A small circular loop of radius R is placed concentrically and coplanarly inside a much larger square loop of side L, where L is much greater than R. What is the mutual inductance between the two loops? Take the permeability of free space as μ₀.
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Question

A small circular loop of radius R is placed concentrically and coplanarly inside a much larger square loop of side L, where L is much greater than R. What is the mutual inductance between the two loops? Take the permeability of free space as μ₀.

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G1 Physics · Accepted Answer

Because R is much smaller than L, the magnetic field produced by the square loop may be taken as nearly uniform over the small circular loop. Magnetic field at the centre due to one side of the square is B(side) = [μ₀I ÷ (4πa)](sin45°+sin45°), where the perpendicular distance from the centre to a side is a = L ÷ 2. Thus, B(side) = [μ₀I ÷ (4π(L ÷ 2))](2 sin45°) = μ₀I√2 ÷ (2πL). There are four sides, so B(square) = 4B(side) = 2√2 μ₀I ÷ (πL). Area of the circular loop is A(circle) = πR². Therefore…