Two coaxial circular loops of radii $0.5\text{m}$ and $5\times {10}^{-2}\text{m}$ are separated by a distance $0.5\text{m}$ and carry currents $2\text{A}$ and $1\text{A}$ respectively. Calculate the mutual inductance.

The coil $2$ can be considered to be very small as compared to $1$; so it can be considered on the axis of coil $1$. Magnetic field due to coil $1$ at a distance $x$ on its axis is given by

$B_1=\frac{{\mu }_0{I}_1{R}_1^2}{2(R_1^2+x^2{)}^{3/2}}$

Flux linked with coil $2$ due to magnetic field of coil $1$, 

${\varphi }_2=B_1\left(\pi {R_2}^2\right)=\frac{{\mu }_0\pi R_1^2\pi R_2^2}{2(R_1^2+x^2{)}^{3/2}}$

So the mutual inductance 

$M=\frac{{\varphi }_2}{I_1}=\frac{{\mu }_0\pi R_1^2{R}_2^2}{2(R_1^2+x^2{)}^{3/2}}=\frac{\left({10}^{-7}\right)\left(2{\pi }^2\right)(0.5{)}^2(0.05{)}^2}{\left[\right(0.5{)}^2+(0.5{)}^2{]}^{3/2}}$

$M\cong 3.48\times {10}^{-9}\text{H}$ Ans.