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: Field from the big coil.
At its centre the larger coil makes a field $B = \tfrac{\mu_0 I}{2R}$. Since the small coil is tiny compared to $R$, this field is nearly uniform over it.

Step 2: Flux through the small coil.
\[ \Phi = B\cdot\pi r^2 = \frac{\mu_0 I}{2R}\cdot\pi r^2. \]

Step 3: Get the mutual inductance.
$M = \tfrac{\Phi}{I} = \tfrac{\mu_0\pi r^2}{2R}$.

Step 4: Read the dependence.
So $M \propto \tfrac{r^2}{R}$. \[ \boxed{M \propto \tfrac{r^2}{R}} \]