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 between two concentric conducting loops of radii \(R_1\) and \(R_2\), where \(R_1\) (radius of the larger loop) is much greater than \(R_2\) (radius of the smaller loop), we must understand the concept of mutual inductance:

Mutual inductance (\(M\)) between two loops is a measure of the magnetic linkage between them. When one loop carries a current, it generates a magnetic field that induces an EMF in the other loop.

Given the configuration:

• The larger loop (with radius \(R_1\)) produces a magnetic field that is approximately constant over the area of the smaller loop (with radius \(R_2\)).

The mutual inductance can be expressed using the following approximation due to the large ratio:

• The magnetic field \(B\) at the center of a loop carrying a current \(I\) is given by: \(B = \frac{\mu_0 I}{2R_1}\) (only considers the larger loop for the magnetic field calculation).

The magnetic flux \(\Phi\) through the smaller loop is then given by: \(\Phi = B \times \pi R_2^2 = \frac{\mu_0 I \pi R_2^2}{2R_1}\)

By definition, mutual inductance \(M\) is: \(M = \frac{\Phi}{I} = \frac{\mu_0 \pi R_2^2}{2R_1}\)

Therefore, the mutual inductance is directly proportional to: \(\frac{R_2^2}{R_1}\)

Thus, the correct answer is: \(\frac{R_2^2}{R_1}\)

This approach assumes the radius \(R_1\) of the larger loop is much larger than that of the smaller loop \(R_2\), thus justifying the approximation used for the uniform magnetic field over the smaller loop area.