Force Between Parallel Currents

Consider wire 1 carrying current $I_1$, treated as an infinitely long straight conductor. Using Ampère's circuital law, the magnetic field at a perpendicular distance $d$ from wire 1 is $$B_1 = \frac{\mu_0 I_1}{2\pi d}$$ where $\mu_0 = 4\pi \times 10^{-7}\,\text{T}\,\cdot\,\text{m/A}$ is the permeability of free space. The field circles around wire 1 according to the right-hand rule: curl the right hand around the wire with the thumb pointing in the direction of $I_1$, and the fingers show the direction of $\vec{B}_1$.

Wire 2, carrying current $I_2$ and having length $L$, sits inside the magnetic field $B_1$ created by wire 1. The force on a current-carrying conductor in a magnetic field is given by $F = BIL\sin\theta$. Because wire 2 runs parallel to wire 1, the field $\vec{B}_1$ is perpendicular to $I_2$ (i.e., $\theta = 90°$, so $\sin\theta = 1$). Therefore the magnitude of the force on length $L$ of wire 2 is $$F = B_1 I_2 L = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$

Dividing both sides by $L$ gives the force per unit length, which is the most useful form because it does not depend on the arbitrary length $L$ chosen: $$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$ This quantity has SI units of N/m. By Newton's third law, wire 1 experiences an equal and opposite force due to the field of wire 2, confirming the symmetry of the interaction.

Apply the right-hand rule to $B_1$ at the location of wire 2, then use $\vec{F} = I_2(\vec{L}\times\vec{B}_1)$. If $I_1$ and $I_2$ point in the same direction (parallel currents), the force on wire 2 points toward wire 1 — the wires attract. If $I_2$ is reversed (anti-parallel currents), the cross product reverses, so the force points away — the wires repel. This is a firm experimental result, first confirmed by André-Marie Ampère in 1820.

Before 2019, the SI ampere was officially defined using this formula: one ampere was the constant current that, when flowing in each of two infinitely long parallel straight wires of negligible cross-section placed 1 metre apart in vacuum, produces a force of exactly $2\times10^{-7}$ N per metre of length between them. Substituting $d = 1\,\text{m}$ and $I_1 = I_2 = 1\,\text{A}$ into the formula verifies this: $F/L = (4\pi\times10^{-7}\times1\times1)/(2\pi\times1) = 2\times10^{-7}\,\text{N/m}$. Since 2019, the ampere is redefined by fixing the elementary charge $e$, but the physics is unchanged.

• F/L — Force per unit length (N/m): The magnetic force acting on each wire per metre of its length.
• \mu_0 — Permeability of free space (T·m/A): Fundamental constant equal to $4\pi \times 10^{-7}\,\text{T}\,\cdot\,\text{m/A}$; describes how easily a magnetic field is established in vacuum.
• I_1 — Current in wire 1 (A): Electric current flowing through the first conductor.
• I_2 — Current in wire 2 (A): Electric current flowing through the second conductor.
• d — Separation distance (m): Perpendicular distance between the two parallel wires.

Problem. Two parallel straight wires are separated by $d = 5.0\,\text{cm}$ and carry currents of $I_1 = 8.0\,\text{A}$ and $I_2 = 12.0\,\text{A}$ in the same direction. Calculate (a) the force per unit length between the wires and (b) the total force on a 30 cm segment of wire 2. State whether the force is attractive or repulsive.

Answer. Force per unit length = $3.84 \times 10^{-4}$ N/m; total force on 30 cm segment = $1.15 \times 10^{-4}$ N, attractive.

• Magnetic Field Around a Straight Current-Carrying Wire — The force between parallel wires is derived directly from the magnetic field produced by each wire; understanding $B = \mu_0 I / (2\pi d)$ is an essential prerequisite.
• Force on a Current-Carrying Conductor in a Magnetic Field — The second step of the derivation uses $F = BIL\sin\theta$; mastering this concept shows why a wire in an external magnetic field experiences a force.
• Ampère's Circuital Law — Ampère's law is the elegant tool used to derive the magnetic field of an infinite straight wire, making it central to the entire force-between-wires calculation.
• The Biot–Savart Law — An alternative and more general approach to calculating the magnetic fields of current elements; it can reproduce the parallel-wire result for finite wire lengths.
• Electric Current and Conventional Flow — A clear understanding of current direction is essential to correctly applying the right-hand rule and determining whether the inter-wire force is attractive or repulsive.