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# A conducting loop of area 5.0 cm2 is placed in a magnetic field which varies sinusoidally with time as B = B0 sin ωt where B0 = 0.20 T and ω = 300 s−1. The normal to the coil makes an angle of 60° with the field. Find (a) the maximum emf induced in the coil, (b) the emf induced at τ = (π/900)s and (c) the emf induced at t = (π/600) s.

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Solution

## Given: Area of the coil, A = 5 cm2 = 5 × 10−4 m2 The magnetic field at time t is given by B = B0 sin ωt = 0.2 sin (300t) Angle of the normal of the coil with the magnetic field, θ = 60° (a) The emf induced in the coil is given by $e=\frac{-d\theta }{dt}=\frac{d}{dt}\left(BA\mathrm{cos}\theta \right)\phantom{\rule{0ex}{0ex}}=\frac{d}{dt}\left[\left({B}_{0}\mathrm{sin}\mathrm{\omega }t\right)×5×{10}^{-4}×1/2\right]\phantom{\rule{0ex}{0ex}}={B}_{0}×\frac{5}{2}×{10}^{-4}\frac{d}{dt}\left(\mathrm{sin}\mathrm{\omega }t\right)\phantom{\rule{0ex}{0ex}}=\frac{{B}_{0}5}{2}{10}^{-4}\omega \left(\mathrm{cos}\mathrm{\omega }t\right)\phantom{\rule{0ex}{0ex}}=\frac{0.2×5}{2}×300×{10}^{-4}×\mathrm{cos}\mathrm{\omega }t\phantom{\rule{0ex}{0ex}}=15×{10}^{-3}\mathrm{cost}\mathrm{\omega }t$ The induced emf becomes maximum when cos ωt becomes maximum, that is, 1. Thus, the maximum value of the induced emf is given by ${e}_{max}=15×{10}^{-3}=0.015\mathrm{V}$ (b) The induced emf at t = $\left(\frac{\mathrm{\pi }}{900}\right)\mathrm{s}$ is given by e = 15 × 10−3 × cos ωt = 15 × 10−3 × cos $\left(300×\frac{\mathrm{\pi }}{900}\right)$ = 15 × 10−3 × $\frac{1}{2}$ $=\frac{0.015}{2}=0.0075=7.5×{10}^{-3}\mathrm{V}$ (c) The induced emf at t = $\frac{\mathrm{\pi }}{600}\mathrm{s}$ is given by e = 15 × 10−3 × cos $\left(300×\frac{\mathrm{\pi }}{600}\right)$ = 15 × 10−3 × 0 = 0 V

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