Q16 of 18 Page 231

Obtain an expression for the mutual inductance between a long straight wire and a square loop of side a as shown in Fig. 6.21.

Let a small element dy in the loop at a distance of y as shown in the given figure:



Magnetic field associated with the the element taken i.e. dy can be calculated as follows:


dФ = BdA ………………..(i)


where dA is the area of the element dy i.e. ady


B is the magnetic field intensity of the element dy at a distance of y and can be written as:



Putting in the equation (i)




Putting limits of y from x to a + x




Mutual conductance is given by the relation:


Ф = MI


Where MI =


Or M =


More from this chapter

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14

Figure 6.20 shows a metal rod PQ resting on the smooth rails AB and positioned between the poles of a permanent magnet. The rails, the rod, and the magnetic field are in three mutual perpendicular directions. A galvanometer G connects the rails through a switch K. Length of the rod = 15 cm, B = 0.50 T, resistance of the closed loop containing the rod = 9.0 mW. Assume the field to be uniform.


(a) Suppose K is open and the rod is moved with a speed of 12 cm s–1 in the direction shown. Give the polarity and magnitude of the induced emf.


(b) Is there an excess charge built up at the ends of the rods when K is open? What if K is closed?


(c) With K open and the rod moving uniformly, there is no net force on the electrons in the rod PQ even though they do experience magnetic force due to the motion of the rod. Explain.


(d) What is the retarding force on the rod when K is closed?


(e) How much power is required (by an external agent) to keep the rod moving at the same speed (=12cm s–1) when K is closed? How much power is required when K is open?


(f) How much power is dissipated as heat in the closed circuit? What is the source of this power?


(g) What is the induced emf in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?

 15

An air - cored solenoid with length 30 cm, area of cross - section 25 cm2 and number of turns 500, carries a current of 2.5 A. The current is suddenly switched off in a brief time of 10–3 s. How much is the average back emf induced across the ends of the open switch in the circuit? Ignore the variation in magnetic field near the ends of the solenoid.

 16

Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity, v = 10 m/s.

Calculate the induced emf in the loop at the instant when x = 0.2 m.


Take a = 0.1 m and assume that the loop has a large resistance.

 17

A line charge λ per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R. The wheel has light non - conducting spokes and is free to rotate without friction about its axis (Fig. 6.22). A uniform magnetic field extends over a circular region within the rim. It is given by,

B = – B0 k (r ≤ a; a < R)


= 0 (otherwise)


What is the angular velocity of the wheel after the field is suddenly switched off?