Question

Magnetron is a device consisting of a filament of radius a and a coaxial cylindrical anode of radius b which are located in a uniform magnetic field parallel to the filament. An accelerating potential difference V is applied between the filament and the anode. Find the value of magnetic induction at which the electrons leaving the filament with zero velocity reach the anode.

Answer 1

This differs from the previous problem in and the magnetic field is along the z-direction. Thus Bx = By = 0, Bz = B

Assuming as usual the charge of the electron to be - | e |, we write the equation of motion

The motion is confined to the plane z = 0. Eliminating B from the first two equations,

so, as expected, since magnetic forces do not work,

On the other hand, eliminating V, we also get,

The constant is easily evaluated, since v is zero at ρ = a. Thus,