Q52 of 104 Page 306

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.

Given:


Radius = r


Angular velocity = w


Magnetic field = B


Diagram:



Formula used:


In this case, the velocity will increase radially.


Let us consider a strip of width dx at a distance x from the centre.


Hence, induced emf of this portion will be , where B = magnetic field, dx = width of the element, x = distance of the element from the centre, w = angular velocity


Hence, integrating on both sides using proper limits, we get



=> Total motional emf (Ans)


More from this chapter

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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.



 51

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.

 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.


 54

The magnetic field in a region is given by where L is a fixed length. A conducting rod of length L lies along the Y-axis between the origin and the point (0, L, 0). If the rod moves with a velocity v = v0, find the emf induced between the ends of the rod.