Question

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

• A. \(\frac{R_2^2}{R_1}\)
• B. \(\frac{R_1}{R_2}\)
• C. \(\frac{R_2}{R_1}\)
• D. \(\frac{R_1^2}{R_2}\)

Answer 1

The Correct Option is A

To determine the mutual inductance \( M \) between two concentric loops with radii \( R_1 \) and \( R_2 \) where \( R_1 \gg R_2 \), we employ the formula for mutual inductance between two coaxial loops. The mutual inductance between such loops in the planar arrangement is largely influenced by the area of the smaller loop, and the magnetic field at the location of the smaller loop due to current in the larger loop.

The mutual inductance \( M \) is given by: 

\(M = \frac{\mu_0 \pi R_2^2}{2R_1}\)

This formula arises because the magnetic field \( B \) produced by the larger loop at the center is \(B = \frac{\mu_0 I_1}{2R_1}\), where \( I_1 \) is the current in the larger loop. The magnetic flux \( \Phi \) through the smaller loop (area \( A = \pi R_2^2 \)) is \(\Phi = B \cdot A = \frac{\mu_0 I_1 \pi R_2^2}{2R_1}\). The mutual inductance is the flux per unit current: \(M = \frac{\Phi}{I_1}\).

This directly implies that the mutual inductance is directly proportional to \(\frac{R_2^2}{R_1}\), i.e., the ratio of the area of the smaller loop to the radius of the larger loop.

Let's rule out the other options:

• \(\frac{R_1}{R_2}\): Implies the mutual inductance increases with the radius of the larger loop, which contradicts the formula since it actually contributes to the denominator.
• \(\frac{R_2}{R_1}\): Correct in identifying the relation with radii but misses squaring the smaller loop radius's effect (area).
• \(\frac{R_1^2}{R_2}\): Suggests incorrectly about dependence on the larger loop's squared radius.

Thus, the correct option is \(\frac{R_2^2}{R_1}\).

Answer 2

Magnetic field at the center of primary coil
\(B=\frac{\mu_0i_1}{2R_1}\)
Now, considering it to be uniform, magnetic flux passing through secondary coil is
\(\phi_2=BA=\frac{\mu_0i_1}{2R_1}(\pi R_{2}^2)\)
Now, \(M=\frac{\phi_2}{i_1}\)
\(=\frac{\mu_0\pi R_{2}^2}{2R_1}\)
\(\therefore\ \ M \propto \frac{R_2^{2}}{R_1}\)
Therefore, the correct option is (A) : \(\frac{R_2^2}{R_1}\).