Classical Mechanics (Goldstein) · Problem 34

The Central Force Problem — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · truncated-coulomb-scattering-flat-top

Problem

Consider a truncated repulsive Coulomb potential defined as

V = \frac{k}{r} \quad r > a

= \frac{k}{a} \quad r \leq a.

For a particle of 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

. (The formulas can be given in closed form but they are not simple!) Make a numerical plot of

\Theta

versus

s/s_0

for the special case

E = 2k/a

. What can you deduce about the angular scattering cross section from the dependence of

\Theta

s/s_0

for this particular case?