# Lecture 9 - Dielectrics

## The Parallel Plate Capacitor Configuration

This consists of two metal plates with area $A$ carrying equal and opposite charges $Q$. The field due to one of these sheets of charges can be found from Gauss's law

$E=\frac{Q}{\varepsilon_{0}A}$

The potential due to this sheet of charge is

$V=-\frac{Qd}{\varepsilon_{0}A}$

Here we have defined the zero of potential as the position of the positive sheet of charge.

## The Definition of Capacitance

$V=\frac{Qd}{\varepsilon_{0}A}$

The amount of charge $Q$ separated when a given voltage $V$ is applied is obtained from the equation above

$Q=\frac{\epsilon_{0}A}{d}V$

The capacitance $C$ is the amount of charge separated per volt applied

$C=\frac{Q}{V}$

and we can see for a parallel plate capacitor

$C=\frac{\epsilon_{0}A}{d}$

The unit for capacitance is the farad, $\mathrm{F}$ (named after Michael Faraday) which is equivalent to $\mathrm{\frac{C}{V}}$

## Dielectric materials

Dielectrics are insulating materials. They do not contain significant amounts of free charge, so charges cannot pass from one side of the capacitor to the other.

However, when a dielectric material is placed in an electric field it develops an electrical polarization. This can arise through several different physical mechanisms, for example, if the material has molecular dipoles they will orient themselves to line up with the field.

## Electric Polarization

The polarization $P$ of a material is the dipole moment per unit volume. At the surface of the dielectric there is a charge density equal to the polarization, but this charge is not free to move, often called the bound charge $\sigma_{B}$

## Dielectric in a field

Since the dipoles reduce the electric field strength inside a dielectric, one can write generally:

$E_{dielectric} = \frac{1}{K}E_{vacuum}$

where $K$ is a constant (specific to each material) greater than 1.

At the boundary of the dielectric the boundary condition is that

$E_{vacuum}=\frac{P}{\varepsilon_{0}}+E_{dielectric}$.

For most dielectric materials we can say that the polarization is given by $\chi \varepsilon_{0} E_{dielectric}$ where $\chi$ is the dielectric susceptibility.

The quantity $K=1+\chi$ is called the dielectric constant of the material.

As there is no way for dipoles to occur in vacuum it has susceptibility $\chi=0$, giving $K=1$

Air has very little susceptibility and so $K$ for air is just a little bit greater than 1

For a dielectric material where $K>1$ the field is less in the dielectric than in the air. A key characteristic of a dielectric is the partial exclusion of the external field from the interior of the material.

## Dielectric in a capacitor

$E_{dielectric}=\frac{Q}{\varepsilon_{0} A}-\frac{P}{\varepsilon_{0}}=\frac{Q}{\varepsilon_{0} A}-\chi E_{dielectric}$

$KE_{dielectric}=\frac{Q}{\varepsilon_{0} A}$

$E_{dielectric}=\frac{V}{d}$

$K\frac{V}{d}=\frac{Q}{\varepsilon_{0} A}$

$C=\frac{Q}{V}=K\frac{\varepsilon_{0}A}{d}$

In general, the capacitance of a configuration with a dielectric is related to the same geometry with a vacuum gap:

$C_{dielectric} = KC_{vacuum}$.

## Partially filled capacitor

$E_{gap}=\frac{Q}{\varepsilon_{0}A}$

$KE_{dielectric}=\frac{Q}{\varepsilon_{0} A}$

$2E_{gap}d_{gap}+E_{dielectric}d_{dielectric}=V$

$2\frac{Q}{\varepsilon_{0}A}d_{gap}+\frac{Q}{\varepsilon_{0}A}\frac{d_{dielectric}}{K}=V$

$\frac{V}{Q}=\frac{1}{C}=\frac{2d_{gap}}{\varepsilon_{0}A}+\frac{d_{dielectric}}{K\varepsilon_{0}A}$

This is the same result as we would have got if we had considered the system to be 3 capacitors in series.

## Effect of the gap

We can consider the effect of a gap for an example of a very high dielectric constant material, strontium titanate, which has a room temperature dielectric constant of ~300.

A capacitor with an area of $1\mathrm{cm^{2}}$ and thickness $0.5\mathrm{mm}$ should have a capacitance of ~500$\mathrm{pF}$ as compared to a capacitor of the same size filled with air which should have a capacitance of $1.8\mathrm{pF}$.

However if we look at the equation we derived previously.

$\frac{1}{C}=\frac{2d_{gap}}{\varepsilon_{0}A}+\frac{d_{dielectric}}{K\varepsilon_{0}A}$

We can see that only if the gap is very small will we get a substantial enhancement from the insertion of a dielectric in to a capacitor.

## Dielectric breakdown

All dielectrics have a maximum electric field that they can withstand; once this field is exceeded conduction can occur and the charge dissipates through the dielectric.

Typically the whole dielectric does not breakdown, instead the current will find pathways through the material. This can lead to a dendritic tree like structure of broken down materials called a Lichtenberg figure.

In a Jacob's ladder, or climbing arc, demonstration when the air breaks down it is heated up and rises. The ionized air remains the best path for charge to flow from one side of the ladder to the other so the current rises with the ionized air.

## Electric Current

So far we have dealt with electrostatics, situations in which electric charges are stationary. Now we will look at situations involving continuous flow of charge from one point to another, which we describe as a current.

## Batteries

In order to have a constant flow of current we need to have a constant amount of charge flowing through a constant potential difference. This implies that we have a constant amount of energy driving the current.

In a battery this energy is provided by a conversion of chemical energy into electrical energy.

A battery actually refers to a collection, of electrochemical cells. (The term battery actually predates the electrochemical cell and was first used by Benjamin Franklin, to describe a number of connected capacitors, which at this time were Leyden Jars).

The electrochemical cell was invented by Alessandro Volta, but he did not fully appreciate the chemical nature of it's operation, which was understood somewhat later by Michael Faraday.