $(a/b)^n$ terms and integrate over the entire surface of the solenoid, and check that you recover the correct total power. (20 points)

4)

Suppose a point charge $q$ is constrained to move along the $x$ axis with a time-dependent trajectory

$${\epsilon (\omega)\over \epsilon_0}-1 =\chi (\omega)= {ne^2\over m}{1\over \omega_0^2-\omega^2 -i\gamma \omega}~.$$ (19)

$$x(t)= x_0 \cos \omega t~.$$ (20)

(a) What are the electric and magnetic fields at position ${\bf r} = (x,0,0)$ for $x>x_0$? (10 points)

(b) What is the total power radiated from this charge? (10 points)

(c) What is the radiation damping force felt by the charge? (10 points)

(d) Explain the relation between the damping force in (c) and the answer to (b). (10 points)