Find the mutual inductance in the arrangement, when a small circular loop of wire of radius is placed inside a large square loop of wire of side . The loops are coplanar and their centres coincide. [2023]

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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. [2023]

Solution

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. [2023]

1 $M = \frac{\sqrt{2} μ_{0} R^{2}}{L}$

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

3 $M = \frac{2 \sqrt{2} μ_{0} R^{2}}{L}$

4 $M = \frac{\sqrt{2} μ_{0} R}{L^{2}}$

Ans.
(3)

$ϕ = M i also ϕ = B A$

$ϕ = π R^{2} (4 \frac{μ_{0}}{4 π} \frac{i}{(L / 2)} \sqrt{2})$

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

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Solution

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. [2023]

1 M=2μ0R2L

2 M=22μ0RL2

3 M=22μ0R2L

4 M=2μ0RL2

Ans. (3) ϕ=Mi also ϕ=BA ϕ=πR2(4μ04πi(L/2)2) ⇒ M=22 μ0R2L

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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. [2023]

1 $M = \frac{\sqrt{2} μ_{0} R^{2}}{L}$

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

3 $M = \frac{2 \sqrt{2} μ_{0} R^{2}}{L}$

4 $M = \frac{\sqrt{2} μ_{0} R}{L^{2}}$

Ans.
(3)

$ϕ = M i also ϕ = B A$

$ϕ = π R^{2} (4 \frac{μ_{0}}{4 π} \frac{i}{(L / 2)} \sqrt{2})$

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