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A Betatron (pictured above with field out of the page) works by increasing a uniform magnetic field in such a way that electrons of charge $e$ and mass $m$ inside the ``doughnut'' tube are accelerated by the $E$-field produced by induction from the average time-dependent magnetic field $B_1(t)$ inside $r$ (via Faraday's law) while the average magnitude of the magnetic field at the radius $B_2(t)$ bends the electrons around in the constant radius circle of radius $r$.

This problem solves for the ``betatron condition'' which relates $B_1(t)$ to $B_2(t)$ such that both things can simultaneously be true.

a) First, assuming that the electrons go around in circles of radius $r$ and are accelerated by an $\vec{E}$ field produced by Faraday's law from the average field $B_1$ inside that radius, solve for that induced $E$ field in terms of $B_1$ and $r$.

b) Second, assuming that the electrons are bent into a circle of radius $r$ by the average field at that radius, $B_2$, relate $B_2$ to the momentum $p = mv$ and charge $e$ of the electron, and the radius $r$.

c) Third, noting that the force $F$ from the $E$-field acting on the electron with charge $e$ in part a) is equal to the time rate of change of $p$ in the result of b) substitute, cancel stuff, and solve for $dB_1/dt$ in terms of $dB_2/dt$. If you did things right, the units will make sense and the relationship will only involve dimensionless numbers, not $e$ or $m$.

Cool! You've just figured out how to build one of the world's cheapest electron accelerators! Or perhaps not....