Q51 of 104 Page 306

A bicycle is resting on its stand in the east-west direction and the rear wheel is rotated at an angular speed of 100 revolutions per minute. If the length of each spoke is 30.0 cm and the horizontal component of the earth’s magnetic field is 2.0 × 10–5 T, find the emf induced between the axis and the outer end of a spoke. Neglect centripetal force acting on the free electrons of the spoke.

Given:


Angular speed(w) = 100 revolutions/minute x 2π = 100 revolutions/60 sec x 2π= 10π/3 revolutions/sec


Length of each spoke(l) = 30 cm = 0.3 m


Magnetic field(B) = 2.0 × 10–5 T


Formula used:


Induced emf … (i), where B = magnetic field, l = length of spoke, v = velocity


Now, linear speed of the spoke v = ωr, where ω = angular speed, r = distance from the axis to the outer end.


Here, , where l = length of spoke


Hence,


Therefore, emf induced (from (i))


⇒ V = 9.42 x 10-6 V (Ans)


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Figure shows a smooth pair of thick metallic rails connected across a battery of emf ϵ having a negligible internal resistance. A wire ab of length ℓ and resistance r can slide smoothly on the rails. The entire system lies in a horizontal plane and is immersed in a uniform vertical magnetic field B. At an instant t, the wire is given a small velocity v towards right.

(a) Find the current in it at this instant. What is the direction of the current?


(b) What is the force acting on the wire at this instant?


(c) Show that after some time the wire ab will slide with a constant velocity. Find this velocity.



 50

A conducting wire ab of length ℓ, resistance r and mass m starts sliding at t = 0 down a smooth, vertical, thick pair of connected rails as shown in figure. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the rails.

(a) Write the induced emf in the loop at an instant t when the speed of the wire is v.


(b) What would be the magnitude and direction of the induced current in the wire?


(c) Find the downward acceleration of the wire at this instant.


(d) After sufficient time, the wire starts moving with a constant velocity. Find this velocity vm.


(e) Find the velocity of the wire as a function of time.


(f) Find the displacement of the wire as a function of time.


(g) Show that the rate of heat developed in the wire is equal to the rate at which the gravitational potential energy is decreased after steady state is reached.



 52

A conducting disc of radius r rotates with a small but constant angular velocity ω about its axis. A uniform magnetic field B exists parallel to the axis of rotation. Find the motional emf between the center and the periphery of the disc.

 53

Figure shows a conducting disc rotating about its resistance R is connected between the centre and the rim. Calculate the current in the resistor. Does it enter the disc or leave it at the centre? The radius of the disc is 5.0 cm, angular speed ω = 10 rad/s, B = 0.40 T and R = 10 Ω.