Question

Two conducting circular loops of radii $R_1$ and $R_2$ are placed in the same plane, with their centers coinciding. If $R_1>>R_2$, the mutual inductance $M$ between them will be directly proportional to -

• A. $\frac{R_2^2}{R_1}$
• B. $\frac{R_1}{R_2}$
• C. $\frac{R_2}{R_1}$
• D. $\frac{R_1^2}{R_2}$

Answer 1

The correct option is A $\frac{R_2^2}{R_1}$

Two concentric coils are of radius $R_1$ and $R_2$ as shown in the figure.


Let current in bigger loop be $i$.

Magnetic field at the center,

$B=\frac{{\mu }_{0}i}{2R_1}$

Magnetic flux through smaller coil,

$\varphi =BA=\frac{{\mu }_{0}i}{2R_1}\times \pi R_2^2$

$\Rightarrow \varphi =\frac{{\mu }_0\pi i{R}_2^2}{2R_1}$

As per definition, $\varphi =Mi$

$\Rightarrow M=\frac{\varphi }{i}=\frac{\frac{{\mu }_0\pi i{R}_2^2}{2R_1}}{i}$

$\Rightarrow M=\frac{{\mu }_0\pi R_2^2}{2R_1}$

$\therefore M\propto \frac{R_2^2}{R_1}$

Hence, option $\left(A\right)$ is the correct answer.