Two protons with different velocity enter a region having a uniform magnetic field that is perpendicular to their velocities. The region is large enough that the protons can execute complete circular trajectories. How do the radii of their circular paths compare? Which particle takes longer to complete one revolution?

Explanation

The two protons have the same charged and mass and are in the same magnetic field. They do have different velocities however, so the proton with the higher speed will move in a circular path with a greater radius.
The time to complete one revolution (T) can be found from the velocity (v) and the circumference `(2pir)`
`v = (2pi r)/(T), T = (2pi r)/(v)`
Substituting for the radius, we find that
`T = (2pi r)/(v) = (2pi mv)/(v|q|B) (2pi m)/(|q|B)`
This expression shows us that the time required to complete one revolution (the period) is independent of the velocity. Both protons complete one revolution in the same amount of time