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André-Marie Ampère investigated the mutual mechanical action of electric currents. Parallel currents in the same direction attract, while reversing one reverses the force; changing the circuit geometry provides further tests. James Clerk Maxwell's contemporary account distinguishes Ampère's experimental facts from the assumption used to model the interaction between individual current elements. The contribution therefore links an observable force between complete electrical circuits with a mathematical description of the interaction, rather than identifying the model's smallest parts directly with separately measured forces.^[ref]

Portrait source. Julien Léopold Boilly; Wellcome Collection, image V0000147, library reference ICV No 329. Creative Commons Attribution 4.0 International.

For a steady current density $J$, the magnetic field is

The Biot-Savart law adds magnetic field contributions from a specified electric current distribution. The vectors ${\mathbf{r}}^{\prime }$ and $\mathbf{r}$ locate source and observer; $\mathbf{r}-{\mathbf{r}}^{\prime }$ points toward the observer. Here ${\mu }_0$ is vacuum permeability. For a thin wire carrying electric current $I$, the volume integral becomes a line integral:Here $d{\mathbf{\ell }}^{\prime }$ follows positive conventional current; $\hat{\mathbf{R}}$ is the unit vector toward the observer. With fixed path orientation, reversing the electric current replaces $I$ by $-I$. The cross product and right-hand rule fix the direction. Reversing the source-to-observer displacement would give the wrong sign.^[ref]

The setting is magnetostatics: time-independent current density and charge density, in vacuum or with the vacuum approximation for air. Electric charge conservation then gives $\nabla \cdot \mathbf{J}=0$. A finite segment is part of a complete electrical circuit; its endpoints cannot continuously accumulate electric charge in a steady state. The instantaneous integral above does not describe switching transients or electromagnetic radiation. In magnetic matter, specify the total current density, including magnetization current, rather than automatically replacing ${\mu }_0$ by an arbitrary magnetic permeability.^[ref]

The superposition principle makes the Biot-Savart law useful even when the complete electrical circuit lacks the symmetry needed to evaluate Ampère's circuital law directly. Integrate a straight piece, a curved piece, and the return path separately, then add their magnetic fields as vectors. The beauty is that a local oriented current element produces both the circulating magnetic field of a finite straight wire and the reinforcing on-axis magnetic field of a circular current loop.

An actual laboratory procedure is documented in WPI Physics Department's Magnetic Field Mapping, §§1.2–1.2.3 and the single-coil checklist on p. 9. Preparation locates the active area of a Hall effect sensor using a small magnet. A field meter reads the component normal to that area. The wire apparatus is a wall-mounted square loop selectable between one and five conductors. The coil connects to a power supply and ammeter; its diameter and turn count are recorded. Align the sensor and scan distance. The control is a current-off background reading at each position, subtracted from the current-on reading. The inference compares corrected magnetic field with position and the predicted geometry dependence. Active-area displacement, sensor tilt, background variation, and finite coil width limit that comparison. These are instructions, not reported measurements.^[ref]

A further proposed control is to reverse $I$ at fixed sensor position and orientation. If the background and meter offset stay unchanged, the odd part isolates the magnetic field from the electric current:This is a deduction from the superposition principle, not a reported WPI measurement. The measured component, sensor position, and complete return path must still match the model.

A current element is the oriented infinitesimal wire contribution $Id{\mathbf{\ell }}^{\prime }$ inside the Biot-Savart law. For uniform axial current density across wire cross-sectional area $A$, $I=JA$ and $d^3{r}^{\prime }=Ad{\ell }^{\prime }$ imply $\mathbf{J}{d}^3{r}^{\prime }=Id{\mathbf{\ell }}^{\prime }$. This reduces a volume integral to a line integral in the thin-wire approximation. It is an integration contribution, not an isolated steady source. Conventional current runs opposite the drift of negatively charged electrons.

A uniform electrically short current element of length $\ell$ with sinusoidal peak current $I_0$ has electric dipole moment amplitude $p_0=I_0\ell /\omega$. The mean electric dipole radiation power is ${\omega }^4{p}_0^2/(12\pi {ϵ}_0{c}^3)$, so fixing current makes it proportional to ${\ell }^2{\omega }^2$, rather than ${\ell }^2{\omega }^4$. Current, charge excursion and dipole moment are different quantities to hold fixed when comparing antennas. The approximation requires $\ell \ll \lambda$ and does not impose the nonuniform standing-current profile of a long antenna.

For a circular current loop, the Biot-Savart law iswhere $\mathbf{s}$ points from a current element to the field point. Choose the loop in the $yz$ plane, with observation point $x\hat{\mathbf{x}}$. A parametrization consistent with positive axial magnetic field isThen $s=\sqrt{a^2+x^2}$ andThe transverse contributions cancel around the circle, leaving the on-axis magnetic field of a circular current loopThe right-hand rule fixes its direction; reversing the electric current reverses the magnetic field. At the centre this becomes ${\mu }_{0}I/(2a)$, and far away it decays as $|x{|}^{-3}$, as expected for a magnetic dipole.