**Problem 1** [20 points]
In lecture we wrote down expressions for the electric and magnetic field amplitudes, **E**_{EI} and **B**_{EI}, 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/r^{2}.

Explain how the non-radial terms, and the terms which decay faster than 1/r^{2}, 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

**A**_{E2} = | μ_{0} 4π |
e^{ikr} r | ik | iω 6 |
**e**_{r}⋅**Q** |

where the subscript "E2" denotes "electric quadrapole", **e**_{r} 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 **B**_{E2} and **E**_{E2}.

b) Find the instantaneous, and time averaged, Poynting vectors, **S**_{E2} and <**S**_{E2}>

c) FInd the radiated power cross-section dP_{E2}/dΩ.

d) As a simple case, assume that all elements of the tensor **Q** vanish except for Q_{zz}. 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 dP_{E2}/dΩ.

e) Find the total power radiated P_{E2}. Compare the magnitude of P_{E2} with that of the electric dipole power P_{E1}.

**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) = λ_{0}sinφ, where λ_{0} 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)e^{iωt}], where **p**(ω) is an appropriate *complex* number.