Electromagnetic Induction-06-Subjective Unsolved

(BRUSH UP YOUR CONCEPTS) 1. A straight rod translates with the uniform speed v on a V-shaped conductor immersed in a uniform magnetic field. If the resistance v per unit length of all conductors is  , calculate the induced current as a function of x, the position of the rod from the vertex of the V. 2. An infinite wire carries a current I. A “S” shaped conducting rod of B B two semi-circles each of radius r is placed at an angle  to the wire. The centre of the conductor is at a distance d from the wire. I If the rod translates parallel to the wire with a velocity v as shown in the figure, calculate the emf induced across the ends OB of the rod. d Y 3. A closed loop of wire consists of a pair of equal semi-circles of radius a lying in mutually perpendicular planes. Auniform magnetic field B is directed perpendicular to the axis AA’ and makes an angle 45º with the planes of the semi-circles. Calculate the flux through this closed loop. Z 4. A copper wire in the form of a square of side a lies on a rough B horizontal surface. The mass of the wire is m. A uniform magnetic field B is in the horizontal plane parallel to one of the sides of the square. What is the minimum current that should pass through the wire to make one end lift off the table. Assume friction is sufficient to prevent slip of the other edge. 5. A equilateral triangle of side a is placed in the magnetic field with one side AC along a diameter and its center coinciding with the centre of the mag- × ×B × × × × netic field as shown in the figure. If the magnetic field varies with time as B = kt; then × × × A × O × × × × C × (a) show by vectors, the direction of the induced electric field → at the three sides of the triangle. (b) What is the induced emf in side AB ? (c) What is the induced emf in side BC ? (d) What is the induced emf in side CA ? (e) What is the total emf induced in the loop ? × × × × × × × × aA × × × × B (f) Does the above result tally with the induced emf computed using Faraday’s Law? 6. A circular coil A of 20 turns and radius 2 cm is placed coaxially at the center of another circular coil B of 40 turns and radius 20 cms. Calculate (a) Their mutual inductance (b) If current in A increases from 0 to 4A in 2s, calculate the emf induced in B (c) Calculate the charge that passes through coil B in the same time if the resistance of coil B is 10 ohm. 7. A conductor has a resistance R at one of its sides and has zero resis- tance elsewhere. The width of the conductor is l. A conducting rod is placed on the wire and given the velocity 0 as shown in figure. If the whole setup is placed in a uniform magnetic field B, find the velocity of the rod as a function of time and position. 8. A wire shaped as a semi-circle of radius a rotates about an axis OO’ with an angular velocity  in a uniform magnetic field of induction B (Figure). The rotation axis is perpendicular to the field direction. The total resistance of the circuit is equal to R. Neglect- ing the magnetic field of the induced current, find the mean amount of thermal power being generated in the loop during a rotation pe- riod. 9. A uniform magnetic field B exists in a cylindrical region of radius a d 10 cm as shown in figure. A uniform wire of length 80 cm and resistance 4.0  is bent into a square frame and is placed with one side along a diameter of the cylindrical region. If the magnetic field increases at a constant rate of 0.010 T/s, find the current induced in the frame. b c 10. An inductor-coil of inductance 20 mH having resistance 10  is joined to an ideal battery of emf 5.0 V. Find the rate of change of the induced emf at (a) t = 0, (b) t = 10 ms and (c) t = 1.0 s. (CHECK YOUR SKILLS) 1. A plane loop shown in figure is shaped as two squares with sides a = 20 cm and b = 10 cm and is introduced into a uniform magnetic field at right angles to the loop’s plane. The magnetic induction varies with time as B = B0 sin t , where B0 = 10 mT and  = 100 s-1. Find the amplitude of the current induced in the loop if its resis- tance per unit length is equal to ρ  50 103 / m . The inductance of the loop is to be neglected. 2. In the middle of a long solenoid there is a coaxial ring of square cross-section, made of conducting material with resistivity  . The thickness of the ring is equal to h, its inside and outside radii are equal to ‘a’ and ‘b’ respectively. Find the current induced in the ring if the magnetic induction produced by the solenoid varies with time as B =  t, where  is a constant. The inductance of the ring is to be neglected. 3. A wire loop enclosing a semi-circle of radius a is located on the boundary of a uniform magnetic field of induction B as shown in figure (Figure). At the moment t = 0 the loop is set into rotation with a constant angular acceleration  about an axis O coinciding with a line of vector B on the boundary. Find the emf induced in the loop as a function of time t. Draw the approximate plot of this function. The arrow in the figure shows the emf direction taken to be positive. 4. A closed circuit consists of a source of constant emf  and a choke coil of inductance L connected in series. The active resistance of the whole circuit is equal to R. After a long time, at the moment t = 0 the choke coil inductance was decreased abruptly  times. Find the current in the circuit as a function of time t. 5. Consider the situation as shown in the figure. Suppose the wire con- necting O and C has zero resistance but the circular loop has a resis- tance R uniformly distributed along its length. The rod OA of length l is made to rotate with a uniform angular speed as shown in the figure. Find the current in the rod when  AOC = 90º. 6. The rectangular wire-frame, as shown in figure has a width d, mass m, resistance R and a large length. A uniform magnetic field B exists to the d 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 . (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 0 . (c) Show that the velocity at time t is given by : v  v (1 eFt / mv0 ) 7. A long-straight wire carries a current I. At a distance a and b from it, there are two long wires parallel to the first. They are connected by a resistor R at one end. A connector is moved with constant velocity away from R. Neglecting friction, resistance and induc- tance of the loop, find (a) the magnitude and the direction of the current induced. (b) the force required to maintain the speed of the connector. 8. A wire loop in the form of a sector of a circle of radius a = 20 cm is located on boundary of a uniform magnetic field of strength B = 1 tesla as shown in figure. Angle of sector is 0 = 45º and resistance O B of the loop is R = 20  . The loop is rotated about axis O which is parallel to the magnetic field and lies on its boundary, with a con- stant angular velocity . If rate of heat generation in the loop is 3.14 mj/revolution, calculate (a) angular velocity and (b) average thermal power. 9. Show that self-inductance for a length 𝑙 of a long wire associated with the flux inside the wire only is 0𝑙 / 8 , independent of the wire diameter. 10. A square loop of side a has a mass m and it is pivoted about a horizontal axis OO' that passes through one of its sides. The whole system is placed in a uniform magnetic field directed downwards. When a current flows through the loop, it deflects and reaches an equilibrium position with the plane of the loop making an angle  with the magnetic field. Calculate the current in the loop.