Using your knowledge of circular (centripetal) motion, derive an equation for the radius r of the circular path that electrons follow in terms of the magnetic field B, the electrons' velocity v, charge e, and mass m. You may assume that the electrons move at right angles to the magnetic field.

As the electron enters the field it starts experiencing a force due to magnetic field which always acts perpendicular to the velocity of electron which is in accordance with the Lorentz force. This force [tex]F_B[/tex] is given by the equation:

[tex]F_B=q.v.B[/tex]...........................................(1)

where q is the charge on the particle.

Due to the this force the path of the electron becomes circular under the influence of the magnetic field.

Now as we know that the force acting on the particle moving in the circular path is a centripetal force which is given as:

[tex]F_c=m.r.\omega^2[/tex]....................................(2)

where:

m = mass of the particle

r = radius of the particle

[tex]\omega[/tex]= angular velocity of the particle

Also, the relation between the angular and the linear velocity is as :

[tex]\omega=\frac{v}{r}[/tex]........................................(3)

From the eq. (2) & (3)

[tex]F_c=m.\frac{v^2}{r}[/tex]...................................(4)

from eq. (1) & (4)

[tex]F_c=F_B[/tex]

[tex]m.\frac{v^2}{r}=q.v.B[/tex]

[tex]r=\frac{m.v}{q.B}[/tex]