InterviewSolution
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InterviewSolution
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Demonstarate that electronsmove in a betatronalong around orbitof constantradiusprovidedthe magnetic induction on the orbitof constantradiusprovidedthe magnetic inductionon the orbitis equalto halfthe meanvalue of that indide the orbit (the betatron condition). |
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Answer» Solution :On the one hand, `(dp)/(dt) = eE = (e)/(2pi r) (d Phi)/(dt) = (e)/(2pi r) (d)/(dt) int_(0)^(r) 2pi r' B (r') dr'` On the other, `p = B (r) er, r` = constant. so, `(dp)/(dt) = er (d)/(dt) B(r) = er dotB (r)` Hence, `er dotB (r) = (e)/(2 pi r) pi r^(2) (d)/(dt) LT B gt` So, `dotB (r) = (1)/(2) (d)/(dt) lt B gt` This is most EASILY SATISFIED by taking`B(r_(0)) = (1)/(2) lt B gt` . |
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