Last updated: 2026-07-01

Force Between Two Parallel Wires

Two long, straight wires that carry electric current exert a magnetic force on each other. Each current sets up a magnetic field, and the current in the neighbouring wire, sitting in that field, feels a force. For wires a distance $d$ apart carrying currents $I_1$ and $I_2$, the force per unit length is $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$, where $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ is the permeability of free space. The total force over a parallel length $L$ is $F = \frac{F}{L} \cdot L$.

This calculator returns both the force per unit length and the total force, and states whether the wires attract or repel based on the relative direction of the two currents.

Same direction attracts, opposite directions repel

The direction of the force depends on how the two currents flow relative to each other. Currents in the same direction pull the wires together — the force is attractive. Currents in opposite directions push the wires apart — the force is repulsive. This is the reverse of the rule for static electric charges, where like signs repel. The result follows from combining the circular magnetic field of one wire (right-hand rule) with the $\vec{F} = I\vec{L} \times \vec{B}$ force on the second wire.

Formula

Because the force per unit length falls off as $1/d$, doubling the separation halves the force. It scales directly with each current, so doubling both currents multiplies the force by four.

Worked example

Two parallel wires each carry a current of 10 A in the same direction, separated by 5 cm (0.05 m), and run parallel for 1 m. The force per unit length is:

Over the 1 m parallel length the total force is $F = (4.0 \times 10^{-4})(1) = 4.0 \times 10^{-4}\ \text{N}$. Because the currents flow in the same direction, the wires attract. The force is small at everyday currents, but it grows quickly with current: at 1000 A each, the same geometry gives 4 N/m, enough to matter in high-power busbars.

The old definition of the ampere

This force once defined the ampere itself. From 1948 to 2019 the ampere was the constant current that, in two infinitely long parallel wires one metre apart in a vacuum, produces a force of exactly $2 \times 10^{-7}$ N per metre. Setting $I_1 = I_2 = 1\ \text{A}$ and $d = 1\ \text{m}$ in the formula reproduces this: $\frac{F}{L} = \frac{\mu_0}{2\pi} = 2 \times 10^{-7}\ \text{N/m}$. That definition fixed $\mu_0$ at exactly $4\pi \times 10^{-7}\ \text{T·m/A}$. Since the 2019 SI redefinition the ampere is defined from a fixed elementary charge, and $\mu_0$ is now a measured quantity extremely close to its former exact value.

Limitations

The formula assumes long, thin, straight wires — long compared with their separation — so that end effects are negligible and each field is well approximated by that of an infinite wire. It also assumes steady currents in a vacuum or, to good approximation, in air. Wires that are short, curved, or close enough that their finite thickness matters require the full Biot–Savart integral, and time-varying currents add induction effects the static form does not capture.