Capacitors and Capacitance

 

A capacitor has one job: to store electric charge. Connect one across a battery and charge flows onto it until it is "full"; disconnect it and it holds onto that charge, ready to release it later.

 

In that sense a capacitor is a little like a small reservoir for charge — but the comparison only takes you so far, and capacitance is one of those topics where a few key ideas clear away most of the confusion.

 

This topic builds directly on the ideas of current and potential difference from the earlier electricity posts, so if those still feel shaky it is worth a quick look back before reading on.

 

 

WHAT CAPACITANCE ACTUALLY MEASURES

Capacitance is a measure of how good a component is at storing charge. There are two ways to get more charge onto a capacitor: use a bigger or better-designed capacitor, or connect it to a supply with a greater EMF (voltage).

The capacitance tells you how much charge is stored for each volt across the plates:

 

Capacitance (farads) = Charge stored (C) / p.d. across capacitor (V)

C = Q / V which rearranges to Q = CV and V = Q / C

 

The unit is the farad (F). One farad turns out to be an enormous capacitance — a whole coulomb of charge for every single volt — so in practice you will almost always meet microfarads (µF), nanofarads (nF) and picofarads (pF) rather than whole farads.

 

 

WHERE STUDENTS GO WRONG: THE ENERGY STORED

Here is the classic trap. If a capacitor holds charge Q at a p.d. of V, then surely the energy stored is simply Q × V? It is not — and the reason is well worth understanding rather than just memorising.

 

When the capacitor is empty, the very first trickle of charge flows on almost without resistance, because there is barely any p.d. to push against. As charge builds up, the p.d. across the capacitor rises, and every extra coulomb has to be pushed on against a larger and larger voltage. So the charge is not all delivered against the full voltage V — on average it is delivered against only half of it.

You can see this straight away from a graph. Plot charge Q against p.d. V and you get a straight line (because Q = CV). The energy stored is the area under that line, and the area of a triangle is ½ × base × height. Hence:

E = ½ QV = ½ CV² = ½ Q² / C

That stray factor of ½ is not a fudge — it comes from the graph.

 

 

CHARGING AND DISCHARGING: WHY IT ISN'T INSTANT

Connect a capacitor to a battery through a resistor and it does not charge up in an instant. As charge piles onto the negative plate, it becomes progressively harder to add any more: the electric field from the charge already sitting there pushes back on the electrons trying to arrive, slowing them down. The current therefore starts off large and dies away, while the charge and p.d. on the capacitor climb towards their final values.

The capacitor is said to be fully charged when its p.d. exactly opposes the battery EMF, leaving zero net EMF in the circuit — at which point the current has fallen to zero and no more charge flows. The very same reasoning works in reverse when the capacitor discharges.

 

The result, in both cases, is exponential. While charging:

Q = Q₀(1 − e^−t/RC)

V = V₀(1 − e^−t/RC)

I = I₀ e^−t/RC

 

and while discharging everything simply decays away:

Q = Q₀ e^−t/RC

V = V₀ e^−t/RC

I = I₀ e^−t/RC

A small but useful detail: the gradient of the Q–t graph is the current (dQ/dt), and the area under the I–t graph is the charge that has flowed.

 

 

THE TIME CONSTANT

You will have noticed the quantity RC cropping up in every one of those equations. It has a name — the time constant — and its own symbol, the Greek letter tau, τ (not to be confused with the t for time sitting right next to it!):

τ = RC

There are two ways to read what it means. It is the time the capacitor would take to discharge completely if it kept up its initial, fastest rate. More usefully, it is the time taken for the charge to fall to 1/e of its starting value — that is, to about 37%. Put the other way round, after one time constant roughly 63% of the charge has already gone.

A small RC gives a quick charge and discharge; a large RC gives a slow, drawn-out one. This is the principle behind any number of timing circuits — and behind the flash on a camera, which quietly charges a capacitor over a second or two and then dumps all of that stored energy through the flash tube in an instant.

 

None of this is as forbidding as it first looks. Capacitance is just charge per volt; the energy has its factor of ½ because the voltage rises as you fill the capacitor; and the charging and discharging are exponential, governed by the single quantity RC. Get comfortable with those three ideas and the rest of the topic falls into place.