Inductors and the magnetic field

The capacitor stores energy in an electric field and cares about changing voltage. Its twin, the inductor, stores energy in a magnetic field and cares about changing current, and the symmetry between the two runs so deep that most of what was learned about one can be read across to the other by swapping voltage for current. The inductor is the last of the three basic components, and with it the subject has everything it needs to reach alternating current.

Current makes a field

A current in a wire creates a magnetic field looping around it, a fact discovered when a compass needle twitched beside a current-carrying wire. A single straight wire makes a weak field, but wind the wire into a coil and the loops of field from each turn add together down the middle, concentrating into a strong, roughly uniform magnetic field, the same principle as an electromagnet. An inductor is simply such a coil, made to have a useful, known amount of this effect, often wound on an iron core that greatly strengthens the field. Where a capacitor is defined by the charge it stores per volt, an inductor is defined by the magnetic field it sets up per unit of current, and the measure of that is its inductance, $L$ , in henries.

The field resists change

The key to the inductor is a law of nature that the field, once established, resists being changed. Faraday found that a changing magnetic field induces a voltage in the coil it threads, and Lenz found the crucial detail of its direction: the induced voltage always opposes the change that caused it. So when the current in an inductor is made to change, the field changes, and the changing field induces a voltage that pushes back against the change. Speed the current up and the inductor develops a voltage fighting the increase; slow it down and it develops a voltage trying to keep it going. The relation is

V = L \frac{d I}{d t}

the voltage across an inductor is its inductance times the rate of change of the current through it. This is the exact mirror of the capacitor's $I = C d V / d t$ , with the roles of voltage and current swapped. A capacitor's voltage cannot change instantly because charge takes time to move; an inductor's current cannot change instantly because the field resists it. Try to break the current in an inductor suddenly, by opening a switch, and the inductor answers with a large voltage spike as it fights to keep the current flowing, which is why switching off an electromagnet or a motor throws a spark, and why circuits that drive coils need protection against the kickback.

Example. The current through a 2 millihenry inductor is increasing at 500 amperes per second. What voltage appears across it?

$V = L d I / d t = 2 \times 10^{- 3} \times 500 = 1$ V, opposing the increase. A modest inductor and a modest rate give a volt of opposition; a fast switch-off, with a huge rate of change, gives the far larger spike that makes the spark.

Now you. A 10 millihenry inductor has its current changing at 200 amperes per second. What voltage develops across it?

Answer

$V = 10 \times 10^{- 3} \times 200 = 2$ V.

Energy in the magnetic field

Establishing the current in an inductor takes work, because the induced voltage opposes the build-up and must be pushed against, and that work is stored in the magnetic field, ready to be returned. The stored energy is

E = \frac{1}{2} L I^{2}

the exact mirror of the capacitor's $\frac{1}{2} C V^{2}$ , with current in place of voltage. The energy lives in the magnetic field in and around the coil, and it is real and retrievable: it is what keeps the current flowing for an instant when the source is removed, and what a switching power supply shuttles back and forth many thousands of times a second to convert one voltage to another efficiently. The inductor, like the capacitor, is a temporary store, giving back what it was given.

Passing the steady, blocking the changing

The inductor's response to steady and changing signals is the precise opposite of the capacitor's, and for a reason that follows straight from its law. A steady current has zero rate of change, so $V = L d I / d t$ is zero: a fully established steady current meets no opposition from the inductor at all, and the inductor behaves like a plain wire to direct current. But a fast-changing current has a large rate of change and meets a large opposing voltage, so the inductor impedes rapid changes strongly. Where the capacitor blocks the steady and passes the changing, the inductor passes the steady and blocks the changing. This makes the two natural partners and natural opposites: a capacitor and an inductor together can pass one band of frequencies and reject another, the basis of tuning a radio to one station and rejecting the rest, which the resonance at the end of the course explains. It also completes the symmetry that has been building. The resistor dissipates energy and cares only about the present. The capacitor stores energy in an electric field and resists changes in voltage. The inductor stores energy in a magnetic field and resists changes in current. Three components, one that spends and two that store, and between them they can build any linear circuit there is. What none of them has needed yet is a source that itself changes with time, and introducing one, the alternating supply, is the step that turns this static theory into the electricity that actually comes out of a wall.