Self-Inductance

Relates current change, linked flux, induced emf, and stored magnetic energy.

Conceptual context

Self-inductance appears when a coil changes its own current and therefore changes the magnetic flux it links. It is not an added external effect: the coil creates a field, that field links its turns, and any change in $I$ modifies $\lambda_B$.

This leaf explains why a coil resists abrupt changes, why it can generate overvoltage when a circuit is opened, and why it stores energy in its magnetic field. The dominant quantity is $L$, which summarizes geometry, number of turns, and magnetic properties of the medium.

[!NOTE] Self-induction does not oppose the existence of current; it opposes current change.

🟢 Essential level

A coil carrying current creates a magnetic field. If current increases, its own field increases; if current decreases, its own field decreases. Self-induction is the coil response to that change: an $\varepsilon_L$ appears and tends to slow the variation of $I$.

The intuitive idea is similar to electromagnetic inertia. An ideal coil can carry steady current without producing self-induced emf, but it reacts strongly when current changes rapidly. That is why switches, relays, and motors with coils need protection against voltage spikes.

In an everyday reading, the coil does not treat the absolute current value as the problem. It responds to its own field changing. If the change is slow, the response may be small; if the change is abrupt, the response may be large even when the initial current is not huge.

[!WARNING] If you look only at instantaneous $I$, you may miss the phenomenon. The key is $\Delta I$ during $\Delta t$.

🔵 Formal level

Basic formulas

In a linear coil, linked flux and current keep a proportional relation. The operational definition of $L$ is:

Definition of self-inductance

This relation separates the magnetic state $\lambda_B$ from the electrical cause $I$. If the slope is constant, the coil has a well-defined $L$ over that segment. The same idea can be written as flux linkage produced by a current:

Flux linkage in a linear coil

Real student questions

Does inductance always slow current? No. In the ideal model it opposes changes in $I$. A steady current produces no self-induced emf.

Why does a spark appear when a coil is opened? Because stored $U_B$ must leave. If the current path is cut, voltage may rise until another path opens.

Why does the change time matter so much? Because the same $\Delta I$ produces larger $\varepsilon_L$ when it occurs over smaller $\Delta t$.

Is the coil energy stored in the wire? Not mainly. It is stored in the magnetic field associated with current.

Final synthesis

Self-induction is the response of a coil to changes in its own current. $L$ measures magnetic linkage per current, $\varepsilon_L$ expresses opposition to change, and $U_B$ reveals energy stored in the field.