Charge oscillating on a spring

[Spriteling]

Homework Statement

(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) = [itex]\frac{\omega \cos\omega(t-r/c))\bf{p_0}}{4\pi r c} [/itex]

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.

Homework Equations

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

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.

 

[mark726]

Thank you for your post. It appears that you have already made good progress on part (a) of the problem. To answer your question regarding part (b), yes, you can treat the particle as an oscillating dipole with a moment of dqsinωt. This is because the particle is oscillating with a small amplitude, and therefore can be approximated as a dipole.

To calculate the intensity on the floor, you can indeed set r = h and then average out the power. However, instead of dividing by pi*R^2, you should divide by the surface area of the floor directly below the particle, which is given by 4πR^2. This will give you the average intensity at a radial distance R from the point on the floor directly below the particle.

I hope this helps. Keep up the good work!