Self-Inductance
When the current flowing through a coil changes, the magnetic flux threading that coil changes too. By Faraday's law, any change in flux induces an electromotive force (emf). Remarkably, this induced emf appears in the very coil whose current is changing — the coil is, in a sense, fighting its own change. This phenomenon is called self-inductance, and it is the electrical equivalent of inertia: just as a massive object resists changes in velocity, an inductor resists changes in current. Self-inductance arises from a simple chain of cause and effect. A current I creates a magnetic field B around and inside the coil. That field produces a magnetic flux Φ through the coil's own turns. If I changes with time, Φ changes, and Faraday's law demands an induced emf. Lenz's law tells us the direction: the induced emf always opposes the change that caused it. Increase the current and the coil pushes back; decrease it and the coil tries to keep it flowing. Self-inductance is not merely a nuisance — it is the operating principle of inductors used in filters, oscillators, transformers, and power supplies found in virtually every piece of electronic equipment. Understanding it quantitatively requires connecting the geometry of a coil to its inductance L, measured in henries (H), and then applying the fundamental emf equation to circuit problems.
- Current Creates Magnetic Flux
- Faraday's Law Gives the Induced emf
- Inductance of a Solenoid
- Energy Stored in an Inductor
- Behaviour in a Circuit
Current Creates Magnetic Flux
A coil of N turns carrying current I produces a magnetic field through its own cross-section. The total magnetic flux linkage is defined as NΦ, where Φ is the flux through one turn. For a linear magnetic material, NΦ is directly proportional to I, so we write NΦ = LI. The constant of proportionality L is the self-inductance of the coil.
Faraday's Law Gives the Induced emf
Faraday's law states that the induced emf is ε = −d(NΦ)/dt. Substituting NΦ = LI (with L constant for a given coil geometry and linear core), we get ε = −L(dI/dt). The negative sign embodies Lenz's law: the induced emf opposes the rate of change of current. If dI/dt > 0 (current increasing), ε is negative, meaning it drives current in the direction that would decrease the current.
Inductance of a Solenoid
For a tightly wound solenoid with N turns, length ℓ, cross-sectional area A, and core permeability μ (= μ₀μᵣ), the magnetic field inside is B = μ(N/ℓ)I. The flux through one turn is Φ = BA = μ(N/ℓ)IA, so flux linkage NΦ = μN²A·I/ℓ. Therefore L = μN²A/ℓ = μ₀μᵣN²A/ℓ. This shows that inductance grows with the square of the number of turns and with core permeability — winding more turns or inserting an iron core dramatically increases L.
Energy Stored in an Inductor
Because the inductor opposes changes in current, work must be done to establish a current through it. The energy stored in the magnetic field of an inductor carrying current I is U = ½LI². This energy is real — it can be recovered when the current is switched off. A collapsing magnetic field can drive large voltage spikes (since ε = −L·dI/dt with large dI/dt), which is why inductors must be handled carefully in switching circuits.
Behaviour in a Circuit
In a DC circuit, after a long time the current is steady (dI/dt = 0), so an ideal inductor acts as a short circuit (zero resistance wire). During the transient phase — when current is building up — the inductor limits how fast the current grows. In an RL circuit (resistor R and inductor L in series with battery V), the current rises as I(t) = (V/R)(1 − e^(−Rt/L)), approaching its final value V/R with time constant τ = L/R.
The Key Equation
- \varepsilon — Induced emf (V (volts)): The electromotive force induced in the coil by its own changing current; negative sign indicates opposition to the change (Lenz's law).
- L — Self-inductance (H (henries)): A geometric property of the coil that quantifies how much emf is induced per unit rate of change of current; 1 H = 1 V·s/A.
- \frac{dI}{dt} — Rate of change of current (A/s (amperes per second)): How rapidly the current through the coil is changing at a given instant.
Worked Example
Problem. An inductor has a self-inductance of L = 40 mH. The current through it decreases uniformly from 5.0 A to 1.0 A in 2.0 ms. (a) Calculate the magnitude of the self-induced emf. (b) If this inductor is a solenoid with 500 turns and cross-sectional area 4.0 cm², find its length.
Answer. The self-induced emf has magnitude 80 V; the solenoid length is approximately 3.1 mm.
Origins · 1 minJoseph Henry and how this was discovered
Joseph Henry discovered self-inductance independently of Faraday in the early 1830s, observing that a changing current in a coil induced a voltage in the same coil. He built powerful electromagnets and constructed early telegraph-like devices. The SI unit of inductance, the henry (H), is named in his honour. Henry discovered self-inductance before Faraday published his results, but Henry was slow to publish — a lesson physicists still cite about the importance of timely scientific communication. He later became the first Secretary of the Smithsonian Institution.
Related Concepts
- Faraday's Law of Induction — Self-inductance is a direct application of Faraday's law — the induced emf in the coil equals the negative rate of change of its own flux linkage.
- Mutual Inductance — Mutual inductance extends the self-inductance concept to two coupled coils; a changing current in one induces an emf in the other, forming the basis of transformers.
- RL Circuits — In RL circuits, self-inductance determines the time constant τ = L/R, governing how quickly current builds up or decays after a switch event.
- Lenz's Law — Lenz's law provides the physical reason for the negative sign in ε = −L dI/dt, explaining why the self-induced emf always opposes the change in current.