All Problems Motion of Charged Particles in Electric and Magnetic Fields
Problem 3.403
Demonstrate that electrons move in a betatron along a round orbit of constant radius provided the magnetic induction on the orbit is equal to half the mean value of that inside the orbit (the betatron condition).
\(3.403 .\) On the one hand, \[ \frac{d p}{d t}=e E=\frac{e}{2 \pi r} \frac{d \Phi}{d t} \] where \(p\) is the momentum of the electron, \(r\) is the radius of the orbit, \(\Phi\) is the magnetic flux acting inside the orbit. On the other hand, \(d p / d t\) can be found after differentiating the relation \(p=e r B\) for \(r=\) const. It follows from the comparison of the expressions obtained that \(d B_{0} / d t=1 / 2 d\langle B\rangle / d t .\) In particular, this condition will be satisfied if \(B_{0}=1 / 2\langle B\rangle .\)