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Unifrom electric and magnetic fields with strengthE andinduction B respectively are directedalong they axis (Fig). A particlewith specific charge q//mleaves the origin O in the direction of the z axiswith an initialnon-realtive velocity v_(0) find: (a) the coordinatey_(n) of theparticlewhen it crossses then y axis for thenth time, (b) the angle alpha between the particle's velocityvectorand the y axis at that moment. |
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Answer» Solution :The equaction of MOTION are, `m (dv_(X))/(DT) = -q Bv_(z), m (dy_(y))/(dt) = q E` and `m(dv_(z))/(dt) = q v_(x) B` These equactions can be solved easily. First, `v_(y) = (qE)/(m) t, y = (qE)/(2m) t'^(2)` Then, `v_(x)^(2) + v_(z)^(2)` = constant `= v_(0)^(2)` as before. Intergatingagainand using `x = z = 0`, at `t = 0` `x = (v_(0))/(OMEGA) sin omega t, z = (v_(0))/(omega) (1 - cos omega t)` Thus, `x = z = 0` for `t = t_(N) = n (2pi)/(omega)` At what instant, `y_(n) = (qE)/(2m) xx (2pi)/(qB//m) xx n^(2) xx (2pi)/(qB//m) = (2pi^(2) m E n^(2))/(qB^(2))` Also, `tan alpha_(n) = (v_(x))/(v_(y))`, (`v_(z) = 0` at this moment) `= (mv_(0))/(q E t_(n)) = (mv_(0))/(qE) xx (qB)/(m) xx (1)/(2pi n) = (B v_(0))/(2pi E n)` |
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