Betatron, Electron accerating at a constant radius

[forceface]

Homework Statement

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?

 

[vela]

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

 

[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.