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Question

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=-dθ dt=ddt(BA cos θ) =ddtB0 sin ωt×5×10-4×1/2 =B0×52×10-4ddt(sin ωt) =B05210-4 ωcos ωt =0.2×52×300×10-4×cos ωt =15×10-3cost ωt

The induced emf becomes maximum when cos ωt becomes maximum, that is, 1.
Thus, the maximum value of the induced emf is given by
emax=15×10-3=0.015 V
(b) The induced emf at t = π900 s is given by
e = 15 × 10−3 × cos ωt
= 15 × 10−3 × cos 300×π900
= 15 × 10−3 × 12
=0.0152=0.0075=7.5×10-3 V
(c) The induced emf at t = π600 s is given by
e = 15 × 10−3 × cos 300×π600
= 15 × 10−3 × 0 = 0 V

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