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

The magnetic field at the centre of a loop is given by
\(B=\frac{\mu_0 N I}{2 R}\)
\(\therefore \quad\) Magnetic field produced by ring \(A, B_A=\frac{\mu_0 I}{2 r_1}\)
\(\therefore \quad\) Magnetic flux produced in ring B due to \(\mathrm{B}_{\mathrm{A}}\), \(\phi_B=B_A A_B\)
\(\begin{aligned}
& A_B=\pi r_2^2 \\
\therefore \quad & \phi_B \\
= & \frac{\mu_0 I}{2 r_1} \times \pi r_2^2=\frac{\mu_0 \pi r_2^2}{2 r_1} I
\end{aligned}\)
Mutual Inductance \(\mathrm{M}=\frac{\phi}{\mathrm{I}}\) (from MI \(\phi=\mathrm{M}\) )
\(\therefore \quad\) We can write,
\(M=\frac{\phi_B}{I}=\frac{\mu_0 \pi r_2^2 \cdot I}{2 r_1 \cdot I}=\frac{\mu_0 \pi r_2^2}{2 r_1}\)