## Capacitors

Capacitors are common components of electronic circuits, used almost as frequently as resistors.A capacitor is an electronic device for storing charge. Capacitors can be found in almost all but the most simple electronic circuits. There are many different types of capacitor but they all work in essentially the same way. A simplified view of a capacitor is a pair of metal plates separated by a gap in which there is an insulating material known as the dielectric.

Various types of capacitor

This simplified capacitor is also chosen as the electronic circuit symbol for a capacitor is a pair of parallel plates as shown in Figure below.

The symbol for an unpolarised capacitor.

Normally, electrons cannot enter a conductor unless there is a path for an equal amount of electrons to exit. However, extra electrons can be "squeezed" into a conductor without a path to exit if an electric field is allowed to develop in space relative to another conductor. The number of extra free electrons added to the conductor (or free electrons taken away) is directly proportional to the amount of field flux between the two conductors.

In this simplified capacitor the dielectric is air. When a voltage, *V* is applied to the terminals of the capacitor, electrons flow on to one of the plates and are taken off the other plate. The total number of electrons in the capacitor remains the same. There are just more on one the negative plate and fewer on the positive plate.

Charging a capacitor with a battery

If the volage were increased the increased potential difference between the plates would push more electrons on to the negatively charged plate. We could measure the charge stored on the plate as a function of different applied voltages.

At zero voltage, the capacitor plates are neutral and so no charge is stored. (we assume that we started with a fully discharged capacitor), at a voltage V the charge on the plates is Q and at twice the voltage, the charge is doubled. We find that for increasing voltage the charge increases linearly. We can plot this as a straight line.

Suppose that we go away and do some research and come back with a better capacitor which stores more charge for a given voltage we can plot the result of the charge stored as a function of applied voltage

This would be represented as another line with a steeper slope. If we plotted lots of graphs for different capacitors we would get many straight lines. We can say that a measure of the capacitance is the how much the charge is stored for a given voltage. This is sometimes expressed as ** Q=CV**.

Of course in charging the capacitor work must be done to move the charge. Therefore energy must be supplied and this energy is available when the capacitor is discharged.

The work done is given by *W*=*qV*. Initially the charge is easily moved onto the plates of the capacitor, however as more charge is moved onto the capacitor plates the repulsive force between the charges makes it harder to add charge, when the repulsive force of the charges equals the power of the battery, no more charge can be moved onto the plates. Therefore the average work is **1/2 qV**. If we look at our graph of charge against voltage we can recognise this is the same as the area under the curve. In general, the work done is equal to the energy transferred.

Mathematically:

## Factors Affecting Capacitance

How can we increase the capacitance of a parallel plate capacitor? There are three factors which affect the capacitance of a parallel plate capacitor.

### Area

By increasing the area of the plates we can put more charge on the plates before the repulsive forces becomes a problem. Therefore, the capacitance is proportional to the overlapping surface area of the plates.

A variable capacitor

In a variable capacitor the overlapping area can be increased or decreased by rotating interpenetrating plates thus increasing or decreasing the capacitance. Electrolytic capacitors have their plates etched to produce a rough surface which increases the surface area still further.

### Separation

Decreasing the separation of the plates, decreases the voltage of the capacitor since the electric-field is not affected by the distance between the plates. The voltage on the capacitor is *V=Ed*. Therefore the voltage increases. For a constant charge Q

**C = Q/V = Q/Ed**.

### Dielectric Constant

The capacitance of a parallel plate capacitor is given by *C=ε*_{r}* A/d*, where *A* is the area of the plates, *d* is the separation between the plates and ε_{r} is the relative permiliability of the dielectric between the plates. The relative permiability is some factor, *K* multiplying the permiability of free-space ε_{0}. ε_{0} has a value of **8.85x10 ^{-12} F.m^{-1}**.

A full list of relative permiabilities can be found for almost any dielectric material. The greater the relative permiability the greater the capacitance of the capacitor. Some good materials are Mica, polystyrene, oil.

ε_{r}=K ε_{0}