Two conducting circular loops of radii $R_1$ and $R_2$ are placed in the same plane with their centres coincid

Question

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

Accepted Answer

Magnetic field at centre $B = \frac{\mu_0 i}{2R_1}$. Magnetic flux through inner coil $\phi = B 	imes \pi R_2^2 = \frac{\mu_0 i}{2R_1} 	imes \pi R_2^2$. As per definition, $\phi = Mi$, so $M = \frac{\mu_0 \pi R_2^2}{2 R_1}$. Therefore, $M \propto \frac{R_2^2}{R_1}$.

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