KAHIBARO
Table of Contents
The Idea of Self-Inductance
When electric current flows in a wire, it produces a magnetic field around that wire. If the current changes, the magnetic field also changes. A changing magnetic field can induce an emf. In self inductance, the important point is that the changing current in a circuit induces an emf in that same circuit.
So, a circuit can oppose changes in its own current. This effect is called self inductance.
If the current increases, the induced emf acts to oppose that increase. If the current decreases, the induced emf acts to oppose that decrease. This is a direct consequence of Lenz's law, but here we focus only on the self effect inside one circuit.
Flux Linkage and Definition of Inductance
Consider a coil with $N$ turns carrying current $I$. The current creates a magnetic field, and that field passes through the turns of the coil. The magnetic flux through one turn is $\Phi_B$. The total flux linkage is
$$
N\Phi_B
$$
For many practical coils, the flux linkage is proportional to the current:
$$
N\Phi_B \propto I
$$
This leads to the definition of self inductance $L$:
$$
L = \frac{N\Phi_B}{I}
$$
This definition applies when the medium and geometry remain fixed, so the flux is proportional to current.
Important definition:
$$
L = \frac{N\Phi_B}{I}
$$
Self inductance $L$ measures how much magnetic flux linkage is produced per unit current in the same circuit.
The SI unit of inductance is the henry, written as H.
From the definition,
$$
1 \ \text{H} = 1 \ \frac{\text{Wb}}{\text{A}}
$$
where Wb is the weber, the unit of magnetic flux.
Induced EMF in a Self-Inductor
Faraday's law says that a changing magnetic flux induces an emf. For a coil, the induced emf due to self inductance is
$$
\mathcal{E} = -L\frac{dI}{dt}
$$
This formula shows that the induced emf depends on how fast the current changes.
If the current is constant, then
$$
\frac{dI}{dt} = 0
$$
so there is no self induced emf.
If the current changes rapidly, the induced emf is larger.
Key formula for self inductance:
$$
\mathcal{E} = -L\frac{dI}{dt}
$$
The negative sign means the induced emf opposes the change in current.
Physical Meaning
Self inductance is similar to inertia in mechanics. In mechanics, mass resists changes in velocity. In circuits, inductance resists changes in current.
A circuit with large inductance does not allow current to change easily. A circuit with small inductance allows current to change more easily.
This is why coils and inductors are useful in circuits where smooth current changes are desired.
What Affects Self-Inductance
The value of self inductance depends on the shape and size of the conductor and on the magnetic properties of the surrounding material.
For a coil, inductance becomes larger when there are more turns, when the magnetic flux through the turns is larger, and when a magnetic core helps strengthen the field.
In general, self inductance depends on factors such as coil geometry, number of turns, cross sectional area, length, and magnetic permeability of the core material.
Self-Inductance of a Long Solenoid
A common example is a long solenoid. For an ideal long solenoid,
$$
L = \mu \frac{N^2 A}{\ell}
$$
where $N$ is the number of turns, $A$ is the cross sectional area, $\ell$ is the length of the solenoid, and $\mu$ is the permeability of the core material.
This formula shows several useful trends. Increasing the number of turns greatly increases inductance, because $L \propto N^2$. Increasing area also increases inductance. Increasing length decreases inductance.
For a long solenoid,
$$
L = \mu \frac{N^2 A}{\ell}
$$
Inductance increases with $N^2$ and $A$, and decreases with $\ell$.
Energy Stored in the Magnetic Field
When current is built up in an inductor, work must be done against the opposing induced emf. That work is stored as magnetic energy.
The energy stored in an inductor is
$$
U = \frac{1}{2}LI^2
$$
This energy is not lost if the circuit is ideal. It is stored in the magnetic field and can later be returned to the circuit.
Energy stored in an inductor:
$$
U = \frac{1}{2}LI^2
$$
This is magnetic potential energy associated with the current in the inductor.
Everyday and Circuit Significance
Self inductance appears whenever current changes in coils, wires, motors, transformers, and many electronic devices. For example, when a switch opens in a circuit containing a coil, the current tries to keep flowing. Because the current changes suddenly, a large induced emf can appear. This is why sparks can occur at switches connected to inductive loads.
Self inductance also helps smooth current in many electrical systems.
Summary Table
| Quantity | Symbol | Formula | SI Unit |
|---|---|---|---|
| Self inductance | $L$ | $L = \dfrac{N\Phi_B}{I}$ | H |
| Self induced emf | $\mathcal{E}$ | $\mathcal{E} = -L\dfrac{dI}{dt}$ | V |
| Energy stored in an inductor | $U$ | $U = \dfrac{1}{2}LI^2$ | J |
Final Picture
Self inductance is the property of a circuit by which a changing current in that same circuit produces an induced emf that opposes the change. It connects current, magnetic flux, induced emf, and stored magnetic energy. An inductor is therefore a circuit element that resists sudden changes in current and stores energy in its magnetic field.