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Homework Help: The motion of Guiding Center

  1. Mar 14, 2014 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations
    The field is tangential to the Amperian loop, so the magnetic field is simply:

    [itex]\oint B\cdot dl = \mu_{o}I \Rightarrow \vec{B}=\frac{\mu_{o}I}{2\pi r}\hat{\phi}[/itex]

    3. 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 [itex]\phi[/itex] component of the field, which is simply

    [itex]\nabla B_{\phi} = -\frac{\mu_{o}I}{2\pi r^{2}}\hat{r}[/itex]

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

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

    where [itex]r_{L}[/itex] is the larmor radius. Does this look correct, am I missing anything?
  2. jcsd
  3. Mar 14, 2014 #2


    User Avatar
    Science Advisor

    looks ok to me
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