Question

An electron and a proton enter a magnetic field perpendicularly, both have the same kinetic energy. Which of the following statements are true?
A. Trajectory of proton is less curved
B. Trajectory of electron is less curved
C. Both trajectories are equally curved
D. Both move on straight line path

Answer 1

Hint: Use the formula for that force exerted on a particle with charge q when it enters a uniform magnetic field B with a velocity of v perpendicular to the direction of the magnetic field i.e. $F=qvB$. The curviness of the trajectory depends on the radius of pat. Also use the formula of kinetic energy to relate the radius with kinetic energy of the particle.

Formula used:
$F=qvB$
$a={\displaystyle \frac{v^2}{R}}$
F=ma
$K={\displaystyle \frac{1}{2}}m{v}^2$

Complete step-by-step answer:
When a charged particle enters a uniform magnetic field perpendicularly, a force is exerted on the particle with is equal to $F=qvB$, where q is the charge on the particle, v is the velocity of the particle, and B is the magnitude of the magnetic field in which the particle enters.
Due to this force, the particle undergoes a uniform circular motion with speed v.
Since the particle is in uniform circular motion, it will have a centripetal acceleration equal to $a={\displaystyle \frac{v^2}{R}}$, where R is the radius of the circular path that the particle traces.
According to Newton’s second law, F = ma.
Therefore, $F=m{\displaystyle \frac{v^2}{R}}$
And the value of force F acting on the particle is qvB.
Hence, $qvB=m{\displaystyle \frac{v^2}{R}}$.
$\Rightarrow R={\displaystyle \frac{mv}{qB}}$ …..(i)
We know that when a particle is moving with a speed equal to v, then it possesses an energy called kinetic energy. This kinetic energy (K) is given as $K={\displaystyle \frac{1}{2}}m{v}^2$ ……. (ii).
We can write equation (ii) as $v=\sqrt{{\displaystyle \frac{2K}{m}}}$.
Substitute this value of v in equation (i).
Hence, $R={\displaystyle \frac{m\sqrt{{\displaystyle \frac{2K}{m}}}}{qB}}$
$\Rightarrow R={\displaystyle \frac{\sqrt{2mK}}{qB}}$.
It is given that electrons and the proton have the same kinetic energy. The magnetic field B is also the same for both. We also know that an electron and a proton have equal charge equal to e.
Therefore, the value of R will depend on the mass of the particles.
More the mass, more will be the radius of curvature.
We know that protons have larger mass than an electron.
 Therefore, the radius of the path of the proton will be more, making it less curved.
Hence, the correct option is A.

Note: Remember that how curved is the trajectory of a particle in circular motion depends on the radius of the circular path. It is not that if the radius is more, the trajectory is more curved. Curviness of trajectory is inversely proportional to its radius. Imagine yourself in a car taking a circular turn. If the radius is large, you do not have to turn the steering very much. That means the curviness of the road is less.