Copper, silver and gold are good conductors of electric current. There are large numbers of conduction electrons in a copper wire – as many conduction electrons as there are atoms. The number of conduction electrons per unit volume (for example, in $1{m^3}$ of the metal) is called the number density and has the symbol n. For copper, the value of n is about ${10^{29}}{m^{ - 3}}$.
Figure 8.9 shows a length of wire, with cross-sectional area A, along which there is a current I.

How fast do the electrons in Figure 8.9 have to travel? The following equation allows us to answer this question:
$I = nAvq$
where n = the number density, A = cross sectional area of the conductor, v = mean drift velocity of the charge carriers, q = the charge on each charge carrier.

The length of the wire in Figure 8.9 is l. We imagine that all of the electrons shown travel at the same speed v along the wire.
Now imagine that you are timing the electrons to determine their speed. You start timing when the first electron emerges from the right-hand end of the wire. You stop timing when the last of the electrons shown in the diagram emerges. (This is the electron shown at the left-hand end of the wire in the diagram.) Your timer shows that this electron has taken time t to travel the distance l.
In the time t, all of the electrons in the length l of wire have emerged from the wire. We can calculate how many electrons this is, and hence the charge that has flowed in time t:

$\begin{array}{l}
number\,of\,electrons = number\,density \times volume\,of\,wire\\
 = n \times A \times l
\end{array}$
$\begin{array}{l}
charge\,\,of\,electrons = number\,electron\,charge\\
 = n \times A \times l \times e
\end{array}$

We can find the current I because we know that this is the charge that flows in time t, and 

$current = \frac{{charge}}{{time}}:$

$I = n \times A \times l \times \frac{e}{t}$

Substituting v for $\frac{1}{t}$ gives:

$I = nAve$

The moving charge carriers that make up a current are not always electrons. They might, for example, be ions (positive or negative) whose charge q is a multiple of e. Hence we can write a more general version of the equation as:

$I = nAvq$

Worked example 3 shows how to use this equation to calculate a typical value of v.

You do not need to know how to derive I = nAvq but it is interesting to recognise that the units are homogeneous.
The unit of current (I) is the ampere (A).
The unit of the number of charge carriers per unit volume (n) is ${m^{ - 3}}$.
The unit of area (A) is ${m^{ 2}}$.
The unit of the drift velocity v is $m\,{s^{ - 1}}$.
The unit of charge (q) is the coulomb (C).
All these are in base units except the coulomb and 1 coulomb is 1 ampere second (A s).
Putting the units into the right-hand side of the equation:

${m^{ - 3}} \times {m^{ - 23}} \times m\,{s^{ - 1}} \times As = A$

This is the same as the left-hand side of the equation. Although this does not prove the equation to be correct, it does give strong evidence for it.
This technique is often used for checking the validity of an expression and also to predict a possible formula.

Slow flow

It may surprise you to find that, as suggested by the result of Worked example 3, electrons in a copper wire drift at a fraction of a millimetre per second. To understand this result fully, we need to closely examine how electrons behave in a metal. The conduction electrons are free to move around inside the metal. When the wire is connected to a battery or an external power supply, each electron within the metal experiences an electrical force that causes it to move towards the positive end of the battery. The electrons randomly collide with the fixed but vibrating metal ions. Their journey along the metal is very haphazard. The actual velocity of an electron between collisions is of the order of magnitude ${10^5}\,m\,{s^{ - 1}}$, but its haphazard journey causes it to have a drift velocity towards the positive end of the battery. Since there are billions of electrons, we use the term mean drift velocity v of the electrons.
Figure 8.10 shows how the mean drift velocity of electrons varies in different situations.
We can understand this using the equation:

$v = \frac{I}{{nAe}}$

- If the current increases, the drift velocity v must increase. That is:

$v \propto I$

- If the wire is thinner, the electrons move more quickly for a given current. That is:

$v \propto \frac{I}{A}$

- There are fewer electrons in a thinner piece of wire, so an individual electron must travel more quickly.
- In a material with a lower density of electrons (smaller n), the mean drift velocity must be greater for a given current. That is:

$v \propto \frac{1}{n}$

It may help you to picture how the drift velocity of electrons changes by thinking about the flow of water in a river. For a high rate of flow, the water moves fast – this corresponds to a greater current I. If the course of the river narrows, it speeds up – this corresponds to a smaller cross-sectional area A.
Metals have a high electron number density–typically of the order of ${10^{28}}$ or ${10^{29}}\,{m^{ - 3}}$. Semiconductors, such as silicon and germanium, have much lower values of n–perhaps ${10^{23}}\,{m^{ - 3}}$. In a semiconductor, electron mean drift velocities are typically a million times greater than those in metals for the same current. Electrical insulators, such as rubber and plastic, have very few conduction electrons per unit volume to act as charge carriers.