Animated Solution for Physics - Electromagnetic Induction: A small square loop of wire of side $l$ is placed inside a large square loop of wire of side $L(L>>l)$. The loops are coplanar and their centres coincide. The mutual inductance of the system is proportional to

Imagine you are standing in the center of a massive square arena, and right at your feet is a tiny square box. This is exactly the setup we are dealing with: a large square loop of side $L$ and a much smaller square loop of side $l$ placed concentrically. Our goal is to find the mutual inductance $M$ of this system.

But what exactly is mutual inductance? It is a measure of how effectively a change in current in one coil induces an electromotive force (EMF) in another. Mathematically, it is defined as the ratio of the magnetic flux $\varphi$ linked with the secondary coil to the current $i$ in the primary coil:

Here is a brilliant trick in physics known as the Reciprocity Theorem: $M_{12}=M_{21}$. This means the mutual inductance of coil 1 with respect to coil 2 is exactly the same as coil 2 with respect to coil 1.

Instead of trying to figure out the complex, non-uniform magnetic field produced by the tiny loop over the massive area of the large loop, we do the reverse! We assume a current $i$ flows through the large loop and calculate the flux it creates through the small loop. It is mathematically much more elegant.

Let's find the magnetic field $B$ at the center of the large loop. According to Biot-Savart's law, the magnetic field produced by a straight wire is inversely proportional to the perpendicular distance from it. For a square loop of side $L$, the distance from the wire to the center is $\frac{L}{2}$.

Therefore, the magnetic field $B$ at the center is directly proportional to the current $i$ and inversely proportional to the side length $L$:

Now, we face a potential hurdle. The magnetic field inside a square loop is not perfectly uniform; it gets stronger as you move closer to the wires.

However, the problem gives us a crucial constraint: $l\ll L$. Because the small loop is so incredibly tiny and located exactly at the center, the variation of the magnetic field across its area is negligible. We can safely assume that the magnetic field $B$ is perfectly uniform over the entire area of the small loop.

With a uniform magnetic field, calculating the flux becomes a breeze. The magnetic flux $\varphi$ through the small loop is simply the product of the magnetic field $B$ and the area $A$ of the small loop. Since it's a square of side $l$, its area is $l^2$.

Now, we substitute our proportionality for the magnetic field into the flux equation:

Finally, we return to our definition of mutual inductance. When we divide the flux by the current $i$, the current beautifully cancels out, proving that mutual inductance is purely a geometric property!

And there we have it! The mutual inductance is proportional to $\frac{l^2}{L}$, which corresponds to option (b).