Two conducting circular loops of radii R1 and R2 are placed in the same plane with their centres coinciding. If R1>> R2, the mutual inductance M between them will be directly proportional to :

The correct answer is R₂²/R₁.

To find the mutual inductance between two concentric circular loops, we need to determine the magnetic flux linked with one loop due to the current in the other loop. Let the larger loop have radius R₁ and the smaller loop have radius R₂. We are given that R₁ >> R₂.

Let a current I₁ flow through the larger loop (radius R₁). The magnetic field produced at the center of this circular loop is given by the formula:

$B_1=\frac{{\mu }_0{I}_1}{2R_1}$

Since R₁ is much larger than R₂, the magnetic field B₁ can be considered approximately uniform over the entire area of the smaller loop. The area of the smaller loop is:

$A=\pi R_2^2$

The magnetic flux (Φ₂) linked with the smaller loop due to this uniform magnetic field is the product of the magnetic field B₁ and the area A of the smaller loop:

${\Phi }_2=B_1\times A$

Substituting the expressions for B₁ and A into the flux equation, we get:

${\Phi }_2=\left(\frac{{\mu }_0{I}_1}{2R_1}\right)\times \left(\pi R_2^2\right)$

Rearranging the terms, we have:

${\Phi }_2=\left(\frac{{\mu }_0\pi }{2}\right)\left(\frac{R_2^2}{R_1}\right)I_1$

By the definition of mutual inductance M, the total flux linked with the secondary coil is proportional to the current in the primary coil, which is expressed as:

${\Phi }_2=M{I}_1$

Comparing the two equations for Φ₂, we find the expression for mutual inductance M:

$M=\frac{{\mu }_0\pi }{2}\left(\frac{R_2^2}{R_1}\right)$

In this formula, the term (μ₀π) / 2 is a constant. Therefore, the mutual inductance M is directly proportional to the term (R₂² / R₁).