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Limits of the Path Length of B-field induced gyroradius

  1. Sep 2, 2014 #1
    I am trying to derive the path length of a charged particle in a B-field. I am assuming the particle will travel a distance L along the applied field. Using the following equations for the path length of a helix and gyroradius:
    Helix defined as
    [itex][a*cos(t),a*cos(t),b*t] [/itex]for t on [0,T] has a path length of
    and assuming that bt must equal L at T=1 (parametrize helix from 0 to 1), I get the following expression:


    However, the bounds do not make sense. At zero field, It should simply travel in a straight line, i.e. P=L.
    I am not sure about the "infinite" field, since it is oscillating more rapidly but with an ever decreasing radius. I could argue that the radius being 0 means only the vertical distance is traveled, or that it just rotates infinity at 0 radius?
    In either case, I cannot get the BC for B=0, as this causes the expression to be infinite.

    Can I simply not apply the gyroradius in a zero field condition, or did I derive this incorrectly? What is the limiting case of infinite field? Actually quite an interesting problem!
  2. jcsd
  3. Sep 7, 2014 #2
    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
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