Q135 / 150 · MHT CET 2021 (21 Sep Shift 1) · Physics

A coil of radius ' $r$ ' is placed on another coil (whose radius is ' $\mathrm{R}$ ' and current through it is changing) so that their centers coincide. $(R \gg r)$. If both are coplanar, then the mutual inductance between them is proportional to

  • A
    $\frac{R}{r^2}$
  • B
    $\frac{r}{R}$
  • C
    $\frac{\mathrm{R}}{\mathrm{r}}$
  • Correct answer
    D
    $\frac{r^2}{\mathrm{R}}$

Solution

Magnetic field at the center, $B=\frac{\mu_0 I}{R}$ $\phi=$ Magnetic flux passing through the smaller coil $=\pi \mathrm{r}^2 \mathrm{~B}$ $$ \begin{aligned} & \phi=\pi r^2 \times \frac{\mu_0 I}{R} \\ & M=\frac{\phi}{I}=\frac{\mu_0 \pi r^2}{R} \\ & M \propto \frac{r^2}{R} \end{aligned} $$

Hence, the correct answer is option: 'D'.

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