A Betatron is pictured above (with field out of the page). It works by increasing a non-uniform magnetic field in such a way that electrons of charge and mass inside the ``doughnut'' tube are accelerated by the -field produced by induction (via Faraday's law) from the ``average'' time-dependent magnetic field inside , while the magnitude of the magnetic field at the radius , , bends those same electrons around in the circle of (constant) radius .
This problem solves, in simple steps, for the ``betatron condition'' which relates to such that both things can simultaneously be true.
a) The electrons go around in circles of radius and are accelerated by an field produced by Faraday's law. We will define the (magnitude of the) average field by . What is the induced field (tangent to the circle) in terms of and ?
(Problem continued on next page!)
b) The electrons (at their instantaneous speed tangent to the circle) are bent into the circle of radius by the field . Relate to the magnitude of the momentum , the charge of the electron, and the radius .
c) The force from the -field acting on the electron with charge in the direction of its motion is equal to the time rate of change of the magnitude of its momentum (if Newton did not live in vain). Substitute, cancel stuff, and solve for in terms of . If you did things right, the units will make sense and the relationship will only involve dimensionless numbers, not or .
Cool! You've just figured out how to build one of the world's cheapest electron accelerators! Or perhaps not....