Q48 of 104 Page 306

The rectangular wire-frame, shown in figure has a width d, mass m, resistance R and a large length. A uniform magnetic field B exists to the left of the frame. A constant force F starts pushing the frame into the magnetic field at t =0.

(a) Find the acceleration of the frame when its speed has increased to v.


(b) Show that after some time the frame will move with a constant velocity till the whole frame enters into the magnetic field. Find this velocity v0.


(c) Show that the velocity at time t is given by



Given:


Length of sliding wire = width of frame = d


Mass = m


Resistance = R


Magnetic field = B


Initial force = F


Formula used:


(a) Induced emf(when it attains a speed v) … (i), where B = magnetic field, d = width of frame, v = velocity


Therefore, induced current , where E = induced emf, R = resistance ⇒ … (ii)


Now, magnetic force acting on the wire … (iii), where I = current, d = length of sliding wire = width of frame, B = magnetic field


Substituting (ii) in (iii), … (iv)


Now, as the magnetic force is in opposite direction to applied force, net force = … (v)


But, from Newton’s 2nd law of motion, net force = ma … (vi), where m = mass, a = acceleration


Equating (v) and (vi):


⇒


Acceleration of the frame at speed (Ans)


(b) For the velocity to be constant, acceleration needs to be 0.


Hence, from previous part,


where F = external force, m = mass, B = magnetic field, d = width of frame, v0 = constant velocity, R = resistance


=>


Constant velocity (Ans)


(c) From part (a), acceleration


Now, acceleration a = dv/dt, where v= velocity, t = time


Hence,


=>


Integrating with proper limits, we get



=>>


=>


=>


=>


But, from previous part (b), we found out that ⇒



Hence,


=> (proved)


More from this chapter

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46

The current generator Ig, shown in figure, sends a constant current i through the circuit. The wire ab has a length ℓ and mass m and can slide on the smooth, horizontal rails connected to I4. The entire system lies in a vertical magnetic field B. Find the velocity of the wire as a function of time.


 47

The system containing the rails and the wire of the previous problem is kept vertically in a uniform horizontal magnetic field B that is perpendicular to the plane of the rails figure. It is found that the wire stays in equilibrium. If the wire ab is replaced by


another wire of double its mass, how long will it take in falling through a distance equal to its length?


 49

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.