# Why is one of Maxwell's equations named after Ampère? Who first named it after Ampère?

Ampère never wrote down what is confusingly called "Ampère's circuital law," not even the form without the displacement current term, as Ampère never dealt with the field concept.* Maxwell derived

$$\nabla \times \mathbf{B} = \mu_0\mathbf{J}\qquad(1)$$

in his 1855 paper On Faraday's Lines of Force, based on analogies to hydrodynamics, which he corrected to be

$$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \dfrac{\partial \mathbf{E}} {\partial t} \right)\qquad(2)$$

in his 1861 paper On Physical Lines of Force; he never wrote down Ampère's force law in either paper.

Ampère's force law is completely different from any of Maxwell's equations. It gives the force that current elements $I_1 d\vec {\ell }_1$ and $I_2 d\vec {\ell }_2$ exert on one another to be:

$$d^2\vec{F_{21}^A} = - \frac{\mu _0 }{4\pi }I_1 I_2 \frac{\hat {r}_{12} }{r_{12}^2 }\left[2(d\vec {\ell }_1 \cdot d\vec {\ell }_2) - 3({\hat {r}_{12} \cdot d\vec {\ell }_1 })({\hat {r}_{12} \cdot d\vec {\ell }_2 })\right] = - d^2\vec{F_{12}^A}.$$

Thus, it is appropriate that Equation (2) is one of Maxwell's equations. Gauss and Faraday utilized the field concept, thus Equation (2) is the most "Maxwellian" of the four Maxwell's equations.

So, why are Equations (1) & (2) above named after Ampère? Who first named them after Ampère?

*cf. Assis, André Koch Torres; Chaib, J. P. M. C; Ampère, André-Marie (2015). Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience (PDF). Montreal: Apeiron. ISBN 978-1-987980-03-5. ch. 15 pp. 221ff.

• Crossposted from physics.stackexchange.com/q/270767/2451 – Qmechanic Jul 30 '16 at 21:16
• Here is an explanation given, in German though: lp.uni-goettingen.de/get/text/6627 – Otto Jun 3 '17 at 16:07
• @Claus Thanks, but this is wrong, as Ampère never dealt with fields: "Ampère hatte empirisch gefunden, dass für das Magnetfeld $\oint\limits_{\partial A} \vec{B}\text{d}\vec{s}=\mu_0\cdot I = \mu_0\int\limits_A\vec{j}\text{d}\vec{A}$ gilt." – Geremia Jun 3 '17 at 20:16