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Newton's 2nd Law and Orbital Motion

  1. Sep 27, 2005 #1
    Here's the problem...unfortunately I don't remember much about orbital motion. I'm a bit stuck on where to begin. If somebody could give me a little advice on how to tackle this problem I would appreciate it.

    Recall that the magnetic force on a charge q moving with velocity v in a magnetic field B is equal to qvXB. If a charged particle moves in a circular orbit with a fixed speed v in the presence of a constant magnetic field, use the relativistic form of Newton's 2nd law to show that the frequency of its orbital motion is

    f=((qB)/(2pim))(1-(v^2/c^2))^(1/2)
     
  2. jcsd
  3. Sep 27, 2005 #2

    Tide

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    If the speed is constant then

    [tex]\frac {d \vec v}{dt} = \vec v \times \vec \Omega[/tex]

    where [itex]\vec \Omega = q \vec B / m_0[/itex]. There are a number of ways to proceed from here but it should be apparent that the same analysis you did in the classical case will work except that B is replaced by [itex]B / \gamma[/itex] from which your result follows.
     
  4. Sep 27, 2005 #3
    still stuck

    Still stuck since I don't really remember the classical case.
     
  5. Sep 27, 2005 #4

    Tide

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    In that case, consider ...

    [tex]\frac {d v_x} {dt} = \Omega v_y[/tex]

    and

    [tex]\frac {d v_y} {dt } = - \Omega v_x[/tex]

    Differentiate, say, the first and substitute the second into the first:

    [tex]\frac {d^2 v_x} {dt^2} = - \Omega^2 v_x[/tex]

    from which it should be evident that the motion is sinusoidal with frequency [itex]\Omega[/itex].
     
    Last edited: Sep 28, 2005
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