Self-inductance

What is Self-Inductance?

Self-inductance is the phenomenon where a changing current in a coil induces an emf in the same coil. This happens because the current produces a magnetic flux through the coil; when the current changes, the flux changes, and by Faraday’s law, an emf is induced. This induced emf always opposes the change in current (Lenz’s law) and is called the back emf.

Flux and Self-Inductance

For a coil with $N$ turns, the total flux linkage $N \Phi_B$ is proportional to the current $I$ flowing through it:

Here:

• $N \Phi_B$ is the total magnetic flux linkage (in weber-turns).
• $I$ is the current (in amperes).
• $L$ is the self-inductance (or coefficient of self-induction) of the coil, measured in henry (H).

Induced emf from Self-Inductance

Using Faraday’s law, the induced emf $\varepsilon$ is:

The negative sign indicates that the self-induced emf opposes the change in current (increase or decrease). This is the back emf.

Self-Inductance of a Long Solenoid

Consider a long solenoid of:

• Cross-sectional area $A$
• Length $l$
• Number of turns per unit length $n$

The magnetic field inside (neglecting edge effects) is:

The total flux linkage for $nl$ turns is:

Thus, the self-inductance $L$ is:

If the solenoid is filled with a material of relative permeability $\mu_r$, then:

Key point: $L$ depends only on the geometry (size, shape, number of turns) and the magnetic properties of the medium inside the coil.

Energy Stored in an Inductor

Work must be done against the back emf to establish a current. The rate of work is:

Integrating from $0$ to $I$ gives the total energy stored as magnetic potential energy:

This is analogous to kinetic energy $\frac{1}{2} m v^2$ — $L$ acts like electrical inertia, opposing changes in current.

Magnetic Energy Density

For a solenoid, using $L = \mu_0 n^2 A l$ and $B = \mu_0 n I$, the energy stored is:

The volume containing the field is $V = A l$, so the magnetic energy density is:

$u_B = \frac{U_B}{V} = \frac{B^2}{2 \mu_0}$ …