Question: 33. Two conducting circular loops of radii \mathrm{R}_{1} and \mathrm{R}_{2} are placed in the same plane with their centres coinciding. If R_{1} \gg R_{2}, the mutual inductance M between them will be directly proportional to:
(1) \frac{R_{1}}{R_{2}}
(2) \frac{R_{2}}{R_{1}}
(3) \frac{R_{1}^{2}}{R_{2}}
(4) \frac{R_{2}^{2}}{R_{1}}
Answer: Option (4)
Explanation:
Mutual inductance is defined as the magnetic flux linked with one loop due to unit current in the other loop.
The magnetic field at the centre of a circular loop of radius R_{1} carrying current I is B=\frac{\mu_{0} I}{2R_{1}}.
Since R_{1} \gg R_{2}, the magnetic field over the smaller loop can be taken as nearly uniform and equal to the field at the centre.
The area of the smaller loop is A=\pi R_{2}^{2}.
The magnetic flux linked with the smaller loop is
\Phi = B A = \frac{\mu_{0} I}{2R_{1}} \times \pi R_{2}^{2}.
Mutual inductance is given by M=\frac{\Phi}{I}.
Substituting, we get M=\frac{\mu_{0}\pi R_{2}^{2}}{2R_{1}}.
Hence, M \propto \frac{R_{2}^{2}}{R_{1}}.
Therefore, the correct answer is Option (4).