Q63 of 104 Page 306

A wire of mass m and length ℓ can slide freely on a pair of smooth, vertical rails figure. A magnetic field B exists in the region in the direction perpendicular to the plane of the rails. The rails are connected at the top end by a capacitor of capacitance C. Find the acceleration of the wire neglecting any electric resistance.


Given:


Mass = m


Length = l


Magnetic field = B


Capacitance = C


Formula used:


Induced emf … (i), where B = magnetic field, l = length, v = velocity


Also, we know that … (ii), where q = charge, C = capacitance.


Hence, ⇒ .. (ii)


Therefore, current , where q = charge, t = time


=> … (iii), where a = acceleration


Therefore, net force on rod = weight - magnetic force = mg - ilB .. (iv), where m = mass, g = acceleration due to gravity, i = current, l = length, B = magnetic force


From newton’s second law of motion, F = ma … (v), where f= force, m = mass, a = acceleration


Therefore, (from (iii))


⇒


=>acceleration (Ans)


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61

Consider a variation of the previous problem figure. Suppose the circular loop lies in a vertical plane. The rod has a mass m. The rod and the loop have negligible resistances but the wire connecting O and C has a resistance R. The rod is made to rotate with a uniform angular velocity ω in the clockwise direction by applying a force at the midpoint of OA in a direction perpendicular to it. Find the magnitude of this force when the rod makes an angle θ with the vertical.

 62

Figure shows a situation similar to the previous problem. All parameters are the same except that a battery of emf ϵ and a variable resistance R are connected between O and C. Neglect the resistance of the connecting wires. Let θ be the angle made by the rod from the horizontal position (shown in the figure), measured in the clockwise direction. During the part of the motion 0 < θ < π/4 the only forces acting on the rod are gravity and the forces exerted by the magnetic field and the pivot. However, during the part of the motion, the resistance R is varied in such a way that the rod continues to rotate with a constant angular velocity ω. Find the value of R in terms of the given quantities.


 64

A uniform magnetic field B exists in a cylindrical region, shown dotted in figure. The magnetic field increases at a constant rate dB/dt. Consider a circle of radius r coaxial with the cylindrical region.

(a) Find the magnitude of the electric field E at a point on the circumference of the circle.


(b) Consider a point P on the side of the square circumscribing the circle. Show that the component of the induced electric field at P along ba is the same as the magnitude forum in part (a).



 65

The current in an ideal, long solenoid is varied at a uniform rate of 0.01 A s–1. The solenoid has 2000 turns/m and its radius is 6.0 cm.

(a) Consider a circle of radius 1.0 cm inside the solenoid with its axis coinciding with the axis of the solenoid. Write the change in the magnetic flux through this circle in 2.0 seconds.


(b) Find the electric field induced at a point on the circumference of the circle.


(c) Find the electric field induced at a point outside the solenoid at a distance 8.0 cm from its axis.