91Ó°ÊÓ

Mutual inductance is defined as the measure of the ability of one circuit to induce electromotive force (EMF) in another circuit due to their magnetic field. Mathematically, it is given as:$$ M = \frac{\mu_0 N_1 N_2 A}{l} $$Where, \(M\) = mutual inductance, \(\mu_0\) = permeability of free space (\(4\pi\times10^{-7}\ Tm/A\)), \(N_1\) = number of turns in the first loop, \(N_2\) = number of turns in the second loop, \(A\) = cross-sectional area, and \(l\) = mean length of the magnetic pathways. In this case, we have concentric rings with radii \(r_1\) and \(r_2\), and since they are planar, the number of turns in each loop is 1. Also, since both loops are in air and very close to each other, we can assume that the length \(l\) of the magnetic pathway is approximately equal to the distance between the loops, \(r_{1}\).

The cross-sectional area, 'A', is the area through which the magnetic field lines pass. For the smaller loop with radius \(r_{2}\), this area can be given as:$$ A = \pi r_{2}^2 $$

Now we will plug the values into the formula for the mutual inductance given in step 1. $$ M = \frac{\mu_0 N_1 N_2 A}{l} = \frac{\mu_0 \cdot 1 \cdot 1 \cdot \pi r_{2}^2}{r_{1}} $$After simplifying, we have:$$ M = \frac{\mu_{0} \pi r_{2}^2}{r_{1}} $$ Comparing with the given options, the answer is: (A) \(\frac{\mu_{0} \pi r_{2}^{2}}{2 r_{1}}\)