Capacitors, storing charge in a field

Every component so far has responded to voltage and current instantly, with no memory of what came before. The capacitor is the first component with a past. It stores charge, and because charge takes time to move in and out, a capacitor makes a circuit's voltages and currents depend on their own history, which is what turns a circuit from a static divider into something that can time, filter and smooth. Its construction is almost absurdly simple, two conductors held close together but not touching, yet from that a great deal follows.

Charge in proportion to voltage

A capacitor is two conducting plates separated by a thin insulating gap. Connect it to a voltage and charge flows onto one plate and off the other, until one plate holds a positive charge and the other an equal negative charge, with the applied voltage across the gap. The plates never touch, so no charge crosses the gap; it simply piles up on the two faces. The amount that piles up is proportional to the voltage applied:

where $Q$ is the charge on each plate, $V$ the voltage across them, and $C$ the capacitance, measured in farads. One farad is one coulomb stored per volt, which is an enormous capacitance; real capacitors are microfarads, millionths, or smaller, and even a microfarad is a substantial part. Capacitance is a property of the geometry: larger plates hold more charge at a given voltage, and a thinner gap holds more still, because the opposite charges on the two plates attract across a smaller distance and pack more densely. Filling the gap with an insulating material, a dielectric, raises the capacitance further, because the material's own charges shift to partly cancel the field and let more charge pile on for the same voltage.

Example. A 10 microfarad capacitor is charged to 5 V. How much charge does it hold?

$Q=CV=10\times {10}^{-6}\times 5=5\times {10}^{-5}$ C, or 50 microcoulombs. Small, but stored and retrievable, which is the point.

Now you. A 100 microfarad capacitor holds 2 millicoulombs of charge. What voltage is across it?

Energy stored in the field

Charging a capacitor takes work, because each additional bit of charge must be pushed onto a plate that already repels it, and that work is stored, ready to be given back. The stored energy is

The factor of a half appears because the voltage was not the full $V$ throughout the charging; it climbed from zero as the charge accumulated, so the average voltage the charge was pushed against was half the final value. The energy lives in the electric field in the gap, the same field of the first lesson, now put to work as a store. A charged capacitor is a reservoir of energy that can be released far faster than a battery can, which is why capacitors run a camera flash and smooth the violent current demands the drone's ESCs make on the battery. The battery holds far more energy; the capacitor gives up its smaller store in an instant, and the two roles are different.

Example. How much energy is stored in a 470 microfarad capacitor charged to 16 V?

$E=\frac{1}{2}C{V}^2=0.5\times 470\times {10}^{-6}\times {16}^2=0.5\times 470\times {10}^{-6}\times 256=0.060$ J. Modest, but delivered in a flash it is a large power.

Now you. A 1000 microfarad capacitor is charged to 10 V. How much energy does it store?

Blocking the steady, passing the changing

The rule that no charge crosses the gap has a striking consequence for how a capacitor treats different signals. Once a capacitor is charged to a steady voltage, the flow stops: the plates are full for that voltage, and a steady direct voltage drives no lasting current through a capacitor at all. It blocks steady current. But if the applied voltage changes, charge must rush onto or off the plates to match the new voltage, and that rush is a current, so a capacitor passes a changing signal. The faster the voltage changes, the larger the current needed to keep up, a relationship written as

the current is the capacitance times the rate of change of voltage. A capacitor is therefore a component that ignores what is constant and responds to what varies, the exact opposite of a plain resistor's indifference to time. This single behaviour, blocking direct current while passing changes, is the seed of filtering, of coupling one stage of a circuit to the next while keeping their steady levels apart, and of the smoothing capacitor that turns a lumpy supply into a steady one by charging on the peaks and discharging into the dips. It also means that a capacitor and a resistor together do something neither can alone: they take time. Put a resistor in the path that charges a capacitor and the charge cannot arrive instantly, because the resistor limits the current, and the voltage climbs on a curve with a definite pace. That pace, the most useful single number in timing circuits, is the subject of the next lesson.