# Scattering angle as a function of impact parameter

Tags:
1. Nov 4, 2016

### Silviu

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. Nov 9, 2016