Question

The mutual inductance between two coplanar concentric rings A and B of radii \(R_1\) and \(R_2\) placed in air when a current \(I\) flows through ring A is (\(R_1 \gg R_2\)). (\(\mu_0\) = permeability of free space)

• A. \( \dfrac{\mu_0 \pi R_2}{R_1} \)
• B. \( \dfrac{\mu_0 \pi R_1}{R_2} \)
• C. \( \dfrac{\mu_0 \pi R_2^2}{2R_1} \)
• D. \( \dfrac{\mu_0 \pi R_1^2}{2R_2} \)

Hint:

When one loop is much smaller than the other, assume a uniform magnetic field over the smaller loop for easier calculation.

Answer 1

The Correct Option is C

Step 1: Magnetic field due to current in ring A.
For a circular loop of radius \(R_1\) carrying current \(I\), the magnetic field at its center is given by:
\[ B = \frac{\mu_0 I}{2R_1} \]
Step 2: Magnetic flux through ring B.
Since \(R_1 \gg R_2\), the magnetic field over ring B can be assumed uniform.
Area of ring B is \(A = \pi R_2^2\).
\[ \Phi = B \times A = \frac{\mu_0 I}{2R_1} \times \pi R_2^2 \]
Step 3: Mutual inductance calculation.
Mutual inductance is defined as:
\[ M = \frac{\Phi}{I} \] \[ M = \frac{\mu_0 \pi R_2^2}{2R_1} \]
Step 4: Conclusion.
The correct expression for mutual inductance is option (C).