Charge, Current and Potential Difference

Electric current is the rate of flow of charge: $I=\frac{\Delta Q}{\Delta t}$. One ampere is one coulomb per second, and the ampere is an SI base unit — the coulomb is defined from it, not the other way round.

In a metal the charge carriers are free electrons, each carrying $-1.60\times 10^{-19}\mathrm{C}$. $\text{Conventional current}$ is defined as flowing from positive to negative, which is opposite to the actual electron flow — a historical convention that predates the discovery of the electron and is now too entrenched to change.

Charge is $\text{quantised}$: any charge is a whole-number multiple of the elementary charge $e$. That is why the number of electrons is always a whole number in these calculations.

$\text{Potential difference}$ across a component is the energy transferred $\text{from}$ the charge to that component, per coulomb: $V=\frac{W}{Q}$. One volt is one joule per coulomb.

$\text{Electromotive force}$ is the same quantity for a source: the energy transferred $\text{to}$ each coulomb by the cell. Despite its name it is not a force — it is measured in volts, and the misnomer is another historical leftover.

The distinction is direction of energy transfer. A cell gives energy to the charge (e.m.f.); a resistor takes energy from it (p.d.). Both are joules per coulomb.

Current depends on how many charge carriers there are, how fast they move, and how much charge each carries. Combining these gives $I=nAvq$, where $n$ is the number density of carriers (per cubic metre), $A$ the cross-sectional area, $v$ the mean drift velocity, and $q$ the charge per carrier.

The derivation is worth seeing: in time $t$ the carriers travel $vt$, so the volume passing a point is $Avt$, containing $nAvt$ carriers and therefore a charge $nAvtq$. Dividing by $t$ gives the current.

The equation explains the difference between conductors, semiconductors and insulators. A metal has an enormous $n$, around $10^{28}{\mathrm{m}}^{-3}$; a semiconductor has far fewer carriers; an insulator has almost none — which is why the same p.d. drives vastly different currents through them.

At that drift velocity an electron would take hours to travel the length of a mains cable, yet a lamp lights the moment the switch closes. The resolution is that the electrons do not have to travel from the switch to the lamp.

The conductor is already full of free electrons everywhere, including inside the lamp. Closing the switch establishes an electric field along the wire at close to the speed of light, and every electron in the circuit begins drifting almost simultaneously.

So what propagates quickly is the $\text{field}$, not the charge carriers. The electrons themselves also move rapidly in random thermal motion — around $10^5\mathrm{m}{\mathrm{s}}^{-1}$ — with the drift velocity being only a tiny net bias superimposed on that chaos.