A charged particle moves along a circle of radius `r = 100 mm` in a unifrom magnetic field with induction `B = 10.0 mT`. Find its velocity and perios of revolution if that particle is
(a) a non-relativistic proton,
(b) a relativistic electron.

Explanation

(a) For motion along a circle, the magnetic froce, acted on the particle, will provide the centripetal force, neccesary for its circular motion.
`i.e. (mv^(2))/(R) = evB` or, `v = (eBR)/(m)`
and the period of revolutin, `T = (2pi)/(omega) = (2piR)/(v) = (2pi m)/(eB)`
(b) Generally, `(d vec(p))/(dt) = vec(F)`
But, `(d vec(p))/(dt) = (d)/(dt) (m_(0) vec(v))/(sqrt(1 - (v^(2)//c^(2)))) = (m_(0) dotvec(v))/(sqrt(1 - (v^(2)//c^(2)))) + (m_(0))/((1 - (v^(2)//c^(2)))^(3//2)) (vec(v) (vec(v) . dotvec(v)))/(c^(2))`
For transerverse motion, `vec(v). dotvec(v) = 0` so,
`(d vec(p))/(dt) = (m_(0) dotvec(v))/(sqrt(1 - (v^(2)//c^(2)))) = (m_(0))/(sqrt(1 - (v^(2)//c^(2)))) (v^(2))/(r)`, here
Thus, `(m_(0) v^(2))/(r sqrt(1 - (v^(2)//c^(2)))) = B ev` or, `(v//c)/(sqrt(1 - (v^(2)//c^(2)))) = (B er)/(m_(0) C)`
or, `(v)/(c) = (B er)/(sqrt(B^(2) e^(2) r^(2) + m_(0)^(2) c^(2)))`
Finally, `T = (2pi r)/(v) = (2pi m_(0))/(eB sqrt(1 - v^(2)//c^(2))) = (2pi)/(cBe) sqrt(B^(2) e^(2) r^(2) + m_(0)^(2) c^(2))`