Two coils are placed close to each other. The mutual inductance of the pair of coils depends upon
(a

Q: Doesn't Faraday's law state that induced emf depends on the rate of change of current?

Yes, the induced emf $e = -M \frac{di}{dt}$ depends on the rate of change of current. However, $M$ itself is just the constant of proportionality in this equation. It is independent of the current, much like how resistance $R$ in Ohm's law ($V = IR$) is independent of $V$ and $I$.

Q: Why doesn't the material of the wire affect mutual inductance?

The material of the wire affects its electrical resistance, not its ability to produce or link magnetic flux. The magnetic field depends on the current and the geometry, while the flux linkage depends on the geometry and the magnetic permeability of the core medium (like air or iron), not the wire itself.

Q: How exactly does the relative orientation affect $M$?

Magnetic flux is a dot product of the magnetic field vector and the area vector ($\Phi = \vec{B} \cdot \vec{A}$). If two coils are placed perpendicular to each other, the magnetic field lines from one coil will graze parallel to the plane of the second coil, resulting in zero flux linkage and thus $M = 0$.

Q: If we increase the current in the primary coil, won't the mutual inductance increase?

No. Increasing the current will proportionally increase the magnetic field and the magnetic flux linked with the secondary coil. Since $M = \frac{\Phi_2}{I_1}$, both the numerator and denominator increase by the same factor, leaving $M$ constant.

Q: Does the distance between the coils matter?

Absolutely! As the distance between the coils increases, the magnetic field produced by the primary coil spreads out and weakens before reaching the secondary coil. This reduces the flux linkage, thereby decreasing the mutual inductance.

Question

Two coils are placed close to each other. The mutual inductance of the pair of coils depends upon

Options:

(A) the rates at which currents are changing in the two coils

(B) relative position and orientation of the two coils

(C) the materials of the wires of the coils

(D) the currents in the two coils

Sigma Percentile · Accepted Answer

### The Essence of Mutual Inductance

When we place two coils close to each other, a fascinating electromagnetic interaction occurs. If we pass a time-varying current through the first coil (the primary), it generates a changing magnetic field. This magnetic field permeates the surrounding space, and a portion of its magnetic field lines inevitably passes through the second coil (the secondary).

This linkage of magnetic flux forms the basis of **mutual inductance**. Mathematically, the magnetic flux $\Phi_2$ linked with the secondary coil is directly proportional to the current $I_1$ flowing through the primary coil. The constant of proportionality is what we call the mutual inductance, denoted by $M$.

$$ \Phi_2 = M I_1 $$

### Decoding the Formula

To truly understand what mutual inductance depends on, we need to look beyond the basic definition and examine a concrete physical setup. Let's consider the classic example of two long coaxial solenoids. The mutual inductance for this system is given by the formula:

$$ M = \mu_0 n_1 n_2 \pi r_1^2 L $$

Let's break down the terms in this elegant equation:
*   $\mu_0$ is the permeability of free space, representing the medium inside the coils.
*   $n_1$ and $n_2$ are the number of turns per unit length of the inner and outer coils, respectively.
*   $\pi r_1^2$ is the cross-sectional area of the inner coil.
*   $L$ is the length of the overlapping region of the coils.

### The Geometric Truth

Notice what is conspicuously absent from this formula. Do you see any term for the current $I$? Do you see any term for the rate of change of current $\frac{di}{dt}$? Do you see any mention of the electrical resistance or the material of the wire?

**Absolutely not.**

Just like the capacitance of a capacitor depends only on the plate area and separation, and the resistance of a wire depends on its length and cross-section, **inductance is a purely geometric property**. It is determined entirely by the physical dimensions of the coils (area and length), the number of turns, and crucially, their **relative position and orientation**.

If you were to take the secondary coil and rotate it by 90 degrees so that it is perpendicular to the primary coil, the magnetic field lines would graze past it without passing through its area. The flux linkage would drop to zero, and consequently, the mutual inductance $M$ would become zero.

Therefore, the mutual inductance of a pair of coils is fundamentally dictated by their relative position and orientation in space.

Animated Solution for Physics - Electromagnetic Induction: Two coils are placed close to each other. The mutual inductance of the pair of coils depends upon

Select Answer:

Visualized Solution

Concept of Mutual Inductance

Formula for Coaxial Solenoids

Analyzing Dependencies

Conclusion

The Sigma Insight: Self and Mutual Inductance

The Essence of Mutual Inductance

Decoding the Formula

The Geometric Truth

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