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Accelerating Ions through semicircular plates

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Hey guys, this is from Purcell 3.19
    "Ions are accelerated through a potential difference [tex]V_0[/tex] and then enter the space between the semicylindrical electrodes A and B. Show that an ion will follow the semicircular path of radius [tex]r_0[/tex] if the potentials of the outer and inner electrodes are maintained, respectively, at [tex]2V_0*ln(b/r_0)[/tex] and [tex]2V_0*ln(a/r_0)[/tex]."

    2. Relevant equations

    3. The attempt at a solution
    Well i started with
    [tex]q*V_0 = KE = (mv^2)/2[/tex]
    [tex]V^2 = (2qV_0)/m[/tex]
    which combined with the rotational motion eq
    [tex]E = F/q = (mv^2)/(qr_0) = (2V_0)/r_0[/tex]
    but i can't think of any way to pin down what the field is between the two cylinders.
    just looking at the solution provided it seems like i should integrate the final equation for a variable r from b to r-nought and a to r-nought. I just don't know why. Any way to elucidate this for me?

    Attached Files:

    Last edited: Jan 28, 2009
  2. jcsd
  3. Jan 28, 2009 #2
    Attached a diagram (sorry for the paint) if it helps make it clearer
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