The force on the particle is given by F=qvB when the motion is perpendicular to the field.  Here q is the electric charge and B is the magnetic field in tesla.  The resulting motion is circular.  According to Newton's laws of motion an object in circular motion is accelerating radially according to a = v²/r where r is the radius of the circular path.  Since F=ma, we can use this acceleration to get the radial force, which is F=mv²/r.  This force and the force from the magnetic field are equivalent so
mv²/r=qvB.  The particle's velocity is then
v=qBr/m
Note the distance around the circle is 2Ï€r. Â This is a linear function in r, so if you double r, you double the length around the circle. Â Since the velocity equation is directly proportional to r, if you triple v, you must also triple the right hand side of the equation as well. Â But on the right side of the equation, q, B, and m are constant. Â The only thing you can triple is r, meaning that while the particle is moving 3 times as fast as before, it now has 3 times further to travel, meaning
the time to go around is the same. Â As the particle speeds up, the circle it is moving in gets bigger. Â To change the time to go around, you'd need to change the magnetic field.
