Force Between Parallel Wires

This equation is two earlier ones stacked. A long straight wire produces a field circling it at $B = \mu_0 I_1/(2\pi d)$ at distance $d$; a second wire sitting in that field feels $F = B I_2 \ell$. Substitute and you get $F = \mu_0 I_1 I_2 \ell/(2\pi d)$. Each wire sits in the other's field, and by Newton's third law they push on each other equally and oppositely. The direction is the part people find surprising: currents flowing the same way attract, currents flowing opposite ways repel — the reverse of the intuition borrowed from electrostatics, where like charges repel. Note the $1/d$ rather than $1/d^2$: a straight wire's field falls off with the first power of distance because the source is a line rather than a point.

Two wires 10 mm apart, each carrying 10 A, over a 1 m parallel run feel $F = (1.257 \times 10^{-6} \times 10 \times 10 \times 1)/(2\pi \times 0.01) = 2.0\ \text{mN}$ — the weight of a grain of rice, which is why nobody notices it in ordinary wiring. Now put a 20 kA fault through the same pair: the currents appear as a product, so the force scales with the square, and 2 mN becomes 8 kN per metre. That is nearly a tonne of force trying to tear a metre of busbar out of its supports, and it is the reason switchgear bracing is engineered rather than assumed.

From 1948 until 2019 this relation did not merely describe the ampere, it defined it: the ampere was the current which, in two infinitely long parallel conductors one metre apart in vacuum, produced a force of exactly $2 \times 10^{-7}$ newtons per metre. That definition is what made $\mu_0$ exactly $4\pi \times 10^{-7}$ — a defined constant rather than a measured one. The 2019 redefinition of the SI moved the anchor to a fixed value of the elementary charge, and one consequence is that $\mu_0$ is now an experimentally determined quantity with an uncertainty, very slightly different from $4\pi \times 10^{-7}$. This page uses the CODATA value.

The traps are dimensional and geometric. The published version of this law is usually the force per unit length; this page multiplies by $\ell$ to give a total force, so do not apply a per-metre figure and then multiply by the length again. $d$ is the centre-to-centre separation, not the gap between insulation surfaces, and on closely spaced busbars the difference is not small. The result assumes long, straight, parallel conductors — near a bend, a termination or a right-angle crossing the geometry changes and the simple form does not hold. And because the currents enter as a product, an alternating current gives a force that is always attractive or always repulsive but pulses at twice the supply frequency, never reversing: that 120 Hz throb on a 60 Hz system is precisely what makes transformers and reactors hum, and what fatigues busbar supports over years rather than breaking them in an instant.