Given its large variety of applications, the capacitor is probably one of the the most used electronic component. Like inductors,
they can store energy for later use,
for example your camera's flash charges a capacitor that unleashes
current when a snapshot is taken and memory chips use them to keep track
of information (see MOS capacitor). They are also widely used in filters,
for example decoupling capacitors or AM demodulation.

**Capacitor quick facts**

The electronic symbol for a capacitor is one of the following, depending
on it having polarity or not :

`--| |--`

or `--| (--`

The main equations used to describe a capacitor's behaviour is :

I = C.dU/dt

Z = 1/jCω (complex impedance)

` i C`

`-<--| |----`

` ---->`

` U`

where i is the current through the capacitor (in Ampere),
U is the voltage (in Volt) and C is
a constant characteristic of the capacitor called capacity and expressed
in Farad. The main recommendation is to keep in mind that the voltage
of a capacitor is continuous.

Capacitors have voltage recommendations and some (for example electrolytic capacitors for obvious reasons) have polarity. If those recommendations are exceeded by too much the capacitor will probably melt down, leak (electrolytic capacitors contain a liquid) or even explode.

The electric charge and energy stored in the capacitor is given by :

q = C.V

E = 1/2.C.V^{2} = 1/2.q^{2}/C

where q is the charge of half of the capacitor (in Coulomb) and E
is the energy stored in it (in Joule).

Capacitors can be associated in different ways. Serial and parallel
associations of two capacitors can be replaced by one equivalent capacitor :

`---| |---| |--- ~ ---| |---`

1/C = 1/C_{1} + 1/C_{2}

` C`_{1} C_{2}
C

` C`_{1}

` +-| |-+ C`

`--| |-- ~ ---| |---`

C = C_{1} + C_{2}

` +-| |-+`

` C`_{2}

**Usage**

Let's see how a capacitor charges and discharges in a resistor. Consider the following circuit :

i
+->--[ R ]----+ ^
| | |
E+ ----- --- C |
E- --- --- | U
| | |
+-------------+ |

There are two elements in the circuit, let's write two equations :

i = C.dU/dt (the capacitor)

E = U + R.i (the resistor)

Hence di/dt + 1/(R.C).i = 0

i is solution of a linear differential equation with constant coefficients. The
general form of solutions is i = I_{0}.exp(-t/τ), with τ = 1/(R.C).
τ is called the capacitor time constant, it is the characteristic duration
of charge. In about 4 to 5 times τ the capacitor is either charged or discharged at 99% (e^{-4}=1.8%, e^{-5}=0.67%).

U = E - R.i = E - U_{0}.exp(-t/τ).
Below is the rough graph of a capacitor charge and discharge. During the first
period, E>0 so the capacitor is charged. During the second period E=0, the
capacitor is discharged in the resistor.

U
^
E_| _____
| . ¨¨ .
| . .
| . .
| . .
|. - . ____
0 +---------------------------------------> time

As you can see, the voltage slowly decreases or increases to limit values.
Hence one application of a capacitor is temporization. This was used in many christmas tree garlands.

A capacitor has a certain inertia in changing its
state. Because of this, low frequencies will be transmitted when the capacitor shunts the circuit since
the capacitor has time to charge/discharge in one period.
But it will block high frequencies since it will
not have sufficient time to change its state during one period.

The following are two famous filters. The first one shunts low
frequencies (high-pass filter) and the second one shunts high frequencies (low-pass filter). With τ = R.C (capacitor time constant), the turnover frequency for both filters is f = 1/τ = 1/(R.C). For example with the first filter, frequencies way below f will be attenuated, those way above f have an attenuation that is asymptotic to 0 dB.

^ ---| |---+--- ^ ---|R|-+--- ^
| C | | | |
| +-+ | --- |
U| |R| V| --- C |W
| +-+ | | |
| | | | |
| ---------+--- | -------+--- |

Say Z is the complex impedance of the capacitor, we have : V = R.U/(R+Z) hence H_{1} = V/U = R/(R+Z) for the first diagram, and W = Z.V/(R+Z) hence H_{2} = Z/(Z+R) for the second diagram. Since Z = 1/jCω, Z is small for high frequencies (high values of ω) and large for low frequencies. Thus the transfer function H_{1} approaches 0 for low frequencies (this is a high-pass filter) and the transfer function H_{2} approaches 0 for high frequencies (this is a low pass filter).

A famous application of filters is AM demodulation. you can build
a little AM receiver with as less as a diode, a resistor and a capacitor.
Here is the circuit :

antenna ------->|---+-----+ headphone
diode | |
+-+ ---
|R| C
+-+ ---
| |
------------+-----+

This circuit only allows low frequencies and blocks high ones thus
it eliminates the carrier and keeps only the signal.
Of course the signal should be amplified for the receiver to be really useful but you can really have fun with this circuit and a pair of headphones.

**Capacitor geometry**

There are many different kinds of capacitors. The most important ones
are ceramic capacitors and electrolytic capacitors. What is a capacitor anyway ?
It's just two conductors placed in front of eachother, separated by an insulator.
The most basic model is called the plane plate capacitor. It consists of two
metal plates of surface S facing eachother at a distance d in free space.

| |
| |
A| |B
---+ +---
| |
| |
| |
d
<---------->

To find the capacity of such a capacitor, just use the formula C = q/V.
Say plate A has charge +q and plate B has charge -q. Gauss' theorem states
that the electric field between the plates is constant (side effects neglected)
and E = q/(2.S.ε_{0}). Hence V = V_{B} - V_{A}
= E.d = q.d/(2.S.ε_{0}). Finally :

C = q/V = 2.S.ε_{0}/d.

You can play around with this formula for a while : for example how to
increase a capacitor's capacity ? Either use larger plates or put them
closer. You can't do this infinitely though because there is radiation pressure
on the surface of conductors.

Another way to build a capacitor with a variable capacity is to use two 3/4 disks,
place them in front of each other. When you make one turn, a variable surface
faces the other disk which makes the capacity change.
Put this sort of capacitor in the little AM demodulator I've given above and you'll be able to change your radio's tuning.