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Usingthebetatroncondition, find the radiusof a roundorbit of an electron if themagneticinduction is knownas a function of distance r fromthe axisof the field. Examine this problem for the specfic case B = B_(0) - alpha r^(2), where B_(0) and a are positive constants. |
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Answer» Solution :The condtion, `B(r_(0)) = (1)/(2) lt B gt = (1)/(2) int_(0)^(r_(0)) B.2pi r dr//pi r_(0)^(2)` or, `B (r_(0)) = (1)/(r_(0)^(2)) int_(0)^(r_(0)) Br dr` The gives `r_(0)`. In the present case, `B_(0) - ar_(0)^(2) = (1)/(r_(0)^(2)) int_(0)^(r_(0)) (B - ar^(2)) RDR = (1)/(2) (B_(0) - (1)/(2) ar_(0)^(2))` or, `(3)/(4) ar_(0)^(2) = (1)/(2) B_(0)` or `r_(0) = sqrt((2 B_(0))/(3a))` |
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