**Problem 1** [20 points]
Consider the electric and magnetic fields, **E**_{E1} and **B**_{E1} from an oscillating charge distribution in the electric dipole approximation. In lecture we focused on the Radiation Zone Approximation in which one assumes kr >> 1, and so we kept only the terms of lowest power in 1/kr.

a) In this problem we consider the Near Zone Approximation, where we assume d << r << λ, and so kr << 1. Here we keep only the terms of highest power in 1/kr. Using the full expressions presented in lecture (before we made the Radiation Zone approximation) find the near zone electric and magnetic fields. The electric field that you find should look familiar. Can you identify it from your earlier study of electrostatics? Noting that the Near Zone is defined by kr << 1, can you give a physical explanation as to why the electric field has the form it does?

b) Show that in the Near Zone the strength of the magnetic field |**B**| is much smaller than |**E**|/c. Comment on other ways that the fields in the Near Zone are very different from the Radiation Zone fields.

c) Compute the Poynting vector **S** in the Near Zone. What are the key ways it differs from the Poynting vector in the Radiation Zone? Consider the time average <**S**> in the Near Zone. Are your results consistent with energy conservation? Why?

**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.