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(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. |
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Answer» Solution :(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)` |
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