Current as the rate of flow of charge, the equation I = nAve linking current to charge-carrier density and drift velocity, and conductors, semiconductors and insulators.

What this dot point is asking

WJEC wants you to define current as the rate of flow of charge, use the equation $I = nAve$, and explain conduction in conductors, semiconductors and insulators in terms of charge-carrier number density. The $I = nAve$ model is the bridge between the everyday idea of current and what the electrons are actually doing, and the examiners love it because it ties a microscopic picture to a measurable quantity.

The answer

Current as flow of charge

By convention, current is the direction in which positive charge would flow, which is opposite to the actual motion of the electrons in a metal.

The equation I = nAve

Here $n$ is the number density of charge carriers (per cubic metre), $A$ is the cross-sectional area, $v$ is the mean drift velocity and $e$ is the charge on each carrier. The equation is derived by counting carriers: in a time $\Delta t$ the carriers sweep through a volume $Av\Delta t$, containing $nAv\Delta t$ carriers, each of charge $e$, so the charge passing is $\Delta Q = nAve\Delta t$ and $I = \Delta Q / \Delta t = nAve$. For a fixed current, a smaller area or smaller $n$ forces a larger drift velocity.

Conductors, semiconductors and insulators

This difference in $n$ explains why a semiconductor's resistance falls as it warms (more carriers are released), unlike a metal.

Examples in context

Example 1. Why thin filaments glow. In a filament bulb the wire narrows to a tiny cross-section $A$. For the same current, $I = nAve$ forces a much larger drift velocity, the electrons collide with the lattice more violently, and the filament heats to incandescence. The conducting parts of the circuit with large $A$ stay cool.

Example 2. A thermistor in a thermostat. A thermistor is a semiconductor whose carrier density $n$ rises sharply as it warms. As temperature climbs, $n$ increases, resistance falls and more current flows, which a control circuit detects to switch off a heater. The whole device works because $n$ in a semiconductor is strongly temperature-dependent, unlike a metal.

Try this

Q1. A copper wire of cross-sectional area $1.0\times10^{-6}\,\text{m}^2$ carries a current of $2.0\,\text{A}$. If $n = 8.5\times10^{28}\,\text{m}^{-3}$, find the drift velocity. Take $e = 1.6\times10^{-19}\,\text{C}$. [3 marks]

• Cue. $v = \frac{I}{nAe} = \frac{2.0}{8.5\times10^{28}\times1.0\times10^{-6}\times1.6\times10^{-19}} \approx 1.5\times10^{-4}\,\text{m s}^{-1}$.

Q2. Explain, in terms of $n$, why a metal conducts much better than a semiconductor. [2 marks]

• Cue. A metal has a far higher number density of free charge carriers than a semiconductor.