The force between wires calculator above computes the mutual magnetic force between two long parallel conductors carrying steady currents. It is a force each wire exerts on the other, not a force applied from outside, and its size scales with the product of the two currents divided by their separation.
Arb Digital builds free physics calculators that each own a distinct job, and this corner of electromagnetism has three neighbouring pages that are easy to confuse. The magnetic field of a wire calculator gives the field a single conductor produces at a distance, and reports a force figure only for the special case of two wires carrying the same current. The Lorentz force calculator gives the force on a moving charge in a field. The magnetic force on a wire calculator gives the force a conductor feels in an externally supplied field. This page is the mutual case: two independent currents, each generating the field the other sits in, with the direction, the total force over a length and the reverse solutions included.
What This Force Between Wires Calculator Does
Two straight parallel conductors each produce a magnetic field that circles them. Each therefore sits in the other's field and feels a force. Because the geometry is symmetric, the two forces are equal and opposite, which they must be by Newton's third law, and the pair either pulls together or pushes apart depending on whether the currents run the same way.
The tool reports the force per unit length, which is the natural quantity because the total force grows without limit as the parallel run gets longer. It also gives the total across the run length you specify, the field one wire produces at the other's position, and a weight equivalent in grams so you have some feel for whether the force matters mechanically.
Two reverse modes are included. Given a target force per metre you can solve for the separation that produces it, or for the equal currents that would. Both come straight from rearranging the same expression, and both are useful when a clearance or a fault rating is the thing being designed against.
How to Use It
- Enter the two currents as magnitudes. Direction is handled by the selector rather than by signs, which keeps the arithmetic unambiguous.
- Give the centre-to-centre separation. The relation assumes thin wires, so it uses the distance between the conductor axes, not the gap between their surfaces.
- Choose the relative direction. Same direction gives attraction, opposite gives repulsion. This is the one input people most often get backwards.
- Set the length of the parallel run. Only the section where the two conductors actually run parallel counts toward the total force.
- Switch modes if you want a separation or a current. Enter a target force per metre and the tool solves the same relation the other way round.
The Formula: How the Force Is Calculated
The force per unit length between two long parallel conductors is F/L = μ0μrI1I2 ÷ (2πd), where d is the centre-to-centre separation. OpenStax University Physics Volume 2, section 12.3 on the magnetic force between two parallel currents, derives it in two steps: the field of the first wire at the second is μ0I1/(2πd), and the force on a length of the second wire carrying I2 in that field is B I2 L.
The permeability of free space used here is the 2022 CODATA value of 1.25663706127 × 10−6 N/A² published by NIST. Dividing it by 2π gives very nearly exactly 2 × 10−7, which is not a coincidence: until 2019 that number was fixed by definition, because the ampere itself was defined through this force.
NIST's introduction to the ampere sets out the old wording: the current which, maintained in two straight parallel conductors of infinite length and negligible cross-section placed one metre apart in vacuum, produces a force of 2 × 10−7 newtons per metre between them. Since May 2019 the ampere has instead been defined by fixing the elementary charge, and μ0 became a measured quantity with a small uncertainty rather than an exact one.
Work the defaults by hand. Two wires each carrying 10 A, 10 mm apart in air: F/L = (1.25663706 × 10−6 × 10 × 10) ÷ (2π × 0.01) = 1.25663706 × 10−4 ÷ 0.0628319 = 2.000 × 10−3 N/m. Cross-check it the two-step way: wire 1 produces B = 2 × 10−7 × 10 ÷ 0.01 = 2 × 10−4 T at wire 2, and the force on 1 m of wire 2 is 2 × 10−4 × 10 × 1 = 2 × 10−3 N. The two agree, and the force is about the weight of a fifth of a gram.
Why Like Currents Attract and Like Charges Do Not
The sign convention here trips up almost everyone at first, because it runs opposite to the electrostatic case. Two positive charges repel. Two currents flowing the same way attract. There is no inconsistency; the two situations are governed by different terms in the same theory.
The right-hand rule makes it concrete. Point your right thumb along the current in wire 1 and your fingers curl the way its field circles. At the position of wire 2 that field points into the page, say. Now apply the force rule to wire 2's current in that field, and the resulting force points back toward wire 1. Reverse wire 2's current and the force reverses with it.
There is a deeper way to see it. In relativity the magnetic force between currents is what the electrostatic repulsion between charges looks like from a frame in which those charges are moving. A neutral wire has equal positive and negative charge densities, and length contraction acting differently on the moving and stationary populations leaves a small net attraction. The magnetic force is a relativistic correction to Coulomb's law that happens to be large enough to run motors with. Our Coulomb's law calculator handles the electrostatic side of that comparison.
Where This Force Actually Matters
At ordinary currents the force is tiny. Two 10 A conductors a centimetre apart pull on each other with about the weight of a paperclip per metre, which no mechanical design would notice. The reason engineers care is that the force scales with the product of the currents, so it grows as the square when both rise together.
Take the same pair of conductors up to a 30 kA short-circuit current, as switchgear is designed to withstand, and the force per metre becomes 18,000 newtons — nearly two tonnes of force on every metre of run. That is why busbars in a distribution board are clamped at intervals and why the spacing of those clamps is a rated design parameter rather than a convenience. Antiparallel conductors, which is what a supply and return pair is during a fault, are pushed violently apart.
The same effect appears deliberately in railguns and electromagnetic pumps, and destructively in lightning-damaged conductors, which are sometimes found crushed rather than melted. It is also the mechanism behind the pinch effect in plasmas, where a current flowing through a conducting column squeezes it radially inward.
What the Thin-Wire Assumption Costs You
Every number here assumes two infinitely long, infinitely thin, straight parallel conductors in a uniform medium. Three of those assumptions are worth examining before trusting a result.
Finite length matters when the separation is not small compared with the run. For wires whose parallel section is much longer than their spacing the infinite-length result is excellent; when the two are comparable, the true force falls below it because the ends contribute less. Conductor thickness matters when the separation approaches the diameter: at that point the current is no longer concentrated on an axis, and in AC conditions the proximity effect redistributes it toward the facing surfaces, which raises the force above the thin-wire figure.
Non-straight geometry breaks the calculation entirely. Two parallel wires in a twisted pair or a coiled cable have continuously varying orientation, and the net force largely cancels — which is precisely why twisting a pair is such an effective mitigation. For a coil the correct treatment is inductance-based, and our solenoid inductance calculator is the right starting point.
Where This Sits Next to the Other Magnetism Tools
Use the magnetic field of a wire calculator when the field itself is the answer you want, or when you need the distance at which a field falls to some value. Use the magnetic force on a wire calculator when the field comes from a magnet or another source and you want the force on a conductor placed in it. Use the Lorentz force calculator when the moving thing is a single charged particle rather than a current in a conductor.
For related quantities, the magnetic field converter handles tesla and gauss, and the solenoid magnetic field calculator covers the coil geometry that most practical fields actually come from.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Getting the direction backwards — parallel currents attract and antiparallel currents repel, which is the reverse of the electrostatic rule people carry over by habit.
- Using surface-to-surface spacing — the relation needs the distance between conductor axes. For thick busbars close together, that difference is substantial.
- Forgetting the force is per unit length — a value in newtons per metre becomes a large total on a long run, and switchgear ratings depend on exactly that.
- Using nominal current for a mechanical check — the force scales with the product of the currents, so the prospective fault current, not the working current, sets the mechanical design.
- Applying it to coiled or twisted conductors — the derivation assumes straight parallel runs. Twisting largely cancels the force, which is why twisted pairs behave so differently.
Related Free Tools From Arb Digital
For the field a conductor produces, use the magnetic field of a wire calculator; for the force a conductor feels in an external field, the magnetic force on a wire calculator; and for a single moving charge, the Lorentz force calculator. Coil geometry is covered by the solenoid magnetic field calculator and the solenoid inductance calculator, unit questions by the magnetic field converter, and the electrostatic comparison by the Coulomb's law calculator. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
They attract when the currents run in the same direction and repel when the currents oppose. This is the opposite of the electrostatic case, where like charges repel, and it follows from applying the right-hand rule twice.
The force per unit length is the permeability times the product of the two currents, divided by two pi times the centre-to-centre separation. Multiply by the length of the parallel run for the total force.
Before 2019 the ampere was defined as the current that produces a force of two times ten to the minus seven newtons per metre between two wires one metre apart in vacuum. The ampere is now defined by fixing the elementary charge instead.
Because the permeability of free space is a very small number. Two 10 amp wires a centimetre apart pull on each other with about two millinewtons per metre, roughly the weight of a fifth of a gram.
Because the force goes as the product of the currents. At a 30 kiloamp fault current the same geometry gives tens of kilonewtons per metre, which is enough to bend or tear conductors that are not mechanically restrained.
Not while the separation is much larger than the diameter. Once they are comparable, the current can no longer be treated as concentrated on an axis, and in AC conditions the proximity effect pushes it toward the facing surfaces and raises the force.
It is an infinite-length result, so it is accurate when the parallel run is much longer than the separation. When the two are comparable, the real force is smaller because the ends contribute less field.
The calculation does not apply. Orientation changes continuously along a twisted pair and the net force largely cancels, which is one reason twisting is used. Coils need an inductance-based treatment instead.
This tool is provided for educational and study use. It models infinitely long, thin, straight parallel conductors in a uniform linear medium carrying steady currents, and does not account for finite length, conductor cross-section, skin and proximity effects or mechanical restraint, so treat its output as a physics result rather than an electrical design or safety calculation.
