Q47 of 104 Page 306

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?


Given:


Initial mass = m


Magnetic field = B


Length of sliding wire = l


Formula used:


Magnetic force … (i), where i = current, l = length of sliding wire, B = magnetic field.


At equilibrium, this magnetic force balances the weight of the wire mg acting downward, where m = mass, g = acceleration due to gravity.


Therefore, … (ii)


Now, when the wire ab is replaced by another wire of mass 2m, the weight acting downward will be 2mg, where g = acceleration due to gravity.


Hence, net force = 2mg - ilB … (iii) where 2m = mass, g = acceleration due to gravity, i = current, l = length of sliding wire, B = magnetic field.


According to Newton’s law of motion, … (iv), where F = net force, m’ = mass, acceleration


In this case, m’ = 2m.


Therefore, equating (iii) and (iv), we get


2mg - ilB = 2ma


⇒ … (v)


Now, distance travelled can be expressed as … (vi), where s = distance travelled, u = initial velocity = 0(in this case), t = time, a = acceleration.


Now, distance travelled = l (given)


Therefore, from (v), (vi) becomes:



⇒


But, from (i),


Therefore, ⇒


Required time taken = (Ans)


More from this chapter

All 104 →
45

The current generation Ig, shown in figure, sends a constant current i through the circuit. The wire cd is fixed and ab is made to slide on the smooth, thick rails with a constant velocity v towards right. Each of these wires has resistance r. Find the current through the wire cd.


 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.


 48

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



 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.