Question

Derive the formula for electron charge to mass ratio

e/m=2V/(Br)^2: first, explain why formulas

(mv^2)/r=evB and

eV=1/2mv^2 must hold. Then, do what algebra is necessary to show that those formulas imply formula e/m=2V/(Br)^2

Answer 1

The electron is first accelerated by potential V. The magnitude of the work done by the electric field to accelerate an electron by potential difference V is

where e is the electronic charge. By work energy theorem, the change in kinetic energy of a particle is equal to the work done on the particle.

Assuming the electron starts from rest, the initial kinetic energy of the electron is zero.

Let v be the speed attained by the electron

After the electron had been accelerated, it enters a region of uniform magnetic field which is directed perpendicular to the direction of velocity of the electron.

The magnetic force is

The magnitude of the force is

where is the angle between the velocity vector and the magnetic field. Since they are perpendicular, , this implies, . The magnitude of the magnetic force is

The electron undergoes a uniform circular motion. Let r be radius. To move an object in uniform of circular motion of radius r and at speed v, the required centripetal force acting towards the center of the circular motion is

The centripetal force required is supplied by the magnetic force. As you can note that always points towards the center of the circle.

We can solve (1) and (2) for e/m in terms of potential V, magnetic field B, and radius r. We have to eliminate v,

Substituting into (1)