Find the mutual inductance in the arrangement, when a small circular loop of wire of radius R is placed inside a large square loop of wire of side L (L ≫ R). The loops are coplanar and their centres coincide: Image
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Question

Find the mutual inductance in the arrangement, when a small circular loop of wire of radius R is placed inside a large square loop of wire of side L (L ≫ R). The loops are coplanar and their centres coincide: Image

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G1 Physics · Accepted Answer

At the centre of the square loop, the magnetic field due to the four sides is approximately uniform over the small circular loop. For each side, B(side) = μ₀I/(4πr)(sin45° + sin45°), with r = L/2. Summing four sides gives B = 2√2 μ₀I/(πL). Flux through the small loop is Φ = BπR² = (2√2 μ₀IR²)/L. Hence M = Φ/I = 2√2 μ₀R²/L.