Q62 of 104 Page 306

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.


Given:


Emf = ϵ


Resistance = r


Angle made by rod = θ


Angular velocity = w


Formula used:


From the previous questions, induced emf … (i), where B = magnetic field, w = angular velocity, a = radius


Hence, Total emf =


Total current i = total emf/resistance = … (ii), where R = resistance


‘Now, net force on the rod … (iii),


where mgcosθ = component of weight along rod, where m = mass, g = acceleration due to gravity, θ = angle made by rod with horizontal, and ilB = Magnetic force, where i = current, l = length of rod, B= magnetic field.


Since the rod rotates with uniform angular velocity, net torque about O = 0.


Hence, torque = net force x distance from line of action = , where a = radius of rod


Therefore, … (iii)


Hence, ⇒ (Ans)


More from this chapter

All 104 →
60

Consider the situation shown in the figure of the previous problem. Suppose the wire connecting O and C has zero resistance but the circular loop has a resistance R uniformly distributed along its length. The rod OA is made of rotate with a uniform angular speed ω as shown in the figure. Find the current in the rod when ∠AOC = 90°.

 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.

 63

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.


 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).