What the Magnetic Force Between Wires Calculator does and when to use it
Two long parallel wires that carry current exert magnetic forces on each other. Each wire creates a magnetic field, and the other wire, sitting in that field, feels a force. This calculator finds that force per metre of wire, the total force over a chosen length, whether the wires attract or repel, and the field strength that wire 1 produces at wire 2.
It is used in physics courses, in estimating forces between bus bars and heavy conductors, and for understanding the historical definition of the ampere. Enter both currents, the centre-to-centre separation in metres, and optionally the length; leave the length blank for a per-metre answer over 1 m. Enter a negative value for one current to describe currents flowing in opposite directions.
Formula and method
The force per unit length between two long parallel wires is F/L = μ0 × I1 × I2 / (2π × d), where μ0 = 4π × 10⁻⁷ T·m/A. Multiplying by the length gives the total force. The magnetic field of wire 1 at the position of wire 2 is B = μ0 × I1 / (2π × d).
Currents in the same direction attract; currents in opposite directions repel. The calculator shows the magnitude in the force outputs and reports the direction in words.
- I1, I2 currents in amperes; opposite signs mean opposite directions.
- d separation between the wires' centres, in metres.
- L length over which the wires run parallel, in metres.
- μ0 permeability of free space, 4π × 10⁻⁷ T·m/A.
- B field of wire 1 at wire 2, shown in microtesla.
Worked example
Two parallel wires carry 10 A and 20 A in the same direction, 0.05 m apart, and run alongside each other for 2 m.
- Force per metre: (4π × 10⁻⁷ × 10 × 20) / (2π × 0.05) = 2 × 10⁻⁷ × 200 / 0.05 = 8.0 × 10⁻&sup4; N/m.
- Total force: 8.0 × 10⁻&sup4; × 2 = 1.6 × 10⁻³ N.
- Field of wire 1 at wire 2: 2 × 10⁻⁷ × 10 / 0.05 = 4.0 × 10⁻⁵ T = 40 µT.
- The currents are parallel, so the wires attract.
The calculator outputs 8.0000e-4 N/m, 1.6000e-3 N, attractive, and 40.000 µT. The force is tiny, about 0.16 grams-force, which is why the effect only becomes large with very high currents or very close conductors. With both currents at 1 A and a 1 m separation, the formula gives 2 × 10⁻⁷ N/m, the classic value once used to define the ampere.
Common mistakes and how to interpret the result
- Measuring the gap to the wire surface. The separation is between the wire axes, so add the wire radius on each side if you measured the gap between insulation surfaces.
- Using short or curved wires. The formula assumes very long straight parallel wires, so it is accurate when the separation is much smaller than the length. Near the ends or for loops it is only approximate.
- Forgetting direction. The magnitude is always positive in the output; use the interaction line to know whether the force pulls the wires together or pushes them apart.
- Applying it to alternating current without care. For AC the force fluctuates at twice the mains frequency, and this calculator gives the force for steady DC or an instantaneous current value.
Related calculators
- Magnetic Field of a Straight Wire — field around a single conductor.
- Force on a Current-Carrying Wire — force in an external field.
- Lorentz Force Calculator — force on a moving charge.
- Coulomb's Law Calculator — the electric counterpart between charges.