How Does an Electron Move in Non-Uniform Magnetostatic Fields?

[Fernando Mourao]

Homework Statement

Describe semiquantitatively the motion of an electron under the presence
of a constant electric field in the x direction,
E =E₀x^{^}
and a space varying magnetic field given by
B = B₀ a(x + z)x^{^} + B₀ [1 + a(x - z)]z^{^}
where Eo, Bo, and a are positive constants, lαxl « 1 and lαzl « 1.
Assume that initially the electron moves with constant velocity in the
z direction, v(t = 0) = v₀z^{^}. Verify if t his magnetic field satisfies the
Maxwell equation ∇ x B = 0

Homework Equations

[/B]
Equation of motion:
m dv/dt = q[E + v × B]

The Attempt at a Solution

I've proven that the B field satisfies the Maxwell equation.

and considering B(0,0,0) = B₀ z^{^}
I got the B field as a first order approximation about the origin as
B(r) = B₀z^{^} + (B₀αx + B₀αz)x^{^} + (B₀αx - B₀αz )z^{^}

So from the equation of motion I get:
m dv/dt = q[E + v × B] = q[E + v × B₀ + v×[r⋅∇B]]

The first two terms on the right hand side show that the particle would have a uniform acceleration along the x direction and a circular motion (varying with the instantaneous velocity) with x and y components; the last term is a force term and results in a combined gradient-curvature drift of the particle.

I feel like I'm missing something regarding the divergent term of ∇B. How far should i go with the resolution of the equation of motion?

 

[Simon Bridge]

You are only asked for "describe semi-quantitatively" - so that is as far as you go.
You have not been asked for the motion close to the origin, or even for a short time ... so you will need to justify the approximation(s?) you used.
OTOH: the description says to assume an initial constant velocity ... which is not possible without some other applied forces, so it is not clear how the velocity at t<0 is important... this also makes it a bit tricky to assess what the question is asking for. You'll have to use the context of the recent coursework to guide you.
I would have interpreted it a bit like being asked to sketch a function ... I'd look for turning points, inflexions, and asymptotes, and an indication of the kind of curve between ... or their motion analogues.

I'm not sure any part of the motion is best described as "circular" ... but I suspect you should try to be more specific about the directions of deflections from the initial direction of travel. Once deflected, check if the direction of the force changes.

Note: best practise to use LaTeX markup in PF... so you have:
$\vec B = aB_0(x+z)\hat x + bB_0(x-z)\hat y$ and $\vec E = E_0\hat x$
$m \ddot{\vec r} = q\vec E+ q\dot{\vec r} \times \vec B : \vec r = x(t)\hat x + y(t)\hat y + z(t)\hat z$