PHY 415: Electromagnetic Theory I
Prof. S. Teitel email@example.com ---- Fall 2008
Problem Set 7
Due Wednesday, December 10, in lecture
- Problem 1 [15 points]
Consider the electric and magnetic fields from an oscillating charge distribution, making the electric dipole approximation. In lecture we discussed the "radiation zone" approximation in which one assumed the observer is many wavelengths away from the source, kr >> 1. We therefore kept only the terms of lowest power 1/kr in our calculations.
Now consider the "near zone" approximation. Here one still assumes the observer is far from the source r>> d on the length scale of the source d, however one is not far on the length scale of the wavelength, i.e. kr << 1. Using the expressions found in the lecture notes (Lecture 26), find the near zone electric and magnetic fields to leading order (i.e. the terms which are the highest power in 1/kr). Identify the electric field that you find in terms of a familiar result from electrostatics. Show that |B| << |E|, unlike what one finds in the radiation zone.
These results should convince you that the near fields are very different from the radiation zone fields!
- Problem 2 [15 points]
In Lecture 26 I wrote down expression for the electric and magnetic fields in the electric dipole approximation before one makes the radiation zone approximation. Using these results compute the instantaneous Poynting vector S(r, t).
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 3 [30 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:
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 figure out how to write the oscillating dipole moment as p(t)=Re [pωe-iωt], with the correct amplitude pω.