# Plasma physics - single particle in time-varying B-field

quarky2001
It's an intro plasma physics course, and already starting to get to me...

I'm asked to derive an expression for the drift velocity of a particle in a constant uniform E-field, and a slowly time-varying B-field.

I've looked at a few books to see if I can even find the final result, so I have an idea what I'm shooting for, but every resource only seems to use the "time-varying B-field" case as a means to talk about the adiabatic invariance of a particles magnetic moment.

Staff Emeritus
Does Faraday's law or magnetic induction ring a bell?

Can one find a discussion of E(t) and dB(t)/dt?

quarky2001
I think I might have come up with a solution, but I'm a bit unsure, as it came about without as much math as I thought it would take.

Given the constant E-field, and the time varying B-field, we know there will be an E-cross-B drift term in the new drift velocity. The other part I think I might be able to get like this:

We know that the magnetic moment is an adiabatic invariant, i.e.

$$\mu = \frac{m{v_\perp}^2}{2B}=constant$$

And, the phrase "slowly time-varying" in the question suggests to me that this should apply here.

So I should get

$$v_D=\frac{E\times B}{B^2} + ???$$

I'm just not sure how to arrive at the new term that comes out of the adiabatic invariance of the magnetic moment.