Capacitance

A capacitor stores energy in an electric field. It is one of the three basic passive circuit components, alongside resistors and inductors, and almost every electronic circuit contains dozens of them. Capacitors smooth the output of power supplies, filter noise out of signals, set the timing of oscillators, deliver the burst of energy in a camera flash or a defibrillator, correct the power factor of industrial motors, and, as supercapacitors, buffer energy in buses and wind turbines.

This chapter explains what capacitance is, calculates it for the parallel-plate capacitor, shows how dielectrics increase it, works out the energy stored, combines capacitors in series and parallel, and introduces the RC circuit, whose exponential charging curve appears throughout engineering.

What a capacitor is

Any two conductors separated by an insulator form a capacitor. Connect them to a battery and charge flows until one conductor holds \(+Q\) and the other \(-Q\), with a potential difference \(V\) between them. For a given capacitor, the charge is proportional to the voltage. The constant of proportionality is the capacitance:

Capacitance depends only on the geometry of the conductors and the insulating material between them, not on how much charge is present. A farad is very large; practical capacitors range from picofarads (pF, \(10^{-12}\) F) in radio circuits through microfarads (μF) in power supplies to thousands of farads in supercapacitors.

The circuit symbol is two parallel lines, representing the plates.

The parallel-plate capacitor

The simplest capacitor is two flat plates of area \(A\) a distance \(d\) apart. From Lesson 2, the field between them is \(E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A)\), and the voltage is \(V = Ed\). So \(V = Qd/(\varepsilon_0 A)\), and

Larger plates hold more charge at a given voltage; closer plates give a stronger field per volt. Both raise the capacitance.

Real capacitors maximize \(A/d\): electrolytic capacitors use a chemically etched foil (huge surface area) coated with an oxide film only nanometres thick; ceramic capacitors stack hundreds of thin layers; supercapacitors use porous carbon with surface areas of over 1000 m² per gram.

Dielectrics

Filling the gap with an insulating material (a dielectric) increases the capacitance by a factor \(\kappa\), the dielectric constant:

The field polarizes the dielectric's molecules, and their bound charges partly cancel the field inside, so a given charge produces less voltage. Typical values: air 1.0006, paper about 3.5, glass 5 to 10, water 80, and specialist ceramics such as barium titanate over 1000.

A dielectric has a second benefit: it can withstand a much higher field than air before breaking down, so the plates can be closer and the capacitor can work at higher voltage. Every capacitor has a rated working voltage that must not be exceeded.

Energy stored in a capacitor

Charging a capacitor takes work, because each new bit of charge must be pushed onto a plate against the voltage already there. The voltage rises from 0 to \(V\) as the charge builds, so the average voltage is \(V/2\) and the work done is \(\tfrac12 QV\). Using \(Q = CV\), the stored energy can be written three ways:

The energy lives in the electric field between the plates. Dividing by the volume \(Ad\) gives the energy density \(u = \tfrac12\varepsilon_0 E^2\), a result that holds for any electric field. Even at the breakdown field of air (3 MV/m) this is only about 40 J/m³, which is why ordinary capacitors store far less energy than batteries of the same size.

Capacitors in real equipment

Capacitors come in families suited to different jobs. Ceramic capacitors are small and fast and filter high-frequency noise on circuit boards. Electrolytic capacitors pack large capacitance into a small can for power supplies, but they are polarized (they must be connected the right way round) and wear out with heat. Film capacitors are robust and are used for motor starting, power-factor correction in factories and high-voltage pulses. Every capacitor carries two ratings that matter: its capacitance and its maximum working voltage.

Capacitors in parallel and series

In parallel, capacitors share the same voltage, and their charges add. It is as if the plate areas add:

In series, capacitors carry the same charge (whatever leaves one plate arrives on the next), and their voltages add. It is as if the gaps add:

These are the reverse of the rules for resistors in Lesson 4. Series capacitance is always smaller than the smallest capacitor in the chain. For two in series there is a handy shortcut: \(C = C_1C_2/(C_1 + C_2)\).

Notice that the smallest capacitor in a series chain takes the largest share of the voltage. Designers must check that every capacitor in a series string stays within its voltage rating.

RC circuits: charging and discharging

A capacitor never charges instantly. In a real circuit there is always resistance, which limits the current. When a capacitor \(C\) charges from a voltage \(V_0\) through a resistor \(R\), the current starts at \(V_0/R\) and falls as the capacitor fills, because the capacitor's own voltage opposes the source. Applying Kirchhoff's voltage law gives a first-order differential equation whose solution is

The product \(\tau = RC\) is the time constant, measured in seconds (ohms times farads). After one time constant, a charging capacitor reaches 63% of its final voltage; a discharging one falls to 37%. After five time constants it is within 1% of the end value, which engineers usually treat as "fully" charged or discharged.

The same exponential appears in heating and cooling, in draining tanks and in the response of sensors. An engineer who recognizes a first-order system can predict its behaviour from a single number, its time constant.

RC circuits are everywhere: they debounce switches, set the delay before a light turns off, and, combined with transistors, form the timing core of the classic 555 timer chip. Paired with a resistor, a capacitor also makes a simple filter that passes slow signals and blocks fast ones (or the reverse), a topic for circuits courses.