A coil of radius ' $r$ ' is placed on another coil (whose radius is $\mathrm{R}$ and current flowing through…

A coil of radius ' $r$ ' is placed on another coil (whose radius is $\mathrm{R}$ and current flowing through it is changing) so that their centres coincide $(\mathrm{R} \gg \mathrm{r})$. If both the coils are coplanar then the mutual inductance between them is ( $\mu_0=$ permeability of free space)
  1. $\frac{\mu_0 \pi R^2}{2 r}$
  2. $\frac{\mu_0 \pi \mathrm{r}^2}{2 \mathrm{R}}$
  3. $\frac{\mu_0 \pi r^2}{R}$
  4. $\frac{\mu_0 \pi R^2}{r}$

Solution

Magnetic field, $B=\frac{\mu_0 I}{2 R}$ Flux passing through the coil, $\begin{aligned} \phi & =\mathrm{B} \times \pi \mathrm{r}^2 \\ \therefore \quad \phi & =\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}} \times \pi \mathrm{r}^2 \end{aligned}$ Mutual Inductance, $/ \mathrm{M}=\frac{\phi}{\mathrm{I}}$ $\therefore \quad M=\frac{\frac{\mu_0 I}{2 R} \times \pi r^2}{I}=\frac{\mu_0 \pi r^2}{2 R}$

Asked in: MHT CET 2023 (11 May Shift 1)

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