Question

A stationary ball of radius $R$ is irradiated by a parallel stream of particles whose radius is $r$. Assuming the collision of a particle and the ball to be elastic, find:
(a) the deflection angle $\theta$ of a particle as a function of its aiming parameter $b$;
(b) the fraction of particles which after a collision with the ball are scattered into the angular interval between $\theta$ and $\theta +d\theta$;
(c) The probability of a particle to be deflected, after a collision with the ball, into the front hemisphere $(\theta <\frac{\pi }{2})$.

Answer 1

It is implied that the ball is too heavy to recoil.
(a) The trajectory of the particle is symmetrical about the radius vector through the point of impact. It is clear from the diagram that
$\theta =\pi -2\phi$ or $\phi =\frac{\pi }{2}-\frac{\theta }{2}$.
Also $b=(R+r)\sin \phi =(R+r)\cos \frac{\theta }{2}$.
(b) With b defined above, the fraction of particles scattered between $\theta$ and $\theta +d\theta$ (or the probability of the same ) is
$dP=\frac{|2\pi bdb|}{\pi (R+r{)}^2}=\frac{1}{2}\sin \theta d\theta$
(c) This is
$P={\int }_0^{\pi /2}\frac{1}{2}\sin \theta d\theta =\frac{1}{2}{\int }_{-1}^{0}d(-\cos \theta )=\frac{1}{2}$