Electromagnetic Induction-02-OBJECTIVE SOLVED PROBLEMS

OBJECTIVE SOLVED PROBLEMS 1. A uniform but time-varying magnetic field B (t) exists in a circular region of radius a and is directed into the plane of the paper, as shown. The magnitude of the induced electric field at point P at a distance r from the centre of the circu- lar region : (a) is zero (b) decreases as 1/r (c) increases as r (d) decreases as 1/r2. Ans. (b) Solution: Magnitude of induced electric field a2 dB P B(t) E  2r dt Thus, the magnitude of the electric field increases linearly from zero at the center of electric field to (a/2) (dB/dt) at the edge of circular region of radius a, and then decreases inversely with distance. 2. A coil of wire having finite inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time t = 0. so that a time-dependent current I1 (t) starts flowing through the coil. If I2 (t) is the current induced in the ring, and B (t) is the magnetic field at the axis of the coil due to I1 (t), then as a function of time (t > 0), the product I2 (t) B (t) (a) increases with time (c) does not vary with time (b) decreases with time (d) passes through a maximum. Ans. (d) Solution : Current through the coil = I1 (t) Magnetic field at center B (t) = 0 . 2I1 (t) 4 a Current induced in the ring I (t)   2 R  = induced emf in the ring I2 (t)  1 d  A dB R dt R dt A = area of the ring I (t)B(t)  A dB(t) B(t) 2 R dt which passes through the maximum. 3. A coil of inductance 8.4 mH and resistance 6  is connected to a 12 V battery. The current in the coil is 1.0 A at approximately the time. (a) 500 ms (b) 20 ms (c) 35 ms (d) 1 ms. Ans. (d) Solution : Current developed with time in a coil of inductance I  V (l  et /  ) where  = L/R R we have  = 8.4 mH  1.4ms 6 Hence 1 A =  12V (l  et /1.4ms )   or et /1.4ms  1 1  1 2 2 or t /1.4ms  ln  1   0.693   or t  (1.4  0.693)ms  0.97ms  l ms. 4. A small square loop of wire of side l is placed inside a large square loop of wire of side L (L>>l). The loops are co-planar and their centers coincide. The mutual inductance of the system is proportional to (a) l/L (b) l2/L (c) L/l (d) L2/l. Ans. (b) Solution: Magnetic field produced by a current in a large square loop of wire at its center B  2 20i L The magnetic flux 12 that links big loop with the small square loop of side l (l<

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