Force Per Unit Length Between Parallel Wires

Two parallel wires separated by d attract for same-direction currents, repel for opposite. This result defines the SI ampere (pre-2019 definition).

Setup

Two long parallel wires separated by distance $d$, carrying currents $I_1$ and $I_2$.

Step 1 — Field of Wire 1 at Wire 2

From mc10:

$$B_1 = \frac{\mu_0 I_1}{2\pi d}$$

Step 2 — Force on Wire 2

From the force on a current-carrying conductor ($\vec{F} = I\vec{L} \times \vec{B}$), with $B_1 \perp I_2$:

$$F = I_2 L B_1 = I_2 L \cdot \frac{\mu_0 I_1}{2\pi d}$$

Force per unit length:

$$\boxed{\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}}$$

Direction

• Same-direction currents: $\vec{F}$ directed toward the other wire. Attraction.
• Opposite-direction currents: Repulsion.

Definition of the Ampere

Substituting $I_1 = I_2 = 1\,\text{A}$, $d = 1\,\text{m}$:

$$\frac{F}{L} = \frac{4\pi \times 10^{-7} \times 1 \times 1}{2\pi \times 1} = 2 \times 10^{-7}\,\text{N/m}$$

This was the pre-2019 SI definition of the ampere. The 2019 redefinition fixes $e$ numerically; the formula remains exact.