In November 1825, French mathematician, physicist, and natural philosopher André-Marie Ampère (1775–1836) published his definitive 175-page mathematical treatise titled “Mémoire sur la théorie mathématique des phénomènes électrodynamiques uniquement déduite de l’expérience” (“Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience”).

Ampère’s treatise synthesized five years of intensive theoretical and experimental research following Hans Christian Ørsted’s 1820 discovery. Ampère coined the term Electrodynamics to distinguish the physics of moving electric charges from electrostatics, formulated the exact mechanical force law between electric currents, established Ampère’s Circuital Law, and proposed microscopic molecular electric currents—earning him James Clerk Maxwell’s praise as the “Newton of Electricity”.

Ampère’s Force Law Between Current Elements

To place electrodynamics on the same rigorous Newtonian foundation as celestial mechanics, Ampère derived the elemental force $d^2 \mathbf{F}$ exerted by an infinitesimal current element $I_1 d\boldsymbol{\ell}_1$ at position $\mathbf{r}_1$ on a second current element $I_2 d\boldsymbol{\ell}_2$ at position $\mathbf{r}_2$:

$$d^2 \mathbf{F}_{12} = -\frac{\mu_0}{4\pi} \frac{I_1 I_2}{r^2} \left[ 2 (d\boldsymbol{\ell}_1 \cdot d\boldsymbol{\ell}_2) \hat{\mathbf{r}} - 3 (d\boldsymbol{\ell}_1 \cdot \hat{\mathbf{r}})(d\boldsymbol{\ell}_2 \cdot \hat{\mathbf{r}}) \hat{\mathbf{r}} \right]$$

Where $\mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1$, $r = |\mathbf{r}|$, and $\hat{\mathbf{r}} = \mathbf{r} / r$.

Parallel Conducting Wires

Integrating Ampère’s element force along two parallel straight conductors carrying steady currents $I_1$ and $I_2$ separated by distance $d$ yields the classic Ampère Force Law:

$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$

  • Parallel Currents ($I_1 \uparrow \uparrow I_2$): Exert an attractive force.

  • Antiparallel Currents ($I_1 \uparrow \downarrow I_2$): Exert a repulsive force.

This quantitative force law served for over a century as the international scientific definition of the Ampere ($\text{A}$), the base SI unit of electric current.

Ampère’s Circuital Law & Solenoidal Fields

Ampère formulated the fundamental integral theorem connecting line integrals of magnetic field $\mathbf{B}$ along a closed loop $C$ to the total electric current $I_{\text{enc}}$ passing through any surface bounded by $C$:

$$\oint_C \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I_{\text{enc}}$$

In differential vector calculus notation (developed later by Oliver Heaviside), Ampère’s law takes the form:

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$

Where $\mathbf{J}$ is the electric current density vector field.

Invention of the Solenoid

To demonstrate that a helix of current-carrying wire behaves identically to a permanent bar magnet, Ampère invented and named the Solenoid (from Greek solen, meaning tube). He proved that inside an infinitely long solenoid with $n$ turns per unit length carrying current $I$, the magnetic field is uniform and parallel to the axis:

$$B_{\text{solenoid}} = \mu_0 n I$$

Ampère’s Molecular Current Hypothesis

Prior to 1825, magnetism in iron, nickel, and lodestones was attributed to hypothetical “magnetic fluids” ($N$ and $S$ pole fluids) residing inside iron atoms.

Ampère proposed a revolutionary physical hypothesis: all magnetism is electrodynamic in origin. He asserted that permanent magnetic materials contain no magnetic fluids; instead, every atom contains microscopic, frictionless circular loops of electric current—Ampèrian molecular currents ($I_{\text{mol}}$).

When an external magnetic field is applied, these randomly oriented microscopic current loops align parallel to one another, producing macroscopic ferromagnetism:

$$\mathbf{M} = \sum_{i} \boldsymbol{\mu}_i = \sum_{i} I_{\text{mol}} \mathbf{A}_i$$

Where $\mathbf{M}$ is macroscopic magnetization and $\boldsymbol{\mu}_i$ is the atomic magnetic dipole moment.

Bridge to Quantum Mechanics: Atomic Orbitals & Electron Spin

Ampère’s 1825 molecular current hypothesis was a prophetically accurate physical model that directly prefigured 20th-century quantum physics:

1. Quantum Orbital Magnetic Dipole Moment

In 1913, Niels Bohr proved that an electron orbiting a hydrogen nucleus at speed $v$ in a circular orbit of radius $r$ constitutes an Ampèrian molecular current $I = e / T = e v / (2\pi r)$. Quantizing orbital angular momentum $L = m_e v r = n \hbar$ yields the quantum orbital magnetic moment:

$$\boldsymbol{\mu}_L = -\frac{e}{2m_e} \mathbf{L} = - \frac{g_L \mu_{\text{B}}}{\hbar} \mathbf{L}$$

Where $\mu_{\text{B}} = \frac{e\hbar}{2m_e} \approx 9.274 \times 10^{-24}\,\text{J/T}$ is the Bohr Magneton.

2. Intrinsic Electron Spin & Einstein-de Haas Effect

In 1915, Albert Einstein and Wander Johannes de Haas performed their famous rotation experiment, proving experimentally that magnetizing an iron cylinder induces mechanical rotation—confirming Ampère’s 1825 prediction that magnetism arises from microscopic angular momentum.

In 1925 (exactly 100 years after Ampère’s treatise), George Uhlenbeck and Samuel Goudsmit proved that the electron possesses an intrinsic quantum spin $\mathbf{S}$, producing a spin magnetic moment:

$$\boldsymbol{\mu}_S = - g_S \frac{e}{2m_e} \mathbf{S} \approx - 2 \left( \frac{e}{2m_e} \right) \mathbf{S}$$

Fulfilling Ampère’s vision of microscopic electric currents as the fundamental source of all magnetism in the universe.

Key Takeaways

  • Year: 1825

  • Key Figure: André-Marie Ampère (French Mathematician & Physicist)

  • Core Achievement: Published “Mémoire sur la théorie mathématique des phénomènes électrodynamiques”, establishing the mathematical foundation of electrodynamics.

  • Force & Field Laws: Derived the quantitative force between current conductors ($F/L = \mu_0 I_1 I_2 / 2\pi d$) and formulated Ampère’s Circuital Law ($\oint \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I$).

  • Molecular Currents: Postulated that atomic-scale electric currents cause all permanent magnetism, inventing the solenoid.

  • Quantum Relevance: Directly prefigured the Bohr magneton ($\mu_{\text{B}}$), orbital magnetic moments ($\boldsymbol{\mu}_L$), and quantum electron spin ($\boldsymbol{\mu}_S$).