Q46 of 91 Page 230

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?

Given-


Magnetic field, B = 0.20 T


Mass of the particle, m = 0.010 g = 1 × 10−5 kg


Charge of the particle, q = 1.0 × 10−5 C


Given in the question that, if the particle has to move with


uniform velocity in the region of the applied field,


the gravitational force experienced by the particle must be


equal to the magnetic force experienced by the particle.


Gravitational force,



where


m is the massof theobject


g=acceleration due to gravity


And


Magnetic force, we know, Lorentz force F is given by -



where,


q = charge on an electron


v = velocity of the electron


B=magnetic field


θ= angle between B and v


So,






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44

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.



 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.



 47

A particle moves in a circle of diameter 1.0 cm under the action of a magnetic field of 0.40 T. An electric field of 200 V m–1 makes the path straight. Find the charge/mass ratio of the particle.

 48

A proton goes undeflected in a crossed electric and magnetic field (the fields are perpendicular to each other) at a speed of 2.0 × 106 ms–1. The velocity is perpendicular to both the fields. When the electric field is switched off, the proton moves along a circle of radius 4.0 cm. Find the magnitudes of the electric and the magnetic fields. Take the mass of the proton = 1.6 × 10–27 kg.