Capacitance and Stored Energy

A capacitor is simply two pieces of metal, close together but not touching, that hold equal and opposite charges: a little tank for electric energy. The name comes from "capacity", and capacitance is exactly that, how much charge it holds for each volt you put across it. It's good for storing energy and letting it out fast, or for holding a voltage steady. The formula $Q = CV$ is just three pieces: $Q$ (the charge stored), $V$ (the voltage across it) and $C$ (how big the tank is, per volt). It's one of the simplest formulas in this course, and the rest of the lesson is about what sets $C$ and where the energy goes.

Storing charge. Connect two metal plates to a battery. The battery pulls electrons off one plate and pushes them onto the other until the voltage between the plates matches its own. Disconnect it and the charge stays, each plate holding the other's attention across the gap, ready to be released all at once: a camera flash, a defibrillator, the burst of current a processor draws in a nanosecond.

Why $Q \propto V$. Why should charge and voltage be proportional at all? Think back to superposition. Double every charge and, by superposition, the field doubles everywhere, and so does the voltage $V = -\int\mathbf{E}\cdot d\mathbf{l}$. So $Q/V$ doesn't depend on how much charge there is, only on the shapes, sizes and materials. That ratio is the capacitance, $C = Q/V$, in farads (1 F = 1 C/V).

Parallel plates. Plates of area $A$, a distance $d$ apart, holding $\pm Q$: between them $E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0A}$ (two sheets, from Gauss's Law), so $V = Ed = \frac{Qd}{\varepsilon_0A}$ and $C = \frac{\varepsilon_0A}{d}$. So you want big plates, close together. That makes sense: bigger plates spread the same charge thinner, so the field and voltage are smaller, and a smaller gap means less distance for the field to build up voltage over. A farad is enormous: plates of 1 cm² a millimeter apart give 0.885 pF, and a 1 F air capacitor with a 1 mm gap would need plates of 113 km². Supercapacitors reach thousands of farads with porous carbon of huge surface area and layers of charge a nanometer apart.

Cylinders. A coaxial cable (inner radius $a$, outer radius $b$) with $\lambda$ per meter has $E = \frac{\lambda}{2\pi\varepsilon_0s}$ in between, so $V = \int_a^bE\,ds = \frac{\lambda}{2\pi\varepsilon_0}\ln\frac ba$ and $\frac CL = \frac{2\pi\varepsilon_0}{\ln(b/a)}$.

Energy. Why isn't the energy just $QV$? Because the voltage isn't $V$ the whole time you're charging. The first bit of charge crosses for free, and each later bit has to fight the charge already there. Moving a bit of charge $dq$ across the gap when the capacitor already holds $q$ costs $\frac qC\,dq$ of work. Charging from 0 to $Q$:

That's the area under the line $V = q/C$, a triangle (hence the $\frac12$), the same idea as work as an integral. For parallel plates, $U = \frac12\cdot\frac{\varepsilon_0A}{d}(Ed)^2 = \frac12\varepsilon_0E^2\cdot Ad$: an energy density of $\frac12\varepsilon_0E^2$ filling the gap. The energy lives in the field, an idea that returns with the Poynting vector.

Combinations. In parallel, both capacitors see the same $V$ and their charges add: $C = C_1 + C_2$. In series, the same charge $Q$ sits on each (the two inner plates form an isolated conductor, $-Q$ on one end and $+Q$ on the other) and the voltages add: $\frac QC = \frac Q{C_1} + \frac Q{C_2}$.

Dielectrics. Remember the dipoles that twist to line up with a field in The Electric Field? Fill the gap with an insulator and its molecules polarize: they stretch or turn so their negative ends face the positive plate, partly canceling the field. For the same $Q$ the voltage drops by a factor $\kappa$, so the capacitance grows by $\kappa$: $C = \kappa\varepsilon_0A/d$. Every insulator also has a dielectric strength, the field at which it breaks down and conducts (about 3 MV/m for air, tens of MV/m for good plastics), and that caps the capacitor's voltage. A slab that fills only part of the gap acts like two capacitors in series, one filled and one not.

A common mistake. It's tempting to reach for the resistor rules here, since the circuits look the same. They're reversed: capacitors in parallel simply add, and in series they combine by reciprocals. That makes sense once you picture it, because parallel capacitors are like one capacitor with more plate area, while series capacitors are like one with a wider gap. Another trap: when a capacitor is disconnected, it's $Q$ that stays fixed, not $V$, so use $\frac{Q^2}{2C}$ rather than $\frac12CV^2$ with the old voltage. And keep pF, nF and μF straight.

Capacitors appear wherever a circuit needs charge on demand or a voltage held steady. Camera flashes and defibrillators dump stored energy in milliseconds; every processor sits on a bed of decoupling capacitors that feed its nanosecond current spikes; each bit of a DRAM chip is a capacitor of about 10 to 30 fF that must be refreshed many times a second. Touchscreens and fuel gauges sense how a finger or a liquid changes $C$, the accelerometers in phones sense how motion changes a tiny gap $d$, and supercapacitors catch the braking energy of buses and trains. Charging and discharging through a resistor is the subject of RC Circuits.