A, proton, a deuteron and a -an particle having the same...

$A$, proton, a deuteron and $a$ -an particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field. If $\mathrm{r}_{\mathrm{p}} \mathrm{r}_{\mathrm{d}}$ and denote respectively the radii of trajectories of these particles, then (a) $\mathrm{r}_{\alpha}=\mathrm{r}_{\mathrm{p}}<\mathrm{r}_{\mathrm{d}}$ (b) $\mathrm{r}_{\alpha}>\mathrm{r}_{\mathrm{d}}>\mathrm{r}_{\mathrm{p}}$ (c) $\mathrm{r}_{0}=\mathrm{r}_{\mathrm{d}}>\mathrm{r}_{n}$ (d) $\mathrm{r}_{n}=\mathrm{r}_{\mathrm{d}}=\mathrm{r}_{1}$

$A$, proton, a deuteron and $a$ -an particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field. If $\mathrm{r}_{\mathrm{p}} \mathrm{r}_{\mathrm{d}}$ and denote respectively the radii of trajectories of these particles, then
(a) $\mathrm{r}_{\alpha}=\mathrm{r}_{\mathrm{p}}<\mathrm{r}_{\mathrm{d}}$
(b) $\mathrm{r}_{\alpha}>\mathrm{r}_{\mathrm{d}}>\mathrm{r}_{\mathrm{p}}$
(c) $\mathrm{r}_{0}=\mathrm{r}_{\mathrm{d}}>\mathrm{r}_{n}$
(d) $\mathrm{r}_{n}=\mathrm{r}_{\mathrm{d}}=\mathrm{r}_{1}$

Key Concepts

Relationship Between Kinetic Energy and Momentum

For particles in motion, kinetic energy and momentum are closely related. Given a constant kinetic energy, a particle's speed and thus its momentum depend on its mass. This relationship is fundamental when considering circular motion in a magnetic field because the derived radius of the trajectory is a function of the particle's momentum, which is affected by its mass, highlighting differences in trajectory radius for particles of different masses but equal kinetic energy.

Charged Particle Motion in a Magnetic Field

This concept explains that when charged particles enter a uniform magnetic field, they experience a magnetic Lorentz force that acts perpendicular to their velocity, resulting in a circular motion. The balance between the magnetic force and the required centripetal force for circular motion leads to a relationship wherein the radius of the path is determined by the particle’s momentum, charge, and the magnetic field strength.

Mass-to-Charge Ratio (m/q)

The radius of the circular path followed by a charged particle in a magnetic field is directly proportional to the square root of its mass and inversely proportional to its charge. This parameter, the mass-to-charge ratio, is crucial in determining how different particles behave when moving in a magnetic field, particularly when comparing the radii of trajectories of particles with different masses and charges but with the same kinetic energy.

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