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A chargedparticle with specfic charge q//m starts movingin the regionof spacewhere there are unifrommutually perpendicular electricand magnitude fields. The magentic field is constant and has aninduction B while the strength of the electric field varies with time as E = E_(m) cosomega t, where omega = qB//m. For the non-relativistic case find the law of motion x(t) and y(t) of theparticle if at the moment t = 0 it was located at the pointO (see Fig). What is the appoximateshape of thetrajectory of the particle? |
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Answer» Solution :The equactions are as in 3.392. `(dv_(x))/(dt) = (qB)/(m) v_(y), (dv_(y))/(dt) = (qE_(m))/(m) cos omega t - (qB)/(m) v_(x)` and`(dv_(x))/(dt)` with `omega = (qB)/(m), xi = v_(x) + iv_(y)`, we get, `(d xi)/(dt) = I (E_(m))/(B) omega cos omega t - i omegaxi` or mutiplying by `e^(i omega t)` `(d)/(dt) (xi, e^(i omega t)) = i (E_(m))/(2B) omega (e^(i omega t) + 1)` on intergrating, `xi e^(i omega t) + (E_(m))/(4B) e^(i omega t) + (E_(m))/(2B) i omega t` or, `xi= (E_(m))/(4B) (e^(i omega t) + 2 i omega t e^(i omega t)) + C e^(i omega t)` SINCE `xi = 0` at `t = 0, C = - (E_(m))/(4B)`, Thus, `xi = i (E_(m))/(2B) sin omega t + i (E_(m))/(2B) omega t e^(i omega t)` or, `v_(x) = (E_(m))/(2B) omega t sin omega t` and `v_(y) = (E_(m))/(2B) sin omega t + (E_(m))/(2B) omega t cos omega t` Intergrating again, `x = (a)/(2 omega^(2)) (sin omega t- omega t cos omega t), y = (a)/(2 omega) t sin omega t`. where, `a = (qE_(m))/(m)`, and we have used `x = y = 0`, at `t = 0` The TRAJECTORY is an UNWINDING spiral. |
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