Problem 1 [20 points]
In lecture we wrote down expressions for the electric and magnetic field amplitudes, EEI and BEI, 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 terms that point in both the radial direction and in non-radial directions. 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 [25 points]
We found in lecture that the contribution to the radiated vector potential from the electric quadrapole moment of an oscillating charge distribution is
| AE2 = | μ0
4π |
eikr
r | ik | iω
6 |
er⋅Q |
where the subscript "E2" denotes "electric quadrapole", er is the unit vector in the radial direction (usually written as r hat), and Q is the electric quadrapole tensor.
Considering behavior only in the Radiation Zone approximation, in which the electric and magnetic fields are proportional to 1/r, (this considerably simplifies the calculation of all the spatial derivatives):
a) Find the resulting magnetic and electric fields BE2 and EE2.
b) Find the instantaneous, and time averaged, Poynting vectors, SE2 and <SE2>
c) FInd the radiated power cross-section dPE2/dΩ.
d) As a simple case, assume that all elements of the tensor Q vanish except for Qzz. Such would be the case, for example, if one has two oscillating electric dipoles oppositely oriented along the z axis. Make a polar sketch of dPE2/dΩ.
e) Find the total power radiated PE2. Compare the magnitude of PE2 with that of the electric dipole power PE1.
Problem 3 [15 points]
An insulating circular ring of radius R lies in the xy plane, centered at the origin. It carries a linear charge density λ(φ) = λ0sinφ, where λ0 is a constant and φ is the usual azimuthal angle. The ring is now set spinning at constant angular velocity ω about the z axis. Calculate the total power radiated in the electric dipole approximation.
Hint: The trick to doing this problem easily is to compute the electric dipole moment as a function of time, p(t), and see that it can be written in the form p(t) = Re[ p(ω)eiωt], where p(ω) is an appropriate complex number.