Two concentric circular coils having radii $r_{1}$ and $r_{2},\left(r_{2}< < r_{1}\right)$ are placed co-…
Two concentric circular coils having radii $r_{1}$ and $r_{2},\left(r_{2}< < r_{1}\right)$ are placed co-
axially with centres coinciding. The mutual induction of the arrangement is
(Both coils have single turn) $\left(\mu_{0}=\right.$ permeability of free space)
$\frac{\mu_{0} \pi r_{2}^{2}}{r_{1}}$
$\frac{\mu_{0} \pi r_{1}^{2}}{r_{2}}$
$\frac{\mu_{0} \pi r_{1}^{2}}{2 r_{2}}$
$\frac{\mu_{0} \pi r_{2}^{2}}{2 r_{1}}$
Solution
If current $\mathrm{I}_{1}$ flows in the coil of radius $\mathrm{r}_{1}$, the magnetic field at the centre is given by
$\mathrm{B}_{1}=\frac{\mu_{0} \mathrm{I}_{1}}{2 \mathrm{r}_{1}}$
The magnetic flux passing through the coil of radius $\mathrm{r}_{2}$ will be
$\phi_{2}=\mathrm{B}_{1} \times \pi \mathrm{r}_{2}^{2}=\frac{\mu_{\mathrm{o}} \mathrm{I}_{1}}{2 \mathrm{r}_{1}} \times \pi \mathrm{r}_{2}^{2}$
Mutual inductance $\quad M=\frac{\phi_{2}}{I_{1}}=\frac{\mu_{0} \pi r_{2}^{2}}{2 r_{1}}$
.