Question

Two concentric circular coils one of small radius $r_1$ and the other of large radius $r_2$ such that $r_1<<r_2$ are placed co-axially with centres coinciding. Obtain the mutual inductance of the arrangement.

Answer 1

$N_1=$ no. of turns per unit length of inner coil
$N_2=$ no. of turns per unit length of outer coil
$r_1$ and $r_2$ radius
$L=$ length of coil (both)
$L_1=$ current flowing in inner coil
Magnetic field $B$ due to this current
Bis only inside coil $1$
Outside coil $IB=0$
Magnetic flux through $I$ turn of coil $2$ due to $B$ is
$\varphi =B\times Area={\mu }_0{n}_1{i}_1\times z_r^2$
Total no. of turns in coil $2=n_2\times L$
Mutual inductance $M$ is $\varphi =M\times i$
So, $M={\mu }_0{n}_1{n}_{2}z{r}^{2}L$