How Does Mutual Inductance Work Between Two Coils?
Mutual inductance is a fundamental property in electromagnetism that quantifies the ability of one coil or circuit to induce an electromotive force (emf) in another nearby coil due to a changing current. This parameter plays a critical role in the analysis of coupled circuits, transformers, and electromagnetic devices.
Definition and Physical Basis of Mutual Inductance
Mutual inductance is defined as the ratio of the magnetic flux linked with one coil to the change in current in a neighboring coil. When a time-varying current flows through a primary coil, it creates a changing magnetic field, which induces an emf in a secondary coil placed within its influence.
The fundamental relationship for induced emf due to mutual inductance, governed by Faraday’s law of electromagnetic induction, is expressed as $ \varepsilon_2 = -M \dfrac{dI_1}{dt} $, where $M$ is the mutual inductance between the two coils.
The SI unit of mutual inductance is the henry (H). Mutual inductance occurs only when the magnetic field generated by one coil links efficiently with another coil, which depends on geometric factors and relative positioning.
For an overview of related electromagnetic principles, refer to Electromagnetic Induction Notes.
Mathematical Expression and Formula for Mutual Inductance
Mutual inductance between two coils can be calculated using the general formula $M = \dfrac{N_2 \Phi_{12}}{I_1}$, where $N_2$ is the number of turns in the secondary coil, $ \Phi_{12} $ is the magnetic flux through the secondary due to the current $I_1$ in the primary coil.
When the coils are tightly coupled and coaxial, the mutual inductance also depends on physical parameters such as the number of turns, cross-sectional area $A$, relative permeability $ \mu_r $, and length $l$ of the core shared by the coils. The formula in such cases is:
$M = \dfrac{\mu_0 \mu_r N_1 N_2 A}{l}$
Here, $\mu_0$ is the permeability of free space, and $N_1$, $N_2$ are the number of turns in the primary and secondary coils respectively. This formula assumes complete flux linkage and is commonly applied to ideal transformers and coaxial solenoids.
Related concepts are discussed in Coaxial Solenoids and Mutual Inductance.
| Physical Quantity | SI Unit |
|---|---|
| Mutual Inductance $(M)$ | Henry (H) |
| Rate of Change of Current $(dI/dt)$ | A/s |
| Induced EMF $(\varepsilon)$ | Volt (V) |
Factors Affecting Mutual Inductance
Mutual inductance depends on several factors including the number of turns in both coils, the area of cross-section, length of the core, separation distance between coils, and the permeability of the core material. Larger coil overlap area and use of a soft iron core increase the value of $M$.
The efficiency of energy transfer by mutual inductance reduces if the coils are loosely coupled, far apart, or oriented at right angles. Precise coil alignment is essential in transformer design and wireless power applications.
Effects of orientation and coupling are elaborated in Understanding Mutual Inductance.
Coefficient of Coupling and its Significance
The coefficient of coupling $(k)$ quantifies the fraction of magnetic flux produced by one coil that links with the other coil. It is a dimensionless quantity between $0$ (no coupling) and $1$ (perfect coupling), and is defined as $k = \dfrac{M}{\sqrt{L_1 L_2}}$, where $L_1$ and $L_2$ are the self-inductances of the coils.
A high value of $k$ signifies tight coupling, often achieved with a common iron core, while low values indicate loose coupling due to poor alignment or larger distance between coils.
Mutual Inductance in Electric Circuits
In the analysis of circuits involving coupled inductors or transformers, the mutual inductance term appears in Kirchhoff’s laws and influences the total emf in each coil. The polarity of induced emf is determined by the direction of winding and the relative current directions, commonly indicated using the dot convention.
When two inductors are connected in series and their magnetic fields aid each other, the effective inductance is $L_{eq} = L_1 + L_2 + 2M$. For opposing magnetic fields, the formula becomes $L_{eq} = L_1 + L_2 - 2M$.
A detailed introduction to inductors in circuits can be accessed at Introduction to Inductors.
Dot Convention for Coupled Circuits
The dot convention is a symbolic representation in circuit diagrams used to indicate the relative polarity of induced voltages due to mutual inductance. If current enters the dotted terminal of one coil, the induced voltage at the dotted terminal of the other is positive with respect to its undotted end.
Correct use of the dot convention is essential in circuit analysis, especially when applying Kirchhoff’s voltage law for coupled inductors.
For more on electromagnetic concepts and circuit analysis, refer to Electromagnetic Induction Overview.
Comparison between Mutual and Self Inductance
Self inductance ($L$) refers to the property of a single coil inducing an emf in itself due to changes in its own current, whereas mutual inductance ($M$) describes the emf induced in a second coil due to changes in the current of the first coil.
| Aspect | Mutual Inductance |
|---|---|
| Number of coils involved | Two separate coils |
| EMF induced in | Second coil |
| Symbol | $M$ |
| Dependence | Orientation and spacing |
| SI Unit | Henry (H) |
Applications and Practical Examples
Mutual inductance is applied extensively in transformers, wireless energy transfer systems, inductive sensors, and radio frequency circuits. Efficient energy transfer and impedance matching in transformers are achieved by maximizing mutual inductance through core design and coil alignment.
A change in current of $4\,\mathrm{A/s}$ in one coil inducing $0.8\,\mathrm{V}$ in another shows $M = 0.8/4 = 0.2\,\mathrm{H}$. Higher number of turns or use of ferromagnetic cores further increase $M$.
Wireless charging devices for phones, electric vehicles, and certain sensors rely fundamentally on the principle of mutual inductance for contactless power transmission.
For in-depth study, see Magnetic Effects of Current.
Summary of Key Points on Mutual Inductance
- Mutual inductance measures magnetic linkage between two coils
- Expressed as $M = \dfrac{N_2 \Phi_{12}}{I_1}$ or $M = \dfrac{\mu_0 \mu_r N_1 N_2 A}{l}$
- SI unit is henry (H)
- Affected by coil geometry, core, distance, and orientation
- Used in transformers and wireless power transfer
- Dot convention determines induced emf polarity
FAQs on Understanding Mutual Inductance in Physics
1. What is mutual inductance?
Mutual inductance is the phenomenon where a change in current in one coil induces an electromotive force (EMF) in a nearby coil. It is a key concept in electromagnetism and is used in many electrical devices such as transformers and inductors.
Key points:
- It involves two coils placed close to each other.
- A change in the current in the primary coil induces EMF in the secondary coil.
- Measured in henry (H).
- Denoted by the symbol M.
2. What is the SI unit of mutual inductance?
The SI unit of mutual inductance is the henry (H).
- One henry is defined as the mutual inductance between two circuits when an EMF of one volt is induced in one circuit due to a change in current of one ampere per second in the other circuit.
3. State the factors affecting mutual inductance between two coils.
The value of mutual inductance between two coils depends on:
- Number of turns in both coils
- Area of cross-section of the coils
- Distance and orientation between the coils
- Permeability of the core material between the coils
Changing any of these factors alters the amount of EMF induced in the secondary coil.
4. Write the formula for mutual inductance between two coils.
The mutual inductance (M) between two coils is given by:
M = (μ₀ μr N1 N2 A) / l
where:
- μ₀ = permeability of free space
- μr = relative permeability of the core material
- N1, N2 = number of turns in the two coils
- A = area of cross-section
- l = length of the coil
5. Define coefficient of mutual induction.
The coefficient of mutual induction is a measure of how efficiently a change in current in one coil induces EMF in another nearby coil. It is expressed as the ratio of the induced EMF in the secondary coil to the rate of change of current in the primary coil.
6. Explain the principle of mutual induction with an example.
The principle of mutual induction states that when the electric current passing through one coil changes, it creates a changing magnetic field that induces an EMF in a second nearby coil.
Example:
- Transformers work on the principle of mutual induction. When AC flows through the primary coil, it induces a changing magnetic field, which in turn induces an EMF in the secondary coil, enabling voltage transformation.
7. What are some practical applications of mutual inductance?
Mutual inductance has several important applications in daily life and technology.
- Transformers for voltage step-up or step-down in power transmission
- Induction coils used in ignition systems
- Wireless charging of devices such as smartphones
- Electric bells and relays
8. How does mutual inductance differ from self-inductance?
Mutual inductance and self-inductance are related but distinct phenomena:
- Mutual inductance occurs between two different coils when a change in current in one induces EMF in the other.
- Self-inductance is the induction of EMF in the same coil in which the current is changing.
9. When will mutual inductance between two coils be maximum?
Mutual inductance is maximum when:
- The coils are wound on a common core made of a material with high magnetic permeability (like soft iron).
- The coils are placed as close as possible to each other with their axes parallel and maximum overlapping of magnetic fields.
- The number of turns and area of the coils are large.
10. What is the physical significance of mutual inductance in electrical circuits?
The mutual inductance between coils enables the transfer of energy from one circuit to another without direct electrical connection. It is essential for:
- Power transmission through transformers
- Wireless charging techniques
- Operation of relays and signal coupling in electronic devices
