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).

Reveal Answer
3.403.3.403 . On the one hand, dpdt=eE=e2πrdΦdt \frac{d p}{d t}=e E=\frac{e}{2 \pi r} \frac{d \Phi}{d t} where pp is the momentum of the electron, rr is the radius of the orbit, Φ\Phi is the magnetic flux acting inside the orbit. On the other hand, dp/dtd p / d t can be found after differentiating the relation p=erBp=e r B for r=r= const. It follows from the comparison of the expressions obtained that dB0/dt=1/2d⟨B⟩/dt.d B_{0} / d t=1 / 2 d\langle B\rangle / d t . In particular, this condition will be satisfied if B0=1/2⟨B⟩.B_{0}=1 / 2\langle B\rangle .
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