Question

A coil of radius 'r' is placed on another coil (whose radius is 'R' and current through it is changing) so that their centres coincide. (\( R > r \)). If both coplanar, then the mutual inductance between them is proportional to

• A. \(\frac{R}{r^2}\)
• B. \(\frac{r}{R}\)
• C. \(\frac{R}{r}\)
• D. \(\frac{r^2}{R}\)

Hint:

For coaxial coplanar coils with one much larger, mutual inductance \(\propto \frac{r^2}{R}\). If the smaller coil is placed at the centre of the larger, the field is nearly uniform.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Two coplanar coaxial coils: larger coil of radius \(R\) (primary) and smaller coil of radius \(r\) (secondary) with \(R > r\), centres coincide. Mutual inductance \(M\) depends on flux through smaller coil due to larger coil's current.

Step 2: Key Formula or Approach:
Magnetic field at the centre of larger coil: \(B = \frac{\mu_0 I}{2R}\). For the smaller coil, if \(r \ll R\), field is nearly uniform over its area. Flux through smaller coil: \(\Phi = B \cdot \pi r^2 = \frac{\mu_0 I}{2R} \cdot \pi r^2\). Then \(M = \frac{\Phi}{I} = \frac{\mu_0 \pi r^2}{2R}\). Thus \(M \propto \frac{r^2}{R}\).

Step 3: Detailed Explanation:
The approximation holds because \(R > r\) and coils are coplanar with coincident centres. The field is approximately constant over the smaller coil's area. Hence \(M\) is proportional to \(r^2 / R\).

Step 4: Final Answer:
Option (D) \(\frac{r^2}{R}\).