Relativistic-Strength EM Field

[andrewm]

Homework Statement

Demonstrate (qualitatively) that an electron in a linearly polarized EM beam that is driven transversely at \omega will also oscillate longitudinally at 2*\omega (due to the B field, apparently).

Homework Equations

I want to use a phasor form for the fields: [tex]\vec{E} = \vec{E_0} e^{i \omega t} [\tex], etc for B and velocity.

[tex] F = q E + q v \cross B [\tex]

The Attempt at a Solution

A hint given is that v is pi/2 out of phase with B, and to focus on the magnetic term in the Lorentz expression.

My guess was that, when v and B are multiplied as phasors, the frequency becomes 2 \omega. However, this doesn't use the fact that v is pi/2 out of phase! So what's the hint for?

 

[BigJDubs]

My other guess was that the B field is also oscillating at 2 \omega, due to the E field oscillating. But then I don't understand how this applies the fact that v is pi/2 out of phase with B. The hint doesn't seem to help me much in understanding what's going on here. Can someone explain what's going on here qualitatively?