Refocusing of paraxial electron from electron gun

[mmainak]

Homework Statement

Electrons emitted with negligible speed from an electron gun are accelerated through a potential difference v along the X-axis. These electrons emerge from a narrow hole into a uniform magnetic field B directed along this axis. However, some of the electrons emerging from the hole make slightly divergent angles as shown in figure. Show that these paraxial electrons are refocused on the X-axis at a distance

sqrt:(8.pi^2.m.V/e.B^2).

Homework Equations

The Attempt at a Solution

m.w^2.r =B.e.V

w= B.e/m ... cyclotron frequency

T= 2.pi.m/B.e

e.V/m = f(acc).d => f= e.V/m.d

d= 1/2.f.T^2 = 1/2. e.V/m.d . 4.pi^2.m^2/B^2.e^2

=> d=sqrt:(2.pi^2.m.V/B^2.e)

2 should be 8 in the actual answer.

Where did I go wrong?

 

[paulina]

W = qV (W=work, V=potential difference)
(mv^2)/2 = qV (v=velocity)
v = sqrt.(2qV/m)

now, time period, T = 2.pi.m/Be
so, d = v.T
d = sqrt.(8.pi^2.m.V/B^2.e)
:)