Unveiling the Magnetic Force Between Parallel Conductors

Two parallel wires carrying current will either attract or repel each other. This simple setup leads to a clean, exact result for the force per unit length — and, for over a century, it served as the basis for the definition of the ampere itself.

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Problem Statement

Consider two infinitely long, straight, parallel conductors in vacuum, separated by a distance \(d\). Wire 1 carries a steady current \(I_1\) and wire 2 carries a steady current \(I_2\). Both currents flow in the same direction (we will later discuss the opposite case). Calculate the magnetic force per unit length \(F/L\) that one wire exerts on the other.

Hints

This is a two-step problem. First, determine the magnetic field produced by wire 1 at the location of wire 2 using Ampère's law in integral form. If you have not done this before, see our article on the magnetic field of an infinite wire. Second, use the Lorentz force law for a current-carrying conductor, \(\mathbf{F} = I\mathbf{L} \times \mathbf{B}\), to find the force on wire 2. Pay attention to the directions: the right-hand rule will tell you whether the wires attract or repel.

Solution

Step 1: Magnetic Field from Wire 1

We place wire 1 along the \(z\)-axis and wire 2 parallel to it at a distance \(d\). Using Ampère's law with a circular Amperian loop of radius \(d\) centred on wire 1, we have

\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_1 \,. \]

By the rotational symmetry of the infinite wire, the magnetic field is purely azimuthal and constant along the loop:

\[ B_1 \cdot 2\pi d = \mu_0 I_1 \quad \Longrightarrow \quad B_1 = \frac{\mu_0 I_1}{2\pi d} \,. \]

The direction of \(\mathbf{B}_1\) at the location of wire 2 is given by the right-hand rule. If \(I_1\) flows in the \(+z\)-direction and wire 2 sits at position \(+x\) from wire 1, then \(\mathbf{B}_1\) points in the \(-y\)-direction (i.e., toward wire 1 from the perspective of wire 2's plane).

Step 2: Force on Wire 2

A straight conductor of length \(L\) carrying current \(I_2\) in an external magnetic field \(\mathbf{B}\) experiences a force

\[ \mathbf{F} = I_2 \mathbf{L} \times \mathbf{B} \,. \]

Wire 2 carries current \(I_2\) in the \(+z\)-direction (same direction as wire 1), and the field from wire 1 at wire 2's position is \(\mathbf{B}_1 = B_1 (-\hat{y})\). Computing the cross product:

\[ \mathbf{F} = I_2 L\, \hat{z} \times B_1 (-\hat{y}) = -I_2 L B_1\, (\hat{z} \times \hat{y}) = -I_2 L B_1\, (-\hat{x}) = I_2 L B_1\, \hat{x} \,. \]

Wait — let us be more careful with the geometry. Place wire 1 at the origin and wire 2 at \(x = d\). The field \(\mathbf{B}_1\) at \(x = d\) points in the \(-\hat{y}\) direction. Then:

\[ \hat{z} \times (-\hat{y}) = -(\hat{z} \times \hat{y}) = -(-\hat{x}) = +\hat{x}\,? \]

No. Recall \(\hat{z} \times \hat{y} = -\hat{x}\), so \(\hat{z} \times (-\hat{y}) = +\hat{x}\). But wire 2 is at \(x = d\), and \(+\hat{x}\) points away from wire 1. That cannot be right for parallel currents, so let us re-examine the field direction. With wire 1 along \(z\) at the origin and using the right-hand rule, the field at the point \((d, 0, 0)\) curls in the \(\hat{\varphi}\) direction. In Cartesian coordinates, \(\hat{\varphi}\) at the point \((d, 0, 0)\) is \(+\hat{y}\), not \(-\hat{y}\). So:

\[ \mathbf{B}_1 = \frac{\mu_0 I_1}{2\pi d}\,\hat{y} \,. \]

Now the force on wire 2 is:

\[ \mathbf{F} = I_2 L\, \hat{z} \times \frac{\mu_0 I_1}{2\pi d}\,\hat{y} = \frac{\mu_0 I_1 I_2 L}{2\pi d}\,(\hat{z} \times \hat{y}) = \frac{\mu_0 I_1 I_2 L}{2\pi d}\,(-\hat{x}) \,. \]

The \(-\hat{x}\) direction points from wire 2 toward wire 1. The force is attractive, as expected for parallel currents. The magnitude of the force per unit length is:

\[ \boxed{\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}} \,. \]

Direction of the Force

The sign we just worked out confirms a general rule:

• Parallel currents (same direction): the wires attract each other.
• Antiparallel currents (opposite directions): the wires repel each other.

This can be understood intuitively. Between two parallel currents, the magnetic fields partially cancel, creating a region of lower field energy. The system lowers its energy by pulling the wires together. For antiparallel currents, the fields reinforce between the wires, and the system lowers its energy by pushing them apart.

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Background: From Force Law to the Definition of the Ampere

The force between parallel conductors is more than a textbook exercise — it played a central role in the international system of units for over a century.

The Old Definition (1948–2019)

In 1948, the 9th General Conference on Weights and Measures (CGPM) adopted a definition of the ampere based directly on the result we derived above [^1]:

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to \(2 \times 10^{-7}\) newton per metre of length.

This is simply our formula with \(I_1 = I_2 = 1\,\text{A}\) and \(d = 1\,\text{m}\):

\[ \frac{F}{L} = \frac{\mu_0 \cdot 1 \cdot 1}{2\pi \cdot 1} = \frac{4\pi \times 10^{-7}}{2\pi} = 2 \times 10^{-7}\;\text{N/m}\,. \]

Under this definition, the permeability of free space \(\mu_0 = 4\pi \times 10^{-7}\;\text{H/m}\) was exact by construction — it was not measured but fixed by the definition of the ampere.

The 2019 SI Redefinition

On 20 May 2019, the revised SI came into effect following the 26th CGPM's 2018 vote [^2]. The ampere is now defined by fixing the numerical value of the elementary charge:

\[ e = 1.602\,176\,634 \times 10^{-19}\;\text{C}\quad (\text{exact})\,. \]

Since one ampere is one coulomb per second, and one coulomb is now a definite number of elementary charges, the ampere is tied to a fundamental constant of nature rather than to a hypothetical mechanical experiment that nobody could actually perform with infinite wires. The second, in turn, is defined via the caesium-133 hyperfine transition frequency.

A practical consequence of the 2019 redefinition is that \(\mu_0\) is no longer exactly \(4\pi \times 10^{-7}\;\text{H/m}\). It must now be determined experimentally and has a small uncertainty. Current measurements give a value consistent with the old one to about one part in \(10^{10}\), so the numerical change is negligible for any engineering application — but the conceptual shift matters [^3].

Why Change?

The old definition had a serious practical weakness: you cannot build two infinite wires. Real experiments to realize the ampere through force measurements between finite coils (using a Kibble balance or its predecessors) were limited in precision. The new definition anchors the ampere to a countable quantity — the charge of the electron — which can be realized with far greater accuracy using single-electron tunnelling devices and quantum current standards [^2].

Practical Applications

The force between parallel conductors is not just a metrological curiosity. It shows up in a range of engineering contexts.

Busbars in power distribution. In electrical switchgear and power plants, heavy copper or aluminium busbars carry thousands of amperes. During a short-circuit fault, currents can spike to tens of thousands of amperes, and the resulting magnetic forces can be enormous — strong enough to deform or tear apart busbar assemblies if they are not properly braced. Busbar support structures are designed specifically to withstand these electromagnetic forces [^4].

Overhead power lines. The three-phase conductors in a high-voltage transmission line carry large currents and are spaced metres apart. The magnetic forces between phases are modest under normal operation (a few newtons per metre at most), but they contribute to conductor motion and must be considered alongside wind and ice loads in line design. During a fault, the forces increase dramatically and can cause conductors to clash — a phenomenon called "conductor slap" [^5].

Railguns. An electromagnetic railgun consists of two parallel conducting rails connected by a sliding armature. Current flows down one rail, through the armature, and back along the other rail. The antiparallel currents in the two rails repel each other (pushing the rails apart, which must be structurally contained), while the armature — carrying current perpendicular to the field produced by the rails — experiences a large Lorentz force that accelerates it along the barrel. The U.S. Navy's experimental railgun, for example, achieved muzzle energies of 33 MJ [^6].

The pinch effect in plasmas. In a cylindrical plasma carrying a large axial current (as in a Z-pinch), the current can be thought of as a bundle of parallel filaments all flowing in the same direction. By our result, parallel currents attract, so the plasma is compressed radially inward by its own magnetic field. This is the Bennett pinch, and it was one of the earliest approaches to controlled nuclear fusion. The equilibrium condition balances the inward magnetic pressure \(B^2 / 2\mu_0\) against the outward kinetic pressure of the plasma. Unfortunately, Z-pinches are plagued by magnetohydrodynamic instabilities (kink and sausage modes), which is why modern fusion research has moved to more stable configurations like tokamaks [^7].

Summary

Starting from Ampère's law and the Lorentz force, we derived the force per unit length between two parallel current-carrying conductors:

\[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}\,. \]

The force is attractive when the currents flow in the same direction, repulsive when they oppose. This clean result held a distinguished position in metrology as the basis for defining the ampere from 1948 to 2019, when the SI was reformed to anchor the ampere to the elementary charge instead. In engineering, the same force law governs the design of busbars, the dynamics of power line faults, the operation of railguns, and the physics of plasma pinches.

Note: This article was updated in March 2026 using Claude Opus 4.6.

References