Electrons emitted with negligible speed from an electron gun are acceleration through a potential difference VAlong the x-axis. These electrons emerge form 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.

Given-
Electrons are accelerated by applying a potential difference = V

Let the mass of an electron = m

charge of an electron= e

Electric field,

Force experienced by the electron by coulomb’s law is given by,

Acceleration a, of the electron is given by,

where,

e= electronic charge

V= applied potential difference

m= mass of the object

Using the 3rd equation of motion

where,

v= final velocity

u=initial velocity

a= acceleration acting on the ion

S=distance travelled

Since initial velocity is zero,

Here, s = r which is the radius of the curve

From (1)

We know that time taken by electron to cover the curved path is given as,

As the acceleration of the electron is along the y axis only, it travels along the xaxis with uniform velocity.

Velocity of the electron moving along the field remains v.

Therefore, the distance at which the beam is

d = velocity × Time

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