The last lesson took the magnetic field as given and asked what it does to a moving charge. Now turn the question round: where do fields come from? From moving charge. Every current, in every wire, wraps itself in a magnetic field, and for a long straight wire that field has the simplest shape there is: circles, centred on the wire, in planes square to it. Point your right thumb along the current and your fingers curl the way the field runs.

Its strength is $B = \dfrac{\mu_0 I}{2\pi r}$, read aloud B equals mu-nought I over two pi r. $B$ is the magnetic field in tesla, though around ordinary cables it runs in microtesla (µT, a millionth). $I$ is the current in the wire in amperes. $r$ is the distance from the wire's axis to the point where you are measuring, in metres. And $\mu_0$, say mu-nought, is the permeability of free space, $4\pi\times 10^{-7}\ \mathrm{T\cdot m/A}$: a constant of nature that says how much field a given current buys. You will solve this for $B$, for $I$ and for $r$.

Two things to carry. First, the $2\pi r$ underneath is the circumference of the field line you are standing on. The same current has to supply a longer circle as you step back, so the field thins as one over r, not one over r squared. Coulomb's law spread over a sphere; a long wire spreads over a cylinder. Reaching for the square is this lesson's named trap. Second, a gift in the constants: $\mu_0$ carries a $4\pi$ of its own, so $\dfrac{\mu_0}{2\pi}$ is exactly $2\times 10^{-7}\ \mathrm{T\cdot m/A}$. The π cancels before you ever need its value, so nothing is lost to rounding it. Ten amperes at five centimetres gives 40 µT, about the strength of the Earth's own field. That is why a compass near a working cable cannot be trusted.

This is the stepping stone to the next lesson. Put a SECOND wire in this field and it feels the motor-effect force: that is all the force between two busbars is. And the word long is doing honest work: the relation holds while the wire runs straight for many times $r$ either side of you. Units guide, they do not confess: $\mathrm{T\cdot m/A}$ times amperes over metres leaves tesla, but the units cannot see a missing 2π, which carries no unit at all.