How Does a Particle's Guiding Center Move Near a Current-Carrying Wire?

[Demon117]

Homework Statement

What is the motion of the guiding center of a particle in the field of a straight current carrying wire? What happens to the particle energy?

Homework Equations

The field is tangential to the Amperian loop, so the magnetic field is simply:

$\oint B\cdot dl = \mu_{o}I \Rightarrow \vec{B}=\frac{\mu_{o}I}{2\pi r}\hat{\phi}$

The Attempt at a Solution

The drift velocity will be due to the curvature of the magnetic field and also the grad-B drift. So we need to compute the Grad of the $\phi$ component of the field, which is simply

$\nabla B_{\phi} = -\frac{\mu_{o}I}{2\pi r^{2}}\hat{r}$

At this point I think I know what I should do, and that is to calculate $\vec{B}\times \nabla B$, such that

$\vec{v_{d}}=1/2 v_{\bot}r_{L}(\vec{B}\times \nabla B)/B^{2}$

where $r_{L}$ is the larmor radius. Does this look correct, am I missing anything?

 

[Khashishi]

looks ok to me