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.

  • $\begingroup$ Crossposted from physics.stackexchange.com/q/270767/2451 $\endgroup$ – Qmechanic Jul 30 '16 at 21:16
  • $\begingroup$ Here is an explanation given, in German though: lp.uni-goettingen.de/get/text/6627 $\endgroup$ – Otto Jun 3 '17 at 16:07
  • $\begingroup$ @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." $\endgroup$ – Geremia Jun 3 '17 at 20:16

Oliver Heaviside's 1893 Electromagnetic Theory (vol. 1) mentions "Ampere's Rule [or 'formula' or 'law'] for deriving the magnetic force from the current" in a handful of places (cf. p. 64). He calls it "Ampère's 'dodge'" in his 1892 Electrical Papers (vol. 1) p. 261.

Probably the most curious statement by Heaviside on Ampère is in his paper "The Mutual Action of a Pair of Rational Current-Elements" (The Electrician, Dec. 28, 1888 (written: 25 Nov. 1888), p. 230 = Electrical Papers (vol. 2), p. 501); Heaviside ends the short paper with:

It has been stated, on no less authority than that of the great Maxwell[Treatise §528], that Ampère's law of force between a pair of current-elements is the cardinal formula of electrodynamics. If so, should we not be always using it? Do we ever use it? Did Maxwell, in his treatise? Surely there is some mistake. I do not in the least mean to rob Ampère of the credit of being the father of electrodynamics; I would only transfer the name of cardinal formula to another due to him, expressing the mechanical force on an element of a conductor supporting current in any magnetic field; the vector product of current and induction. There is something real about it; it is not like his force between a pair of unclosed elements; it is fundamental; and, as everybody knows, it is in continual use, either actually or virtually (through electromotive force) both by theorists and practicians.


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