First slide

Magnetic flux and induced emf

Question

Plane figures made of thin wires of resistance R = 50 milli ohm/metre are located in a uniform magnetic field perpendicular into the plane of the figures and which decrease at the rate dB/dt = 0.1 m T/s. Then currents in the inner and outer boundary are. (The inner radius a = 10 cm and outer radius b = 20 cm)

Moderate

A

10^{– 4} A (Clockwise), 2 $\times$ 10^{– 4} A (Clockwise)
B

10^{– 4} A (Anticlockwise), 2 $\times$ 10^{– 4} A (Clockwise)
C

2 $\times$ 10^{– 4} A (clockwise), 10^{– 4} A (Anticlockwise)
D

2 $\times$ 10^{– 4} A (Anticlockwise), 10^{– 4} A (Anticlockwise)

Solution

Current in the inner coil $i = \frac{e}{R} = \frac{A_{1}}{R_{1}} \frac{dB}{dt}$
length of the inner coil $=2πa$
so it’s resistance $R_{1} = 50 \times 10^{- 3} \times 2 π (a)$
$∴ i_{1} = \frac{{πa}^{2}}{50 \times 10^{- 3} \times 2 π (a)} \times 0 .1 \times 10^{- 3} = 10^{- 4} A$
According to lenz’s law direction of i1 is clockwise.
Induced current in outer coil $i_{2} = \frac{e_{2}}{R_{2}} = \frac{A_{2}}{R_{2}} \frac{dB}{dt}$
$\Rightarrow i_{2} = \frac{{πb}^{2}}{50 \times 10^{- 3} \times (2 πb)} \times 0 .1 \times 10^{- 3} = 2 \times 10^{- 4} A (CW)$

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