There are two coils $A$ and $B$ separated by some distance. If a current of $2\ \text{A}$ flows through $A$, a magnetic flux of $10^{-2}\ \text{Wb}$ passes through $B$ (no current through $B$). If no current passes through $A$ and a current of $1\ \text{A}$ passes through $B$, what is the flux through $A$?

> [!TLDR] > Mutual inductance is symmetric: the flux through coil A when 1 A flows in coil B equals the flux through coil B when 2 A flows in coil A, scaled by the current ratio. The answer is $5 \times 10^{-3}\ \text{Wb}$. The key idea here is **mutual inductance** — a property of two coils that links the current in one to the flux through the other. When current changes in coil A, it induces a flux in coil B, and vice versa. The constant that relates them, $M$, is the same in both directions. That symmetry is the entire backbone of this problem. Let’s work through it step by step. 1. **Define mutual inductance.** For two coils $A$ and $B$, the mutual inductance $M$ is defined by: - When current $I_A$ flows in $A$, the flux through $B$ is $\Phi_B = M I_A$. - When current $I_B$ flows in $B$, the flux through $A$ is $\Phi_A = M I_B$. The same $M$ appears in both equations — this is a fundamental result from electromagnetic theory (and follows from energy conservation / reciprocity). 2. **Use the first situation to find $M$.** We are told: $I_A = 2\ \text{A}$, and the flux through $B$ is $\Phi_B = 10^{-2}\ \text{Wb}$. From $\Phi_B = M I_A$: $$M = \frac{\Phi_B}{I_A} = \frac{10^{-2}}{2} = 5 \times 10^{-3}\ \text{H}$$ (henry, the unit of inductance). 3. **Now apply the second situation.** Here, $I_B = 1\ \text{A}$ and we want $\Phi_A$. Using the same $M$: …

There are two coils A and B separated by some distance. If a current of 2A flows through A, a magnetic flux of

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Concept understanding — Mutual Inductance

Mutual Inductance: From Intuition to Definition

Imagine you have two separate coils of wire placed near each other. You connect one coil to a battery — current starts flowing through it. Now, something strange happens in the other coil, which isn't connected to anything: a voltage appears across its ends. That voltage can even light a small bulb for an instant.

This is mutual inductance in action. One circuit "feels" the changing current in another circuit, even though they are not physically connected.

The Core Intuition

The key idea is changing magnetic fields. When current flows through a coil, it creates a magnetic field around it. If that current changes (increases or decreases), the magnetic field also changes. This changing field reaches the second coil. And a changing magnetic field, by Faraday's law, induces an electromotive force (emf) in any nearby conductor.

So mutual inductance is simply: how effectively a change in current in one coil induces a voltage in another coil.

Note

Mutual inductance only works when the current is changing. A steady DC current produces a steady magnetic field, which induces nothing in the second coil. That's why the bulb lights only for an instant when you first connect the battery — the current is rising from zero.

The Precise Definition

Let's formalise this. Consider two coils: coil 1 and coil 2. Let $I_1$ be the current in coil 1. This current produces a magnetic flux $\Phi_{21}$ through coil 2 (the flux from coil 1 that passes through coil 2).

The mutual inductance $M$ (also written $M_{21}$ ) is defined as the constant of proportionality between the current $I_1$ and the flux it produces in coil 2:

\Phi_{21} = M \, I_1

Similarly, if current $I_2$ flows in coil 2, it produces a flux $\Phi_{12}$ through coil 1:

\Phi_{12} = M \, I_2

Important

The mutual inductance $M$ is the same for both directions. $M_{21} = M_{12} = M$ . This is a fundamental symmetry property.

Now, by Faraday's law, the induced emf in coil 2 due to a changing current in coil 1 is:

\mathcal{E}_2 = - \frac{d\Phi_{21}}{dt} = - M \frac{dI_1}{dt}

And the induced emf in coil 1 due to a changing current in coil 2 is:

\mathcal{E}_1 = - M \frac{dI_2}{dt}

The negative sign is Lenz's law — the induced emf opposes the change that produced it.

Units

The SI unit of mutual inductance is the henry (H), named after Joseph Henry. From the definition:

1 \, \text{H} = 1 \, \frac{\text{V} \cdot \text{s}}{\text{A}} = 1 \, \frac{\text{Wb}}{\text{A}}

One henry means that a current change of 1 ampere per second induces an emf of 1 volt in the other coil.

What Determines Mutual Inductance?

$M$ depends on:

Geometry: size, shape, number of turns of both coils
Relative position: how close they are and how they are oriented
Core material: if a magnetic material (like iron) is present, $M$ increases dramatically

For two coaxial solenoids of length $l$ , with $N_1$ and $N_2$ turns, and cross-sectional area $A$ , the mutual inductance is: …

Why this formula?

Mutual Inductance: Why the Formula Holds

Mutual inductance is a beautiful example of Faraday's Law in action — it describes how a changing current in one coil can induce an EMF in a nearby coil, without any direct electrical connection.

1. The Core Idea: Flux Linkage

Imagine two coils, Coil 1 and Coil 2, placed close together.

When a current $I_1$ flows in Coil 1, it creates a magnetic field $\vec{B}_1$ .
Some of the magnetic field lines from Coil 1 pass through Coil 2.
The total magnetic flux through Coil 2 due to $I_1$ is called the mutual flux:

\Phi_{21} = \text{flux through Coil 2 due to current in Coil 1}

Key insight: For a fixed geometry (coils not moving), the mutual flux is directly proportional to the current $I_1$ :

\Phi_{21} \propto I_1

Why? Because $\vec{B}_1$ itself is proportional to $I_1$ (Biot–Savart law), and the area of Coil 2 is fixed. So:

\Phi_{21} = M_{21} \, I_1

where $M_{21}$ is the mutual inductance (a constant depending on coil shapes, sizes, turns, and relative positions).

2. Why the EMF Formula Arises

Now, if $I_1$ changes with time, then $\Phi_{21}$ changes with time. By Faraday's Law, a changing flux induces an EMF in Coil 2:

\mathcal{E}_2 = -\frac{d\Phi_{21}}{dt}

Substitute $\Phi_{21} = M_{21} I_1$ :

\mathcal{E}_2 = -M_{21} \frac{dI_1}{dt}

That's the key formula. The negative sign (Lenz's law) tells us the induced EMF opposes the change in flux.

3. Symmetry: $M_{12} = M_{21}$

If we reverse the situation — current $I_2$ in Coil 2 induces flux $\Phi_{12}$ in Coil 1 — we get:

\Phi_{12} = M_{12} \, I_2

and

\mathcal{E}_1 = -M_{12} \frac{dI_2}{dt}

A deep result from energy conservation (or from the reciprocity theorem in electromagnetism) shows:

\boxed{M_{12} = M_{21} = M}

So we simply call it $M$ , the mutual inductance between the two coils.

4. The Complete Formula Set

Quantity	Expression	Why?
Mutual flux (Coil 2 due to Coil 1)	$\Phi_{21} = M I_1$	Proportionality from Biot–Savart
Induced EMF in Coil 2	$\mathcal{E}_2 = -M \frac{dI_1}{dt}$	Faraday's Law
Mutual flux (Coil 1 due to Coil 2)	$\Phi_{12} = M I_2$	Symmetry
Induced EMF in Coil 1	$\mathcal{E}_1 = -M \frac{dI_2}{dt}$	Faraday's Law

5. Physical Intuition (Exam-Ready) …

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Mutual inductance is symmetric: the flux through coil A when 1 A flows in coil B equals the flux through coil B when 2 A flows in coil A, scaled by the current ratio. The answer is $5 \times 10^{-3}\ \text{Wb}$ .

The key idea here is mutual inductance — a property of two coils that links the current in one to the flux through the other. When current changes in coil A, it induces a flux in coil B, and vice versa. The constant that relates them, $M$ , is the same in both directions. That symmetry is the entire backbone of this problem.

Let’s work through it step by step.

Define mutual inductance.

For two coils $A$ and $B$ , the mutual inductance $M$ is defined by:
- When current $I_A$ flows in $A$ , the flux through $B$ is $\Phi_B = M I_A$ .
- When current $I_B$ flows in $B$ , the flux through $A$ is $\Phi_A = M I_B$ .
The same $M$ appears in both equations — this is a fundamental result from electromagnetic theory (and follows from energy conservation / reciprocity).
Use the first situation to find $M$ .

We are told: $I_A = 2\ \text{A}$ , and the flux through $B$ is $\Phi_B = 10^{-2}\ \text{Wb}$ .

From $\Phi_B = M I_A$ :

M = \frac{\Phi_B}{I_A} = \frac{10^{-2}}{2} = 5 \times 10^{-3}\ \text{H}

(henry, the unit of inductance).

Now apply the second situation. Here, $I_B = 1\ \text{A}$ and we want $\Phi_A$ . Using the same $M$ : …

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Concept understanding — Mutual Inductance

Mutual Inductance: From Intuition to Definition

The Core Intuition

The Precise Definition

Units

What Determines Mutual Inductance?

Why this formula?

Mutual Inductance: Why the Formula Holds

1. The Core Idea: Flux Linkage

2. Why the EMF Formula Arises

3. Symmetry: $M_{12} = M_{21}$

4. The Complete Formula Set

5. Physical Intuition (Exam-Ready) …

Method: Using the Reciprocity of Mutual Inductance ( $M_{12}=M_{21}$ )

Steps

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