Q36 of 104 Page 306

Figure shows a long U-shaped wire of width ℓ placed in a perpendicular magnetic field B. A wire of length ℓ is slid on the U-shaped wire with a constant velocity v towards right. The resistance of all the wires is r per unit length. At t = 0, the sliding wire is close to the left edge of the U-shaped wire. Draw an equivalent circuit diagram, showing the induced emf as a battery. Calculate the current in the circuit.


Current will flow from the left edge to the right, that is, in the clockwise direction. Therefore, the induced emf will also flow along the clockwise direction.


Diagram showing induced emf as a battery:



Given:


Resistance per unit length = r


Length of wire = l


Velocity with which the wire moves = v


Formula used:


We know, that … (i), where E = emf, B = magnetic field, v = velocity with which the wire moves, l = length of wire.


Now, total resistance … (ii), where r = resistance per unit length, l’ = total length of loop


Horizontal length of loop = vt, where v = velocity, t = time.


Hence, total length of loop


Therefore, R(total resistance) = … (iii), where r = resistance per unit length, l = length of wire


By Ohm’s law, we know that where E = emf, I = current, R = total resistance. Hence, from (iii), we get, ……….(iv)


Equation (i) and (iv), we get:



=>


Current in the circuit (Answer)


More from this chapter

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34

A circular copper-ring of radius r translates in its plane with a constant velocity v. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the ring. Consider different pairs of diametrically opposite points on the ring.

(a) Between which pair of points is the emf maximum?


(b) Between which pair of points is the emf minimum?


What is the value of this minimum emf?


 35

Figure shows a wire sliding on two parallel, conducting rails placed at a separation ℓ. A magnetic field B exists in a direction perpendicular to the plane of the rails. What force is necessary to keep the wire moving at a constant velocity v?


 37

Consider the situation of the previous problem.

(a) Calculate the force needed to keep the sliding wire moving with a constant velocity v.


(b) If the force needed just after t = 0 is F0, find the time at which the force needed will be F0/2.


 38

Consider the situation shown in figure. The wire PQ has mass m, resistance r and can slide on the smooth, horizontal parallel rails separated by a distance ℓ. The resistance of the rails is negligible. A uniform magnetic field B exists in the rectangular region and a resistance R connects the rails outside the field region. At t = 0, the wire PQ is pushed towards right with a speed v0. Find

(a) the current in the loop at an instant when the speed of the wire PQ is v,


(b) the acceleration of the wire at this instant,


(c) the velocity v as a function of x and


(d) the maximum distance the wire will move.