In the following diagram, the arm PQ of the rectangular conductor is moved from x= 0; outwards. The uniform magnetic field is perpendicular to the plane and extends from x = 0 to x = b and is zero for x >b. Only the arm PQ possesses substantial resistance ‘r’. Consider the situation when the arm PQ is pulled outwards from x = 0 to x = 2b, and is then moved back to x = 0 with constant speed ‘ѵ’. Obtain expressions for the (i) electric flux, (ii) the induced emf, (iii)the force necessary to pull the arm and (iv) the power dissipated as Joule heat. Sketch the variation of these quantities with distance.

OR

Write working principle of cyclotron and with a suitable diagram explain its working. Give any two applications of cyclotron.

Let Width of the rectangle = L

Magnetic field strength = B

(i) Electric flux

Area of the rectangle A= Lx

So flux

But B=0 for x>b. hence for x>b

The flux will be same while moving PQ from right to left

(2) E.m.f. induced =

The flux at any time, for x<b

But, B= 0 for x>b

Thus, for x>b

From moving right to left,

Negative sign represents that flux will decrease with time.

OR

Cyclotron is a device which is used to accelerate the charged particles to very high energy. The principle it works on is that if a charged particle moves perpendicular to the magnetic field, it experiences the magnetic Lorentz force due to which it starts moving in a circular path.

It is having two Dees, D1 and D2 which are enclosed in a chamber. The Dees are separated from each other by a small gap and the source of charged particle is placed in the centre of the gap. They are placed between the two big electromagnet poles to provide strong magnetic field. The magnetic field acts perpendicular to the plane of the Dees and the Dees are connected to a high frequency oscillator.

Working

When a charged particle with charge q having mass m is emitted from the source, it is accelerated towards the Dee which have a negative potential at that time. Due to the perpendicular magnetic field, the particle experiences magnetic Lorentz force and it moves in a circular path. By the time the particle arrives at the gap between the Dees, the polarity of the Dees changes and the D at negative potential becomes positive and the D at positive potential becomes negative. Hence, the particle is once again accelerated and enters into other Dee with a greater velocity in a circle of greater radius. Therefore, the particle moves in a spiral path of increasing radius and when it comes near the edge, deflector plate is placed outside the Dee which helps the charged particle to come out. When the particle moves along a circle of radius r with a velocity v, the magnetic Lorentz force provides centripetal force.

Time taken to cover one half circle

It is clear from the above expression that the time taken by the charged particle is independent of (i) the radius (r) of the path and (ii) the velocity (v) of the particle

Hence, period of rotation T = 2t

Cyclotron is used to accelerate protons, deuterons and α–particles