Question

In a Geiger Marsden experiment, calculate the distance of the closest approach to the nucleus of $Z=75$ , when a $\alpha$ particle of $5MeV$ energy impinges on it before it comes momentarily to rest and reverse its direction. How will the distance of closest approach be affected when the kinetic energy of the $\alpha$ particle is doubled?

Answer 1

Hint: An alpha particle is a positive, sub microscopic matter particle. According to Thomson’s model, if an Alpha particle collided with an atom it would actually go straight forward, and at most a fraction of its direction would be deflected. The definition of "solid matter" is obsolete at the atomic scale. The Thomson atom is an electrically charged sphere, rooted in its mass. The alpha particle therefore does not bounce down like a ball from the atom, but could slip on through when the electric fields of the atom are small enough to make it.

Complete step by step solution:
Geiger and Marsden carried out a series of trials at the invitation of Rutherford to point a beam of alpha particles to a thin metal foil and tested the dispersion pattern with a fluorescent tube. They found alpha particles in all directions and some back to the source, leaping off the metal video. According to Thomson’s model, this should have been impossible; all the alpha particles would have been direct.
Kinetic energy of the $\alpha$ particle is given by,
$$E_k={\displaystyle \frac{1}{4\pi {\epsilon }_0}}{\displaystyle \frac{(Ze)(2e)}{r_0}}$$
$E_k$ is the kinetic energy of the particle
$r_0$ is the distance of closest approach between the nucleus and the alpha particle,
Therefore,
Kinetic energy is inversely proportional to the distance of closest approach.

Hence, if the kinetic energy is doubled, the distance of closest approach will be halved.

Note: The Geiger-Marsden experiments were a landmark series of experiments from which scientists find that each radioactive substance has a nucleus in which all its positive charge and much of its mass accumulate. The experiments were dubbed Rutherford's gold foil experiments. Since calculating the dispersal of an alpha particle beam as it reaches a small metal foil, they deduced this.