Q.The dimensional formula of coefficient of mutual inductance is
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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.
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 I1 be the current in coil 1. This current produces a magnetic flux Φ21 through coil 2 (the flux from coil 1 that passes through coil 2).
The mutual inductance M (also written M21) is defined as the constant of proportionality between the current I1 and the flux it produces in coil 2:
Φ21=MI1
Similarly, if current I2 flows in coil 2, it produces a flux Φ12 through coil 1:
Φ12=MI2
The mutual inductance M is the same for both directions. M21=M12=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:
E2=−dtdΦ21=−MdtdI1
And the induced emf in coil 1 due to a changing current in coil 2 is:
E1=−MdtdI2
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:
1H=1AV⋅s=1AWb
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 N1 and N2 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 I1 flows in Coil 1, it creates a magnetic field B1.
- Some of the magnetic field lines from Coil 1 pass through Coil 2.
- The total magnetic flux through Coil 2 due to I1 is called the mutual flux:
Φ21=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 I1:
Φ21∝I1
Why? Because B1 itself is proportional to I1 (Biot–Savart law), and the area of Coil 2 is fixed. So:
Φ21=M21I1
where M21 is the mutual inductance (a constant depending on coil shapes, sizes, turns, and relative positions).
2. Why the EMF Formula Arises
Now, if I1 changes with time, then Φ21 changes with time. By Faraday's Law, a changing flux induces an EMF in Coil 2:
E2=−dtdΦ21
Substitute Φ21=M21I1:
E2=−M21dtdI1
That's the key formula. The negative sign (Lenz's law) tells us the induced EMF opposes the change in flux.
3. Symmetry: M12=M21
If we reverse the situation — current I2 in Coil 2 induces flux Φ12 in Coil 1 — we get:
Φ12=M12I2
and
E1=−M12dtdI2
A deep result from energy conservation (or from the reciprocity theorem in electromagnetism) shows:
M12=M21=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) | Φ21=MI1 | Proportionality from Biot–Savart |
| Induced EMF in Coil 2 | E2=−MdtdI1 | Faraday's Law |
| Mutual flux (Coil 1 due to Coil 2) | Φ12=MI2 | Symmetry |
| Induced EMF in Coil 1 | E1=−MdtdI2 | Faraday's Law |
5. Physical Intuition (Exam-Ready) …
Mutual inductance M satisfies EMF = M(dI/dt), so [M] = [EMF]·[time]/[current] = [ML²T⁻²I⁻²]. Option (b).
Step 1 — defining relation: The induced emf in the secondary is ε = M(dI/dt), so M = ε/(dI/dt).
Step 2 — dimensions of emf (a potential difference): [ε] = [ML²T⁻³I⁻¹].
Step 3 — dI/dt has dimensions [I T⁻¹].
Step 4 — divide: [M] = [ML²T⁻³I⁻¹]/[I T⁻¹] = [ML²T⁻²I⁻²].
…
- CBSE 2026Set SEM31 markMCQQ.The dimensional formula of coefficient of mutual inductance is(a) [ ML²T⁻²I² ](b) [ ML²T⁻²I⁻² ](c) [ ML⁻²T²I² ](d) [ ML⁻²T⁻²I⁻² ]
›Reveal solutionSolution
Mutual inductance M satisfies EMF = M(dI/dt), so [M] = [EMF]·[time]/[current] = [ML²T⁻²I⁻²]. Option (b).
Step 1 — defining relation: The induced emf in the secondary is ε = M(dI/dt), so M = ε/(dI/dt).
Step 2 — dimensions of emf (a potential difference): [ε] = [ML²T⁻³I⁻¹].
Step 3 — dI/dt has dimensions [I T⁻¹].
Step 4 — divide: [M] = [ML²T⁻³I⁻¹]/[I T⁻¹] = [ML²T⁻²I⁻²].
…
- CBSE 2025Set A1 markQ.Write answer in one sentence: Write the SI unit of mutual inductance.
›Reveal solutionSolution
The SI unit of mutual inductance is the henry (H).
Mutual inductance M between two coils is defined through ε2=−MdtdI1, i.e., the emf induced in the secondary coil per unit rate of change of current in the primary coil. Its SI unit, the henry (H), is defined such that 1 H is the mutual inductance between two coils when a cu …
- CBSE 2024Set 55/5/11 markMCQQ.Two coils are placed near each other. When the current in one coil is changed at the rate of 5A/s, an emf of 2mV is induced in the other. The mutual inductance of the two coils is ______. (A) 0.4mH (B) 2.5mH (C) 10mH (D) 2.5H
›Reveal solutionSolution
The mutual inductance M is defined by the induced emf E=−Mdtdi.
Using the given values: E=2×10−3V, dtdi=5A/s, we get M=0.4×10−3H=0.4mH.
The correct option is (A).
The idea is simple: mutual inductance tells you how effectively a changing current in one coil “induces” an emf in a neighbouring coil. The definition is direct — the induced emf in the second coil is proportional to the rate of change of current in the first coil, and the constant of proportionality is the mutual inductance M.
The formula is:
E=−Mdtdi
The negative sign is Lenz’s law (direction of induced emf), but for magnitude we drop the sign.
Let’s work it out.
-
Write down what’s given
- Rate of change of current in the first coil: dtdi=5A/s
- Induced emf in the second coil: E=2mV=2×10−3V
- We need M.
-
Use the defining relation
From E=Mdtdi (taking magnitude), we get:
M=di/dtE
- Plug in the numbers
M=52×10−3=0.4×10−3H
That’s 0.4 millihenry. …
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- CBSE 2024Set 55/1/11 markMCQQ.For question 14, two statements are given – one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) below. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false and Reason (R) is also false. Assertion (A) : The mutual inductance between two coils is maximum when the coils are wound on each other. Reason (R) : The flux linkage between two coils is maximum when they are wound on each other.
›Reveal solutionSolution
The mutual inductance between two coils depends on the flux linkage between them. Winding the coils on each other places them as close as possible, maximising the flux linkage and therefore the mutual inductance. Both Assertion and Reason are true, and the Reason correctly explains the Assertion.
Mutual inductance M between two coils is defined by the relation M=I1N2Φ21, where Φ21 is the magnetic flux through coil 2 due to current I1 in coil 1, and N2 is the number of turns in coil 2. The key physical idea is simple: mutual inductance measures how effectively a changing current in one coil induces an emf in another. The closer the coils are, and the more their magnetic fields overlap, the larger the mutual inductance.
When two coils are wound directly on each other (like one layer of wire over another on the same core), almost every magnetic field line produced by one coil passes through the other coil. This gives the maximum possible flux linkage — the fraction of flux from one coil that threads the other is nearly 100%. If the coils were separated or placed at an angle, some flux would leak out, reducing the linkage and hence the mutual inductance.
Now let’s examine the statements step by step.
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Assertion (A) says mutual inductance is maximum when coils are wound on each other. This is true. The mutual inductance depends on geometry, distance, and orientation. Winding one coil directly over the other gives the smallest possible separation and the best alignment, so the coupling coefficient k (where M=kL1L2) approaches 1. That is the maximum possible value for a given pair of coils.
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Reason (R) says flux linkage between two coils is maximum when they are wound on each other. This is also true. Flux linkage is the product of the number of turns and the magnetic flux passing through the coil. When coils are wound on each other, the magnetic field lines from one coil almost entirely pass through the other coil’s turns, giving the highest possible flux linkage. …
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