An electron and a proton move in circular orbits in a plane perpendicular to a uniform magnetic field B.

Find the ratio of the radii of their circular orbits when the electron and the proton have the same momentum.

Find the ratio of the radii of their circular orbits when the electron and the proton have the same kinetic energy.

centripetal force=mv^2/r

electron centripetal force=protoncentripetal force.

me*ve^2/re=mp*vp^2/rp

re/rp=me/mp * Ve/vp * Ve/Vp

but if they have the same momentum, then
meve/mpvp=1 so
re/rp=Ve/Vp

You do the energy.

To find the ratio of the radii of their circular orbits in each case, we need to use the equations that relate the radius of a circular path, momentum, and kinetic energy of a charged particle moving in a magnetic field.

1. When the electron and proton have the same momentum:
According to the equation for the momentum of a charged particle moving in a magnetic field, p = qB