Magnetic dipole in two reference frames (special relativity problem)

[quasar_4]

Homework Statement

An EM wave traveling in vacuum has a magnetic field in the lab frame K which is given by $$\vec{B}(\vec{x},t) = \hat{z} B0 \cos{kx-\omega t}$$
where Bo, k are positive constants and omega = ck.

a) A point magnetic dipole, m, where $$m = \hat{y}m0$$ (m0 constant) is at rest in a reference frame K' which is moving at constant velocity $$v = v0 \hat{x}$$ relative to the lab frame. The given m is measured in the rest frame of the magnetic dipole. Find the instantaneous torque on the magnetic dipole as measured in the lab frame K'. Only K' spacetime coordinates must appear in the answer.
b) For the situation of part a, find the instantaneous torque on the magnetic dipole as measured in the lab frame K.

Homework Equations

$$\omega = c k$$
Maxwell's equations
Lorentz transformation

The Attempt at a Solution

I'm pretty sure I got part (a) with no difficulties. I used Ampere's law to find E in the K frame, then used the standard transformation equations for unprimed E and B fields to primed E and B fields (the ones given in Jackson). So, I end up with

$$\vec{B'} &=& \gamma (1-\beta) B0 \cos{(kx-\omega t)} \hat{z}$$

which makes the torque

$$\vec{N'} = \vec{m} \times \vec{B'}$$
$$= m0 \gamma (1-\beta) B0 \cos{(kx-\omega t)} \hat{x}$$

Ok, great... now I just use the Lorentz transformation to make x-> x', t-> t', and I can simplify my answer to $$\vec{N'}= m0 \gamma B0 \cos{(\gamma (1-\beta) \{kx' - \omega t'\})}$$.

The problem is that I have no idea how to do part b. I need to somehow find the dipole m in the frame K, but it's not a 4-vector... so I'm not sure how to do this. Can any of you help?

 

[mark726]

For part b, you can use the Lorentz transformation to transform the magnetic dipole moment from the primed frame to the unprimed frame. This can be done using the transformation equations for the magnetic field and electric field, which can then be used to find the transformed magnetic dipole moment. Once you have the magnetic dipole moment in the unprimed frame, you can use it to calculate the torque on the magnetic dipole in the lab frame K. This should give you the answer for part b.