Force Between Parallel Current-Carrying Wires Explained

Each current-carrying wire creates a magnetic field that circles around it. Place a second wire nearby and that field pushes on its current, so two parallel wires exert force on each other. Currents in the same direction attract; currents in opposite directions repel.

This is not a curiosity: the attraction defines the ampere in the SI system, and the repulsion appears as dangerous mechanical stress in bus bars during short circuits. Power cables, transmission lines and coil windings all have to be braced against these forces.

Key Concepts

Field of a long straight wire: B = μ0 I/(2πd), circles the wire, unit tesla (T). Force on the second wire: F = B I L applied per wire. Force per unit length f: in newton per metre (N/m) — the useful form for long wires. Separation d: distance between wire axes in metre (m). Direction rule: same direction attract, opposite repel.

Formulas

f = F/L = μ0 I1 I2 / (2πd) where f in N/m, currents I1, I2 in ampere (A), d in m and μ0 = 4π × 10⁻⁷ T m/A. So f = (2 × 10⁻⁷ × I1 I2)/d in SI units. Assumption: wires are long compared with d and thin compared with d. Limit: for short wires or close spacing comparable to wire diameter, end effects and internal field distribution change the result.

Worked Example

Two long parallel wires 0.10 m apart each carry 10 A in the same direction. Find force per metre.

1. f = μ0 I1 I2/(2πd) = (4π × 10⁻⁷ × 10 × 10)/(2π × 0.10).
2. f = (4π × 10⁻⁵)/(0.2π) = 2 × 10⁻⁴ N/m.

Each metre of wire is pulled toward the other with 2 × 10⁻⁴ N. Small here, but at a 10,000 A fault current the same geometry gives forces thousands of times larger.

Common Misconceptions

• Parallel currents always repel like charges: Magnetic attraction wins for same-direction currents; they attract, unlike identical electric charges.
• The force depends on wire length absolutely: For long wires we quote force per unit length; total force still scales with overlap length.
• Only one wire exerts force on the other: Newton's third law holds — both wires feel equal and opposite forces.

Real-World Applications

• Definition and realisation of the ampere through the force between standard wires.
• Bracing of bus bars and transformer windings against short-circuit forces.
• Spacing and support design of overhead transmission lines, especially under fault conditions.

Two parallel currents interact through each other's magnetic fields, attracting or repelling with f = μ0 I1 I2/(2πd) — a small laboratory effect that becomes a major engineering force at high currents.