Aproton and an electron have the same kinetic energy upon entering a region of constant magnetic field. what is the ratio of the radii or their circular paths?

The Lorentz force on a particle of charge q produced by a magnetic field is

where v is the velocity of the particle and B the intensity of the magnetic field.
Due to this force, the particle will acquire a circular motion, so this force will be the centripetal force of the motion:

where m is the mass of the particle, and  the radius of the trajectory.
Re-arranging we have


So, we have to find r for both the proton and the electron. We know that the two particles have same kinetic energy:

and so

The mass of the proton is 1833 times the mass of the electron (, ), i.e.

and so from the relationship between the kinetic energies we find:


Now we can calculate the ratio between the radius of the proton and electron trajectories. Keeping in mind that q is the same for proton and electron, and the field B is the same, we have

So, the radius of the proton trajectory is 42.8 times the radius of the electron trajectory.

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26.4 is about the right time