A small square loop of wire of side L is placed inside a large square loop of wire of side . The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is H, where

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Solution

Q.
A small square loop of wire of side L is placed inside a large square loop of wire of side $L (L = l^{2})$ . The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $\sqrt{x} \times 10^{- 7}$ H, where $x =$ ____. [2024]

Ans.
(128)

Flux linkage for inner loop.

$ϕ = B_{center} \cdot l^{2} = 4 \times \frac{μ_{0} i}{4 π \frac{L}{2}} (\sin 45 + \sin 45) l^{2}$

$ϕ = 2 \sqrt{2} \frac{μ_{0} i}{π L} l^{2}$

$M = \frac{ϕ}{i} = \frac{2 \sqrt{2} μ_{0} l^{2}}{π L} = 2 \sqrt{2} \frac{μ_{0}}{π}$

$= 2 \sqrt{2} \frac{4 π}{π} \times 10^{- 7} = \sqrt{128} \times 10^{- 7} H$

$\Rightarrow x = 128$

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