Assignment 4

Problem 1:

The z = 0 plane is perfectly conducting.  A point electric dipole oscillating in time as p(t) = p0(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 qe and velocity v1 (v1 << c) impinges head-on on a fixed nucleus of charge Zqe.  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 v2.  Taking radiation loss into account (but assuming it is small), find v2 as a function of v1 and Z.

Problem 3:

Let E0 = E0 k.  The Abraham-Lorentz force equation for a damped, charged, oscillator driven by an electric field E0exp(-iωt) in the dipole approximation i
d2r'/dt2 + Γ dr'/dt - τ d3r'/dt3 + ω02 r' = (q/m)E0exp(-iωt),
where Γ, τ, and ω0 are constants, q is the charge and m is the mass of the oscillator.
Using this and the expression for the radiation electric field, Erad(r,t) = -(4πε0)-1[(q/(c2r'')]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 = (e2/(mc2))2 (kn)24/((ω02 - ω2)2 + ω2Γt2)].

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 = I0cos(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'))/(4πε0rc).
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.