Question

A coil of radius \( r \) is placed on another coil of radius \( R \) carrying a changing current such that their centres coincide. If both the coils are coplanar, the mutual inductance between them is proportional to

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

Hint:

Mutual inductance depends on magnetic field strength and effective area of the receiving coil.

Answer 1

The Correct Option is D

Step 1: Magnetic field due to a circular coil.
The magnetic field at the center of a circular coil of radius \( R \) is proportional to \( \dfrac{1}{R} \).
Step 2: Magnetic flux through smaller coil.
Flux through the smaller coil depends on magnetic field and area.
\[ \Phi \propto B \times (\pi r^2) \]
Step 3: Mutual inductance relation.
Mutual inductance is proportional to magnetic flux linked per unit current.
\[ M \propto \frac{r^2}{R} \]
Step 4: Conclusion.
The mutual inductance is proportional to \( \dfrac{r^2}{R} \).