Animated Solution for Physics - Electromagnetic Induction: Two circular coils can be arranged in any of the three situations shown in the figure. Their mutual inductance will be

A Tale of Two Coils

When dealing with electromagnetic induction, one of the most fascinating concepts is mutual inductance. It tells us how effectively a changing current in one coil can induce an electromotive force (EMF) in a neighboring coil. But this isn't just about proximity; it's heavily dependent on geometry and orientation.

Let's embark on a journey to understand how the arrangement of two circular coils dictates their mutual inductance, using the three classic situations presented in our problem.

The Magnetic Embrace

At its core, mutual inductance $M$ is defined by the magnetic flux linkage. If we pass a steady current $I_1$ through the first coil, it generates a magnetic field ${\mathbf{B}}_1$. The mutual inductance is the ratio of the magnetic flux ${\Phi }_{21}$ passing through the second coil to the current $I_1$:

The flux itself is calculated using the surface integral:

Here, $\theta$ is the angle between the magnetic field lines and the area vector of the second coil. To maximize $M$, we need to maximize this flux. This means we want the strongest possible magnetic field passing perpendicularly through the plane of the second coil (so that $\theta =0^{\circ }$ and $\cos \theta =1$).

Coaxial Alignment

In Situation (A), the coils are arranged coaxially, meaning they share the same central axis and their planes are parallel.

Imagine the magnetic field generated by the bottom coil. The field lines emerge from its center, shooting straight up along the axis. Because the top coil is positioned directly in this path, these dense, strong field lines pass right through its center.

Crucially, the magnetic field lines are parallel to the area vector of the top coil. This makes $\theta \approx 0^{\circ }$, yielding a maximum dot product. This coaxial arrangement ensures the highest possible flux linkage, creating a perfect "magnetic embrace."

Coplanar Alignment

Now, let's look at Situation (B), where the coils are coplanar (sitting side-by-side in the same plane).

The magnetic field is strongest at the center of the first coil, but these strong lines completely miss the second coil. The field lines must form closed loops, so they eventually curve back down outside the first coil. It is only these weaker, spreading "return" lines that manage to pass through the second coil.

While the flux isn't zero, it is significantly smaller than in the coaxial case. Furthermore, the direction of the field is opposite to the central field, but the sheer lack of field density makes the mutual inductance quite low.

Perpendicular Alignment

Finally, we arrive at Situation (C). Here, the second coil is turned $90^{\circ }$ so that its plane is perpendicular to the first coil.

The magnetic field lines from the first coil travel upwards and graze right past the plane of the second coil. Because the plane of the second coil is vertical, its area vector is horizontal. The magnetic field lines in that region are mostly vertical.

Mathematically, the angle $\theta$ between the magnetic field ${\mathbf{B}}_1$ and the area vector $d{\mathbf{A}}_2$ is $90^{\circ }$. Since $\cos (90^{\circ })=0$, the dot product vanishes. The flux linkage is practically zero, meaning the mutual inductance in this perpendicular arrangement is zero!

Comparing the three scenarios, it is abundantly clear that the coaxial arrangement captures the lion's share of the magnetic field lines.

Therefore, the mutual inductance is maximum in situation (A). Understanding this geometric dependence is crucial not just for solving physics problems, but for designing efficient transformers, wireless chargers, and metal detectors in the real world!