(a) A particle having a charge of `10 muC` and a mass `20 mg` is projected with a speed `2 km//sec` in a region having a uniform magnetic field of `1 kT`. Find the radius of the circle formed by the particle and also time period.
(b) A proton, deuteron and `alpha`-particle enter a region of constant magnetic field perpendicularly. If `r_(p)`, `r_(d)` and `r_(alpha)` denote respectively the radii of the trajectories of these particles, then find ratio `r_(p):r_(d):r_(alpha)` if they enter with same (i) speed, (ii) linear momentum, (iii) kinetic energies.
( c) A charged particle is accelerated through a potential difference `V_(0)` and acquires a speed of `v`. It is then injected perpendicular into a magnetic field `B`. Find the radius of the circle described by it.

Explanation

(a) `q = 10 umC =10^(5)C, m = 20 mg = 20 xx 10^(-6)kg`
`v = 2 km//sec = 2 xx 10^(3) m//sec, B = 1 kT = 10^(3)T`
`R = (mv)/(Bq) = (20 xx 10^(-6) xx 2 xx 10^(3))/(10^(3) xx 10^(-5)) = 4 m`
(b) (i) `r = (mv)/(Bq)`,
proton `(e, m)`, deutron `(e, 2 m), alpha`-particle `(2e, 4m)`
`r_(p):r_(d):r_(alpha) = (mv)/(Be):(2mv)/(Be):(4mv)/(B(2e))`
`= 1:2:2`
(ii) `r = (p)/(Bq)`
`r_(p):r_(d):r_(alpha) = (p)/(Be):(p)/(Be):(p)/(B(2e)) = 1:1:(1)/(2)`
`= 2:2:1`
(iii) `r = (sqrt(2mK))/(Bq)`
`r_(p):r_(d):r_(alpha) = (sqrt(2 xx 2mK))/(Be):(sqrt(2 xx 2mK))/(Be):(sqrt(2 xx 4mK))/(B.2e)`
`= 1:sqrt(2):1`
(c ) When a charged particle of mass `m` having charge `q` is acceleration by a potential `V_(0)`, its kinetic energy `K` is given by
`K = qV_(0) = (1)/(2)mv^(2) rArr (q)/(m) = (v^(2))/(2V_(0))`
`R = (mv)/(Bq) = (v)/(B).(2V_(0))/(v^(2)) = (2 V_(0))/(Bv)`