Q44 of 91 Page 230

Electrons emitted with negligible speed from an electron gun are acceleration through a potential difference V


Along 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


r= radius of the curve


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




More from this chapter

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42

A narrow beam of singly charged potassium ions of kinetic energy 32 keV is injected into a region of width 1.00 cm having a magnetic field of strength 0.500 T as shown in figure. The ions are collected at a screen 95.5 cm away form the field region. If the beam contains isotopes of atomic weights 39 and 41, find the separation between the points where these isotopes strike the screen. Take the mass of a potassium ion = A(1.6 × 10–27) kg where A is the mass number.


 43

Figure shows a convex lens of focal length 12 cm lying in a uniform magnetic field B of magnitude 1.2 T parallel to tis principal axis. A particle having a charge 2.0 × 10–3 C and mass 2.0 × 10–6 kg is projected perpendicular to the plane of the diagram with a speed of 4.8 ms–1. The particle moves along a circle with its centre on the principal axis at a distance of 18 cm from the lens. Show that the image of the particle goes along a circle and find the radius of that circle.


 45

Two particles, each having a mass m are placed at a separation d in a uniform magnetic filed B as shown in figure. They have opposite charges of equal magnitude q. At time t = 0, the particles are projected towards each other, each with a speed v. Suppose the Coulomb force between the charges is switched off.

(a) Find the maximum value vm of the projection speed so that the two particles do not collide.


(b) What would be the minimum and maximum separation between the particles if v = vm/2 ?


(c) At what instant will a collision occur between the particles if v = 2vm?


(d) Suppose v = 2vm and the collision between the particles is completely inelastic. Describe the motion after the collision.



 46

A uniform magnetic field of magnitude 0.20 T exists in space from east to west. With what speed should a particle of mass 0.010 g and having a charge 1.0 × 10–5 C projected from south to north so that it moves with a uniform velocity?