Self Inductance and Mutual Inductance

Definition

Self inductance is the property of a coil or circuit by which a change in current in it induces an emf in the same coil, opposing the change in current.

Mutual inductance is the property of two coils or circuits by which a change in current in one coil induces an emf in the other coil due to magnetic flux linkage.

Mathematically:

Self inductance

$$e = -L \frac{di}{dt}$$ where $e$ is induced emf, $L$ is self inductance, and $\frac{di}{dt}$ is rate of change of current.

Mutual inductance

$$e_2 = -M \frac{di_1}{dt}$$ where $e_2$ is emf induced in coil 2, $M$ is mutual inductance, and $\frac{di_1}{dt}$ is rate of change of current in coil 1.

The negative sign follows Lenz’s law, showing that the induced emf always opposes the cause producing it.

Main Content

1. Self Inductance

Meaning and physical idea

• Self inductance is the tendency of a coil to oppose any change in the current flowing through it. When current increases, the magnetic flux linked with the coil increases, and an induced emf is produced in the same coil opposing that increase. Similarly, when current decreases, the induced emf tries to keep the current flowing by opposing the decrease.

Expression and factors affecting it

• For a coil, self inductance is defined as the flux linkage per unit current: $$L = \frac{N\Phi}{I}$$ where $N$ is number of turns, $\Phi$ is flux through each turn, and $I$ is current. It depends on the number of turns, shape and size of the coil, permeability of the core material, and geometry of the conductor. A coil with more turns and an iron core has greater self inductance.

A common example is a choke coil, which is designed to oppose changes in current and is used in AC circuits to limit current without wasting much power.

2. Mutual Inductance

Meaning and magnetic coupling

• Mutual inductance occurs when two coils are placed close to each other such that the magnetic field produced by one coil links with the other coil. A changing current in the first coil produces a changing magnetic flux, which induces emf in the second coil. This is the principle behind transformers.

Expression and influencing factors

• Mutual inductance is given by: $$M = \frac{N_2\Phi_{21}}{I_1}$$ where $N_2\Phi_{21}$ is the flux linkage of coil 2 due to current $I_1$ in coil 1. It depends on the number of turns in both coils, distance between coils, orientation, core material, and the extent of magnetic flux sharing. The closer and better aligned the coils are, the larger the mutual inductance.

A practical example is a transformer, where alternating current in the primary coil induces emf in the secondary coil.

3. Comparison and Energy Perspective

Difference in induction source

• In self inductance, the induced emf appears in the same coil due to its own changing current. In mutual inductance, the induced emf appears in a different coil due to a nearby coil’s changing current.

Energy and practical significance

• Both phenomena store energy in magnetic fields. For a coil with self inductance $L$, stored magnetic energy is: $$U = \frac{1}{2}LI^2$$ In coupled coils, energy transfer between circuits occurs through mutual inductance. This is highly useful in signal transfer, voltage transformation, and isolation in electrical systems.

These concepts are not only theoretical; they are the foundation of many electromagnetic devices and electrical engineering applications.

Working / Process

1. Current changes in a coil

• When current in a coil increases or decreases, the magnetic field around it changes accordingly.
• This changing magnetic field creates changing magnetic flux.

2. Induced emf is produced

• Due to electromagnetic induction, the changing flux induces an emf.
• In self inductance, the emf is induced in the same coil; in mutual inductance, it is induced in a nearby coil.

3. Induced emf opposes the change

• According to Lenz’s law, the induced emf always opposes the change causing it.
• This opposition produces delayed current growth or decay in circuits, and in transformers it allows power transfer from primary to secondary windings.

Advantages / Applications

Transformers and voltage conversion

• Mutual inductance is used in transformers to step up or step down AC voltages efficiently.

Current control and smoothing

• Self inductance is used in inductors and choke coils to limit sudden changes in current, reduce noise, and smooth current in circuits.

Energy transfer and coupling

• Both concepts are used in wireless power transfer, radio-frequency circuits, relays, induction heating, ignition coils, and signal coupling in electronic systems.

Summary

• Self inductance is the property by which a coil opposes change in its own current.
• Mutual inductance is the property by which one coil induces emf in another nearby coil.
• Both are based on changing magnetic flux and Lenz’s law.
• They are important in inductors, transformers, and many electromagnetic devices.