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Scattering angle as a function of impact parameter

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  1. Nov 4, 2016 #1
    1. The problem statement, all variables and given/known data

    This is the problem 3.34 from Goldstein, 3rd Edition:

    Consider a truncated repulsive Coulomb potential defined as V=k/r, for r>a and V=k/a, for r<=a. For a total energy E>k/a, obtain expressions for the scattering angle ##\Theta## as a function of ##s/s_0##, where ##s_0## is the impact parameter for which the periapsis occurs at the point r=a.

    2. Relevant equations

    ##d^2u/d\theta^2 + u = -m/l^2*dV/du, with u=1/r##
    ##\Theta(s) = \pi -2\int_{0}^{u_m}(s\cdot du/(\sqrt{1-V(u)/E-s^2u^2}))##, with E - the energy of the particle and ##u_m = 1/r_m, r_m## the distance at the closest approach

    3. The attempt at a solution
    I tried to calculate ##r(\theta)## outside and inside the sphere, ##s_0## can be calculated assuming that the potential is k/r all the time, as for ##s_0## the particle doesn't go inside the sphere. But I am not sure how to relate ##\Theta## to s, when I have the potential inside the sphere.
     
  2. jcsd
  3. Nov 9, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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