Charge oscillating on a spring

1. May 9, 2012

Spriteling

1. The problem statement, all variables and given/known data

(a) Consider an oscillating electric dipole of moment p(t)=p0sinωt. At large distances r>>c/ω from the dipole, the magnetic potential in the Lorenz gauge is

A(r,t) = $\frac{\omega \cos\omega(t-r/c))\bf{p_0}}{4\pi r c}$

Calculate the E and B fields, and deduce the Poynting vector, showing that it points radially outwards and vanishes on the axis of the dipole.

(b) A small body of mass m and charge q hangs from the ceiling by a spring with spring constant k. The body is initially at rest, a distance h from a very cold floor, h>>mc^2/k. At time t= 0 it is given a slight downwards kick so that executes tiny oscillations with amplitude d<<h. Calculate the average intensity of the electromagnetic radiation hitting the floor as a function of the radial distance R from the point on the floor directly below the particle.

2. Relevant equations

Clearly the motion of the particle is given by z = dsinωt with ω=sqrt(k/m).

3. The attempt at a solution

I've done all of part (a), which was fairly trivial. However for part (b) I am somewhat confused. Can you just treat the particle as an oscillating dipole, (with moment dqsinωt) with the relevant E and B fields be that as calculated in part (a)? If so, then to calculate the intensity on the floor, do you set r = h, and then average out the power, and divide by pi*R^2?

I hope these questions make sense.