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Question:

A small circle loop of wire of radius a is located at the centre of a much larger circular wire loop of radius b. The two loops are in the same plane. The outer loop of radius b carries an alternating current I + Iocos(ωt). The emf induced in the smaller inner loop is nearly:

πμoIob²aωcos(ωt)

πμoIo2.a²bωcos(ωt)

πμoIo2.a²bωsin(ωt)

πμoIo a²bωsin(ωt)

Solution:

Magnetic field due to outer current loop
Bo = μ₀I/2R = μ₀(I₀cosωt)/2b
Induced emf in inner loop
V = N d(BA)/dt = 1 x d/dt(μ₀I₀cosωt/2b x πa²) = πμ₀I₀a²/2b d/dt(cosωt) = πμ₀I₀a²/2bω(-sinωt) = -πμ₀I₀a²ωsin(ωt)/2b.