Force Between Two Parallel Current-Carrying Conductors and the Definition of Ampere

Setting up the mutual-force calculation. Consider two long, straight, parallel conductors,

separated by a perpendicular distance $d$, carrying currents $I_1$ and $I_2$ respectively. Wire 1,

by the result of Section 4.4, produces a magnetic field at the location of wire 2 of magnitude

$B_1 = \mu_0 I_1/(2\pi d)$, directed (by the right-hand rule) perpendicular to wire 2 -- i.e. exactly

perpendicular to the current $I_2$ flowing in it. Wire 2, carrying current $I_2$ and sitting in this

field $B_1$, then experiences a force (by the result of Section 4.10, with $\theta=90^\circ$ since

the field from a parallel wire is always perpendicular to the second wire's own length) of magnitude,

per unit length $l$,

By Newton's third law (or by repeating the identical argument with the roles of the two wires

reversed), wire 1 experiences an EQUAL and OPPOSITE force due to wire 2's field -- the two wires

exert a genuine, mutual, Newton's-third-law-consistent pair of forces on each other.

Attraction versus repulsion. Careful application of the right-hand rule to the directions

involved shows: two parallel wires carrying current in the SAME direction ATTRACT each other; two

parallel wires carrying current in OPPOSITE directions REPEL each other -- the exact opposite of the

familiar electrostatic rule (like charges repel, unlike charges attract), a contrast worth keeping

firmly in mind, since it is a common source of confusion.

Definition of the ampere. This mutual force formula is precise, reproducible, and depends on

nothing but the two currents and their separation, which is exactly the kind of relationship

suitable for DEFINING a unit. Historically, the SI base unit of current, the ampere, was defined

directly from this force law: one ampere was defined as that constant current which, if maintained

in each of two infinitely long, straight, parallel conductors of negligible cross-section, placed

one metre apart in vacuum, would produce between them a force of exactly $2\times 10^{-7}\ \text{N}$

per metre of length. (Substituting $I_1=I_2=1\ \text{A}$ and $d=1\ \text{m}$ into the force formula …