Using the betatron condition, find the radius of a round orbit of an electron if the magnetic induction is known as a function of distance `r` from the axis of the field. Examine this problem for the specfic case `B = B_(0) - alpha r^(2)`, where `B_(0)` and `a` are positive constants.

Using the betatron condition, find the radius of a round orbit of an e

1.	Using the betatron condition, find the radius of a round orbit of an electron if the magnetic induction is known as a function of distance `r` from the axis of the field. Examine this problem for the specfic case `B = B_(0) - alpha r^(2)`, where `B_(0)` and `a` are positive constants.
Answer» 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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Using the betatron condition, find the radius of a round orbit of an electron if the magnetic induction is known as a function of distance `r` from the axis of 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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