1. Concept Overview
The coefficient of mutual induction tells me how effectively one coil can induce emf in another nearby coil when its current changes. If coil 1 carries a changing current, the magnetic flux linked with coil 2 changes, and this produces an induced emf in coil 2. The strength of this induced emf depends on a property called the coefficient of mutual induction.
I like to think of it as the ‘magnetic coupling strength’ between the two coils — a higher value means coil 2 picks up more of the flux changes created by coil 1.
1.1. Short one-line note
Coefficient of mutual induction = how strongly two coils influence each other through changing magnetic flux.
2. Definition of Coefficient of Mutual Induction
The coefficient of mutual induction (written as M) is defined as the induced emf in one coil per unit rate of change of current in the nearby coil.
Mathematically:
\( M = \dfrac{\varepsilon_2}{\dfrac{dI_1}{dt}} \)
This means: if the rate of change of current in coil 1 is 1 A/s and it induces 1 V in coil 2, the mutual inductance between them is 1 henry.
2.1. Flux linkage perspective
The magnetic flux linked with coil 2 because of current in coil 1 is:
\( \Phi_{21} = M I_1 \)
This tells me that the mutual inductance represents how much flux in coil 2 is produced per unit current in coil 1.
3. Unit and Dimensions
The SI unit of mutual inductance is the henry (H), the same as self-inductance. In many practical circuits, mutual inductance values are in millihenry (mH) or microhenry (µH).
3.1. When M = 1 henry
Two coils have mutual inductance of 1 H if a change of current of 1 A/s in coil 1 induces an emf of 1 V in coil 2.
3.2. Dimensions
The dimensional formula of M is:
\( [M L^2 T^{-2} I^{-2}] \)
4. Induced EMF and Mutual Induction
The induced emf in the second coil due to changing current in the first coil is given by:
\( \varepsilon_2 = -M \dfrac{dI_1}{dt} \)
The negative sign comes from Lenz’s law and shows that the induced emf opposes the change in flux.
4.1. Key point about magnitude
A higher value of M means stronger induced emf for the same rate of current change in coil 1. So coils with strong coupling produce larger responses.
5. Factors Affecting Mutual Inductance
The value of M depends heavily on how the two coils are placed and the materials used. These factors determine how much of the magnetic flux created by coil 1 links with coil 2.
5.1. 1. Number of turns
More turns in either coil increase mutual flux linkage, raising the value of M.
5.2. 2. Distance between the coils
Coils placed close together have higher mutual inductance. Increasing the distance reduces the shared magnetic flux.
5.3. 3. Area of the coils
Larger cross-sectional area gives more space for magnetic field lines to pass through, increasing mutual induction.
5.4. 4. Orientation of the coils
Coils aligned with their axes parallel and close have maximum mutual inductance. Misaligned coils share less flux.
5.5. 5. Core material
A soft iron core between or around the coils increases magnetic flux linkage and significantly increases M. Air-core coils have lower M.
6. Understanding Mutual Induction Through Flux
I find it helpful to imagine the magnetic field lines from coil 1 spreading outward. Some of these lines cut through coil 2. When coil 1's current changes, the number of these shared lines changes, so coil 2 experiences a changing flux and gets an induced emf.
6.1. Ideal vs real conditions
In an ideal transformer, all flux produced by the primary links with the secondary (maximum M). In reality, some flux is always lost, so mutual inductance depends on how well the coils are coupled.
7. Lenz’s Law and Mutual Induction
The direction of induced emf in mutual induction follows Lenz’s law. Coil 2 reacts against the change caused by coil 1. This maintains energy conservation and keeps the induction process stable.
7.1. Opposition explained simply
If current in coil 1 increases, coil 2 produces a magnetic field that tries to oppose the increase. If current in coil 1 decreases, coil 2 tries to oppose the decrease by supporting the field.
8. Applications of Mutual Induction
Mutual induction is the backbone of many electrical systems that transfer energy between coils.
8.1. Transformers
Transformers rely entirely on mutual induction. Alternating current in the primary coil creates changing flux that induces emf in the secondary coil.
8.2. Induction coils
Used for generating very high voltages. A rapid change in current in the primary induces a large voltage in the secondary coil.
8.3. Wireless charging
A changing magnetic field in the transmitter coil induces emf in the receiver coil, allowing energy transfer without wires.