Question

Two concentric circular coils having radii $r_1$ and $r_2$ $(r_2 \ll r_1)$ are placed co-axially with centres coinciding. The mutual induction of the arrangement is (Both coils have single turn, $\mu_0$ = permeability of free space)

• A. $\dfrac{\mu_0 \pi r_2^2}{r_1}$
• B. $\dfrac{\mu_0 \pi r_1^2}{r_2}$
• C. $\dfrac{\mu_0 \pi r_1^2}{2r_2}$
• D. $\dfrac{\mu_0 \pi r_2^2}{2r_1}$

Hint:

For concentric coils, always calculate flux through the smaller coil due to current in the larger coil.

Answer 1

The Correct Option is D

Step 1: Magnetic field at the centre of larger coil.
Magnetic field due to a circular coil of radius $r_1$ at its centre is:
\[ B = \frac{\mu_0 I}{2r_1} \]
Step 2: Magnetic flux through smaller coil.
Area of smaller coil:
\[ A = \pi r_2^2 \] \[ \Phi = BA = \frac{\mu_0 I}{2r_1}\cdot \pi r_2^2 \]
Step 3: Mutual inductance.
\[ M = \frac{\Phi}{I} = \frac{\mu_0 \pi r_2^2}{2r_1} \]
Step 4: Conclusion.
Mutual inductance of the arrangement is $\dfrac{\mu_0 \pi r_2^2}{2r_1}$.