1. Concept Overview
The coefficient of self-induction tells me how effectively a coil can oppose changes in its own current. When current flows in a coil, it produces a magnetic field linked with the coil. If this current changes, the magnetic flux also changes and an emf is induced in the same coil. The strength of this induced emf depends on a property called the coefficient of self-induction.
I like to think of it as the ‘electrical stiffness’ of the coil — a higher value means the coil strongly resists any sudden change in current.
1.1. Short note
Coefficient of self-induction = measure of how strongly a coil opposes changes in its own current.
2. Definition of Coefficient of Self-Induction
The coefficient of self-induction (usually written as L) is defined as the induced emf in the coil per unit rate of change of current in that coil.
Mathematically:
\( L = \dfrac{\varepsilon}{\dfrac{dI}{dt}} \)
This means: if the rate of change of current is 1 ampere per second and the induced emf is 1 volt, the value of L is 1 henry.
2.1. Another useful form
The magnetic flux linked with a coil is proportional to the current:
\( \Phi_B = L I \)
This expression helps me understand L as the flux linkage per unit current.
3. Unit and Dimensions
The SI unit of the coefficient of self-induction is the henry (H). It is a large unit, so in practical circuits we often use millihenry (mH) or microhenry (µH).
3.1. When is L = 1 henry?
A coil has self-inductance of 1 H if a change of current of 1 A/s induces an emf of 1 V in the coil.
3.2. Dimensions
The dimensional formula of L is:
\( [M L^2 T^{-2} I^{-2}] \)
4. Understanding L Through Induced EMF
The induced emf due to self-induction is given by:
\( \varepsilon = -L \dfrac{dI}{dt} \)
The larger the value of L, the larger the induced emf for the same rate of change of current. This makes the coil resist sudden changes more strongly.
4.1. Why the minus sign appears
The negative sign comes from Lenz’s law. It shows that the induced emf always opposes the change in current, not the current itself.
5. Factors Affecting the Coefficient of Self-Induction
The value of L depends on the physical characteristics of the coil and the material inside it. These factors help in designing inductors for different applications.
5.1. 1. Number of turns
More turns mean more flux linkage for the same current. So, L increases with number of turns.
5.2. 2. Area of the coil
L increases with the cross-sectional area. Larger area means more magnetic flux linked with the coil.
5.3. 3. Length of the coil
A longer coil has smaller inductance because the magnetic field becomes weaker for a given current.
5.4. 4. Nature of the core material
A soft iron core increases L because it increases the magnetic flux for the same current. Air-core coils have lower inductance.
5.5. 5. Shape and construction
Tightly wound, uniformly shaped coils have higher inductance compared to loosely wound or irregular coils.
6. Energy Stored in a Coil
The self-inductance of a coil also determines how much magnetic energy it can store when current flows through it. This energy is stored in the magnetic field of the coil.
The energy stored is given by:
\( U = \dfrac{1}{2} L I^2 \)
6.1. Meaning of the formula
Greater inductance or greater current means more magnetic energy stored. This is why inductors are used in energy storage and filtering applications.
7. Everyday Examples
Even though L sounds like a theoretical property, it appears in many practical circuits.
7.1. Chokes in lighting
Chokes use high inductance to control current and prevent sudden jumps in electrical circuits.
7.2. Relay coils
The coils inside relays have significant inductance, which is why a spark appears when the circuit is suddenly broken.
7.3. Smoothening circuits
Inductors with high L values are used to smooth out large fluctuations in current.