Question

A proton, a deuteron and an $\alpha -$particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field. If ${r_p},{r_d}$ and ${r_\alpha }$ denote, respectively, the radii of the trajectories of these particles, then
A. ${r_\alpha } = {r_p} < {r_d}$
B. ${r_\alpha } > {r_d} > {r_p}$
C. ${r_\alpha } = {r_d} > {r_p}$
D. ${r_p} = {r_d} = {r_\alpha }$

Answer 1

When charged particles are moving in a magnetic field, they experience a force which acts at right angle to the velocity of the particles. The right hand rule can be used to determine the direction of the force. When the expression for the magnetic force is combined with the electric force, the combined force is called Lorentz force. Since the magnetic force is perpendicular to the velocity, a charged particle follows a curved path in a magnetic field. The particle continues to follow the path to form a circular path.

Complete step-by-step answer:
Radius of the circular path is, $r = \dfrac{{mv}}{{qB}}$, where $m$ is the mass of charged particle, $q$ is charge and $v$ is the velocity of charged particle.
The kinetic energy of a charged particle in the magnetic field is given by,
$K = \dfrac{1}{2}m{v^2}$ or $v = \sqrt {\dfrac{{2K}}{m}}$
Therefore $r = \dfrac{m}{{qB}}\sqrt {\dfrac{{2K}}{m}} = \dfrac{{\sqrt {2Km} }}{{qB}}$ , where $K$ is the kinetic energy and $B$ is the strength of magnetic field. As $K$ and $B$ are constants. So $r$ is proportional to the $\dfrac{{\sqrt m }}{q}$. ${r_p}:{r_d}:{r_\alpha } = \dfrac{{\sqrt {{m_p}} }}{{{q_p}}}:\dfrac{{\sqrt {{m_d}} }}{{{q_d}}}:\dfrac{{\sqrt {{m_\alpha }} }}{{{q_\alpha }}}$ , where ${r_p},{r_d},{r_\alpha }$ is the radius of proton, deuteron and $\alpha -$particle respectively , ${m_{p,}}{m_d},{m_\alpha }$is the mass of the particles and ${q_p},{q_d},{q_\alpha }$ is the charge of particles. So
$\Rightarrow \dfrac{{\sqrt m }}{e}:\dfrac{{\sqrt {2m} }}{e}:\dfrac{{\sqrt {4m} }}{{2e}} = 1:\sqrt 2 :1$
$\Rightarrow {r_\alpha } = {r_p} < {r_d}$ , the radii of the trajectories of these particles.

So, the correct answer is “Option A”.

Note: We know that magnetic force is always perpendicular to the velocity so that it does not work on the charge particles. Kinetic energy of particles and speed thus remains constant. The direction of motion is only affected not the speed. The particle continues to follow the curved path and forms a complete circle.