Q20 of 104 Page 306

Figure shows a circular coil of N turns and radius a, connected to a battery of emf ϵ through a rheostat. The rheostat has a total length L and resistance R. The resistance of the coil is r. A small circular loop of radius a’ and resistance r’ is placed coaxially with the coil. The center of the loop is at a distance x from the center of the coil. In the beginning, the sliding contact of the rheostat is at the left end and then onwards it is moved towards right at a constant speed v. Find the emf induced in the small circular loop at the instant

(a) the contact begins to slide and


(b) it has slid through half the length of the rheostat.




Given:



Area of coil (2) of radius a’ =


We know that magnetic field due to coil (1) at the center of coil (2) is



Where


N=no. of turns in coil (1)


i= current in coil (1)


a=radius of coil (1)


x=distance of center of coil (2) from center of coil (1)


We know that,


Flux (ϕ) of magnetic field (B) through the loop of cross section area A in the magnetic field is given by




Since the magnetic field due to coil (1) is parallel to axis of coil (2) θ =0° and flux through the coil (2) is given by



Now,


by faraday’s law of electromagnetic induction


…(i)


Where


e =emf produced


ϕ =flux of magnetic field


using eqn.(i) emf induced in the coil (2) is given by


. (ii)


Let y be the distance of sliding contact from its right end


Given,


Total length of rheostat =L


Total resistance of rheostat=R


When the sliding contact is at a distance y from its right end then the resistance (R’) of the rheostat is given by



So the current i flowing through the circuit is given by



Where r is the resistance of the coil and ϵ is the emf of battery


Putting value of i in eqn.(ii) we get,




Since



(a) When the contact begins to slide


Therefore, magnitude of emf induced is




(b) When the contact has slid through half the length of rheostat


Therefore, magnitude of emf induced is




More from this chapter

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18

A uniform magnetic field B exists in a cylindrical region of radius 10 cm as shown in figure. A uniform wire of length 80 cm and resistance 4.0Ω is bent into a square frame and is placed with one side along a diameter of the cylindrical region. If the magnetic field increases at a constant rate of 0.010 T/s, find the current induced in the frame.


 19

The magnetic field in the cylindrical region shown in figure increases at a constant rate of 20.0 mT/s. Each side of the square loop abcd and defa has a length of 1.00 cm and a resistance of 4.00 Ω. Find the current (magnitude and since) in the wire ad if

(a) the switch S1 is closed but S2 is open,


(b) S1 is open but S2 is closed,


(c) both S1 and S2 are open and


(d) both S1 and S2 are closed.



 21

A circular coil of radius 2.00 cm has 50 turns. A uniform magnetic field B = 0.200 T exists in the space in a direction parallel to the axis of the loop. The coil is now rotated about a diameter through an angle of 60.0°. The operation takes 0.100s.

(a) Find the average emf induced in the coil.


(b) If the coil is a closed one (with the two ends joined together) and has a resistance of 4.00 Ω, calculate the net charge crossing a cross-section of the wire of the coil.


 22

A closed coil having 100turns is rotated in a uniform magnetic field B = 4.0 × 10–4 T about a diameter which is perpendicular to the field. The angular velocity of rotation is 300 revolutions per minute. The area of the coil is 25 cm2 and its resistance is 4.0 Ω. Find

(a) the average emf developed in half a turn from a position where the coil is perpendicular to the magnetic field,


(b) the average emf in a full turn and


(c) the net charge displaced in part (a).