Question

An $\alpha$ -particle and a proton moving with the same kinetic energy enter a region of uniform magnetic field at right angles to the field. The ratio of the radii of the paths of $\alpha$ -particle to that of the proton is:
(A) $1:1$
(B) $1:2$
(C) $1:4$
(D) $1:8$

Answer 1

Hint
We know that an alpha particle is a fast-moving packet that contains two protons and two neutrons. The charge an alpha particle carries is $(+2)$ and they interact strongly with the matter. Based on this concept we have to solve this question.

Complete step by step answer
We can denote the Radius of circular path as $(\mathrm{r})=\dfrac{\mathrm{mv}}{\mathrm{qB}}$
Here, $m$ is mass, $v$ is velocity, $q$ is charge of the particle, $B$ is the magnetic field.
With the above assumptions, we can frame an equation for the Kinetic Energy as:
We know that the kinetic energy of an $\alpha$-particle is equal to the kinetic energy of a proton. $\dfrac{1}{2} \mathrm{m}_{\alpha} \mathrm{v}_{\alpha}^{2}=\dfrac{1}{2} \mathrm{m}_{\mathrm{p}} \mathrm{v}_{\mathrm{p}}^{2}$
We know, $\mathrm{m}_{\alpha}=4 \mathrm{m}_{\mathrm{p}}$
$\Rightarrow {{\text{v}}_{\text{p}}}=2{{\text{v}}_{\alpha }}$
We know that the magnetic force acting on a particle in a magnetic field provides the necessary centripetal force that is required for the circular motion.
$\therefore \dfrac{\text{m}{{\text{v}}^{2}}}{\text{R}}=\text{qvB}$
$\Rightarrow \text{R}=\dfrac{\text{mv}}{\text{qB}}$
Therefore, ratio of radii of the paths of $\alpha$ particle to that of proton is:
$\therefore \dfrac{{{R}_{\alpha }}}{{{R}_{p}}}=\dfrac{\dfrac{{{m}_{\alpha }}{{v}_{\alpha }}}{{{q}_{\alpha }}B}}{\dfrac{{{m}_{p}}{{v}_{p}}}{{{q}_{p}}B}}$
${{\text{m}}_{\alpha }}=4{{\text{m}}_{\text{p}}}$
${{\text{v}}_{\text{p}}}=2{{\text{v}}_{\alpha }}$
${{\text{q}}_{\alpha }}=2{{\text{q}}_{\text{p}}}$
Substituting the values, we get,
 $\mathrm{R}_{\alpha}: \mathrm{R}_{\mathrm{p}}=1: 1$
Therefore, the correct answer is Option (A).

Note
It should be known to us that alpha radiation occurs when the nucleus of an atom becomes unstable. By being unstable we mean that the ratio of the neutrons and the protons are too low. The alpha particles are emitted to restore the balance of the chemical reaction.