Physics Q43 – JEE-MAIN 2026

A circular current loop of radius

R

is placed inside square loop of side length

L

(

L >> R

) such that they are co-planar and their centers coincide. The permeability of free space is

μ_{0}

. The mutual inductance between circular loop and square loop is _______.

Clue 1 of 3: Definition of Mutual Inductance

Mutual inductance $M$ between two coils is defined by the relation $Φ = M I$ , where $Φ$ is the magnetic flux through one coil due to current $I$ in the other.

🧠 Full Solution Path

Step 1: Calculate Magnetic Field at the Center of the Square Loop✦ Active

The magnetic field at the center of a square loop of side $L$ carrying current $I_{s}$ is the sum of the fields due to its four sides. For one side, the magnetic field at the center is given by $B_{s i d e} = \frac{μ_{0} I_{s}}{4 π (L / 2)} (\sin 45^{\circ} + \sin 45^{\circ})$ . Summing for all four sides, the total magnetic field at the center is:

B_{c e n t e r} = 4 \times \frac{μ_{0} I_{s}}{2 π L} (2 \times \frac{1}{\sqrt{2}}) = \frac{2 \sqrt{2} μ_{0} I_{s}}{π L}

Step 2: Calculate Magnetic Flux through the Circular Loop○ Expand

Since the circular loop of radius $R$ is much smaller than the square loop ( $L >> R$ ) and concentric, the magnetic field $B_{c e n t e r}$ can be considered uniform over its area. The area of the circular loop is $A_{c} = π R^{2}$ . The magnetic flux $Φ_{c}$ through the circular loop due to the current $I_{s}$ in the square loop is:

Φ_{c} = B_{c e n t e r} A_{c} = (\frac{2 \sqrt{2} μ_{0} I_{s}}{π L}) (π R^{2}) = \frac{2 \sqrt{2} μ_{0} I_{s} R^{2}}{L}

Step 3: Determine Mutual Inductance○ Expand

By the definition of mutual inductance, $Φ_{c} = M I_{s}$ . Equating the two expressions for flux, we get:

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

Therefore, the mutual inductance $M$ is:

M = \frac{2 \sqrt{2} μ_{0} R^{2}}{L}

💡 Teacher's Secret Hint

Ensure to use the correct approximation for the magnetic field when one loop is much smaller than the other and concentric.

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Physics Question 43 – JEE-MAIN 2026

Clue 1 of 3: Definition of Mutual Inductance

🧠 Full Solution Path