Three particles, an electron (e), proton (p) and a helium atom (He) are moving in circular paths with constant speeds in the x-y plane in a region where a uniform magnetic field B exists along the z-axis. The times taken by e, p and He inside the field to complete on revolution are ${{t}_{e}}$, ${{t}_{p}}$and ${{t}_{He}}$ respectively. Then:
A. ${{t}_{He}}$ > ${{t}_{p}}$ = ${{t}_{e}}$
B. ${{t}_{He}}$ > ${{t}_{p}}$ > ${{t}_{e}}$
C. ${{t}_{He}}$ = ${{t}_{p}}$ = ${{t}_{e}}$
D. none of these

Hint: As the particles move perpendicular to magnetic field, they will move in circular motion and the radius of the circle made by them can be found by balancing the Lorentz force due to the magnetic field and the centripetal force needed for the circular motion.

Formula used:
Lorentz force
F = q(v×B)

Complete step by step answer:
In the formula for Lorentz force we will ignore the electrostatic component of force as there are no electrostatic interactions. Here it has been given that the particles move in the x-y plane and the magnetic field exists along the z-axis. So, the cross product of v and B will be equal to their product as they are perpendicular to each other. So, the force provided by the magnetic field will be equal to
F = qvB
This force will be perpendicular to both the magnetic field and the velocity. So, the force will be in the x-y plane acting perpendicular to the velocity this will make the particles move in circles. This force will be the centripetal force acting on the particles.
$\begin{align}
  & \dfrac{m{{v}^{2}}}{r}=qvB \\
 & \Rightarrow r=\dfrac{mv}{qB} \\
\end{align}$
The velocity and the magnetic field is the same for all particles so the radius of their motion will depend only on the ratio of their mass and charge. And as they have the same velocities, the time taken by them to complete one circle will be proportional to the radius of the circle they make. The ratio for electrons will be less than that for a proton as it has less mass but the same charge as proton. And the radius for the helium atom will be infinite as there is no charge on the atom.

So, the correct option is B, i.e. ${{t}_{He}}$ > ${{t}_{p}}$ > ${{t}_{e}}$

Note:
If we take the charge of an electron as negative, we will get negative force and a negative radius but that does not happen because we only consider the magnitude of force and not the direction when calculating the radius. Negative force simply means that the circle made by the electron will be in another direction than given by the right-hand rule.