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A narrow beam of identicalions with specific chargeq//m, possessing differentvelocities, enters the region of space, where thereare unifromparallel electric and marneticfields the strengthE and induction B, at the pointO (see Fig). The beam directioncoincides with thex axisat the point O. A plane screenoriented at rightanglesto the x axisis locatedat a distance l fromthe point O. Find theequation of the tracethat theions leaveon the screen. Demonstrate that at x lt lt l it is theequactionof a parabola. |
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Answer» Solution :The equcation of the trajectory is, `x = (v_(0))/(OMEGA) SIN omega t, z = (v_(0))/(omega) (1 - cos omega t), y = (qE)/(2m) t^(2)` as before SEE (3.384) Now on the screen `x = l`, so `sin omega t = (omega l)/(v_(0))` or, `omega t = sin^(-1) (omega l)/(v_(0))` At that moment, `y = (qE)/(2m omega^(2)) ("sin"^(-1)(omega l)/(v_(0)))^(2)` so,`(omega l)/(v_(0)) = sin sqrt((2m omega^(2) y)/(qE)) = sin sqrt((2q B^(2) y)/(Em))` and`z = (v_(0))/(omega) 2 "sin"^(2) (omega t)/(2) = l "tan" (omega t)/(2)` `= l "tan" (1)/(2) ["sin"^(-1) (omega I)/(v_(0))] = l "tan" sqrt((qB^(2) y)/(2 mE))` For SMALL `z, (qB^(2) y)/(2 mE) = ("tan"^(-1) (z)/(l))^(2) = (z^(2))/(l^(2))` or, `y = (2 mE)/(q B^(2) l^(2)). z^(2)` is a parabola. |
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