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Offline Centres

Q.

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:

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a

$M = \frac{\sqrt{2} μ_{0} R}{L^{2}}$

b

$M = \frac{2 \sqrt{2} μ_{0} R}{L^{2}}$

c

$M = \frac{2 \sqrt{2} μ_{0} R^{2}}{L}$

d

$M = \frac{\sqrt{2} μ_{0} R^{2}}{L}$

answer is D.

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Detailed Solution

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Let current I flows through square loop so,
$\begin{array}{l} B_{center} = 4 \times \frac{μ_{0} I}{4 π \times \frac{L}{2}} [\sin (45^{\circ}) \times 2] \\ = \frac{2 \sqrt{2} μ_{0} I}{πL} \\ ∴ flux through circular loop: \\ ϕ = B_{center} \times {πR}^{2} \end{array}$

$\begin{array}{l} \Rightarrow MI = \frac{2 \sqrt{2} μ_{0} I}{πL} \times {πR}^{2} \\ \Rightarrow M = \frac{2 \sqrt{2} μ_{0} R^{2}}{L} \end{array}$

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