**Problem 1:**

The z = 0 plane is perfectly conducting. A point electric dipole
oscillating in time as **p**(t) = p_{0}(**z**/z)exp(-iωt)
is placed at a height h above the conducting plane at the position (0,0,h).

(a) Calculate the angular distribution of the radiated power at large
distances.

(b) Calculate the total power radiated in the long wave length limit.

**Problem 2:**

A non-relativistic positron of charge q_{e} and velocity **v**_{1}
(v_{1 }<< c) impinges head-on on a fixed nucleus of charge Zq_{e}. The
positron which is coming from far away (∞), is decelerated until it comes to
rest and then accelerated again in the opposite direction until it reaches a
terminal velocity **v**_{2}. Taking radiation loss into account (but
assuming it is small), find v_{2} as a function of v_{1} and Z.

**Problem 3:**

Let **E**_{0} = E_{0} **k**. The Abraham-Lorentz force equation for a damped, charged,
oscillator driven by an electric field **E**_{0}exp(-iωt) in the
dipole approximation i

d^{2}**r**'/dt^{2} + Γ d**r**'/dt - τ d^{3}**r**'/dt^{3}
+ ω_{0}^{2} **r**' = (q/m)**E**_{0}exp(-iωt),

where Γ, τ, and ω_{0} are
constants, q is the charge and m is the mass of the oscillator.

Using this
and the expression for the radiation electric field, **E**_{rad}(**r**,t)
= -(4πε_{0})^{-1}[(q/(c^{2}r'')]**a**_{⊥}(t -
r''/c), where **r**'' = **r** - **r**'(t - |**r **- **r**'|/c), show that the differential cross
section for scattering of radiation of frequency ω and
polarization **n **= (**θ**/θ)** ** is

dσ/dΩ = (e^{2}/(mc^{2}))^{2 }(**k**∙**n**)^{2}[ω^{4}/((ω_{0}^{2}
- ω^{2})^{2} + ω^{2}Γ_{t}^{2})].

**Problem 4:**

(a) Consider the radiation from an oscillating
electric dipole. Find **E**, **B** and the Poynting vector at a large
distance from the dipole and integrate your result to find the total radiation.

(b) From the symmetry of Maxwell’s equations and the form of the electric
and magnetic field of an oscillating electric dipole, deduce the field of an oscillating
magnetic dipole.

The near field must resemble the field of a dipole formed by a small
current loop of radius a (a << c/ω), and
current I = I_{0}cos(ωt).

**Problem 5:**

The vector potential **A**(r,t) of an oscillating dipole **p** at the origin
is **A**(r,t) = -ik **p** exp(i(kr - ωt'))/(4πε_{0}rc).

Note: complex notation, the real part matters.

(a) Let **p** = p **k**. Use this vector potential to calculate the magnetic field.

(b) Use Maxwell's equations to calculate the accompanying electric field.

(c) Find the fields in the radiation zone.