Problem 1 [20 points]
In Notes 6-3 I wrote down expressions for the electric and magnetic fields in the electric dipole approximation before one makes the radiation zone approximation (these are Eqs. (6.3.4) and (6.3.12)). Using these results compute the instantaneous Poynting vector S(r, t). You may assume that the dipole moment pω is a real valued vector.
Show that in S there exist both radial and non-radial terms. Show that there exist terms which decay faster than 1/r2.
Explain how the non-radial terms, and the terms which decay faster than 1/r2, can still be consistent with energy conservation!
Problem 2 [20 points]
Consider a point charge q moving in a circular orbit of radius R, centered about the origin in the xy plane. The charge is orbiting counterclockwise with an angular velocity ω . Working within the electric dipole approximation in the radiation zone:
a) Compute the radiated electric and magnetic fields, expressing your answer in terms of spherical coordinates. Make sure your answers are given as real valued functions!
b) What is the polarization of the outgoing radiation at a general spherical angle (θ,φ)? What is the polarization when θ=0? when θ=π/2?
c) What is the total radiated energy per one orbit of the charge?
Hint: The trick to doing this problem easily is to regard the orbiting charge as an oscilating charge distribution of frequency ω and to figure out how to write the oscillating dipole moment as p(t)=Re [pωe-iωt], with the correct amplitude pω. You should find that you need a complex valued pω.