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

JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Electromagnetic Induction

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

x \times 1 0^{- 7} H

, where

x =

____.

Answer: 128

The magnetic field at the centre of a square loop of side

L

carrying current

i

B = \frac{2 2 μ _{0} i}{π L}

Since the small loop (side

ℓ

) is centred inside the large one and much smaller, this field is essentially uniform over its area, so the flux linked with it is

Φ = B ℓ^{2} = \frac{2 2 μ _{0} i ℓ ^{2}}{π L}

Mutual inductance:

M = \frac{Φ}{i} = \frac{2 2 μ _{0} ℓ ^{2}}{π L}

Using the given condition

L = ℓ^{2}

M = \frac{2 2 μ _{0} ℓ ^{2}}{π ℓ ^{2}} = \frac{2 2 μ _{0}}{π} = 22 \times (4 \times 1 0^{- 7}) = 82 \times 1 0^{- 7} = 128 \times 1 0^{- 7} H

x = 128

JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Integer answer