Betatron, Electron accerating at a constant radius

1. Jul 30, 2014

forceface

1. The problem statement, all variables and given/known data
An electron with speed v, undergoing cyclotron motion in a transverse magnetic field B(r) at cyclotron radius r0, given r0 = mv/[eB(r0)], can be accelerated by ramping the B field in time.
(a) Since magnetic fields do no work,what is increasing thwe kinetic energy of the electron?
(b) Show that if the magnetic field at r0 is half of the average across the orbit,
B(r0,t)= 1/(2πr02)∫B(r,t) da
then the radius r0 of the orbit must be constant in time. Assume nonrelativistic speeds

1. The attempt at a solution
(a) This is the easy part, since the magnetic field is changing in time an electric field is induced and that is what is increasing the kinetic energy of the electron.
(b) I have worked out why the given equation is true but I would rather not write the whole thing out. So given this equation how do I prove that the radius is constant in time?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 30, 2014

vela

Staff Emeritus
I haven't tried working this out, but I'll suggest you could try to show that $dr_0/dt = 0$.

3. Aug 4, 2014

forceface

When I try your suggestions I come up with B dv/dt = dB/dt v. But I am still not sure how this along with the intergral identity helps show what I want.