Was it known before Maxwell's time that the vacuum permitivity was inversely related to the permeability by a factor of $c^2$?

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The relation of the speed of light $c$ to electrodynamics was known before Maxwell.

In 1846, Weber derived his force law between point charges:1

$$F=\frac{ee'}{r^2}\left[1-\frac{1}{2c^2}\left(\frac{dr}{dt}\right)^2+\frac{1}{c^2}r\frac{d^2r}{dt^2}\right]$$

from Ampère's force law2 (not to be confused with one of Maxwell's equations, the Ampère circuital law) between current elements (written in modern vector notations and with modern units):

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

In the form Weber wrote his law, it involved a constant $a$ that is related to the speed of light $c$ by a factor of $\sqrt{2}$.

In 1856, Kohlrausch & Weber experimentally determined the constant.3

In 1857, Kirchoff explicitly tied this constant to the propagation of electricity in a wire:4

The velocity of propagation of an electric wave is here equal to $\frac{c}{\sqrt{2}}$; it is therefore independent of the cross-section of the wire, of its conductivity, and, finally, of the electric density; […] it is thus very near the speed of light in empty space.

See Assis's Weber's Electrodynamics and ch. 8 of Duhem's The Electric Theories of J. Clerk Maxwell: A Historical and Critical Study.

Also, this is a good historical overview:


References

  1. W. Weber, Elektrodynamische Maassbestimmungen [Determinations of Electrodynamic Measure], Leipzig, 1846: p. 142 ff. of Weber's Werke vol. 3 or p. 93 (§21 or PDF p. 94) ff. of this translation (also here). In 1848, Weber wrote a shorter paper, "On the Measurement of Electro-dynamic Forces" (also here); see spec. pp. 32-43 for the derivation of his law.

  2. First published English translation:

  3. R. Kohlrausch and W. Weber, Elektrodynamische Maassbestimmungen, insbesondere Zurückführung der Stromintensitäts-Messungen auf mechanische Maass, Leipzig, 1856. [English translation: Weber and Kohlrauch (2003)].

  4. G. Kirchhoff, Ueber die Bewegung der Elektricität in Drähten [On the motion of electricity in wires] (Poggendorff’s Annalen), Bd., 1857. [English translation: Kirchhoff (1857a)].

  • Wow, thank you for this detailed and informative answer. It has become clear to me that Maxwell's equations are in fact a very neat and pretty way of capturing electrodynamics but they hide a lot of complex machinery that requires some level of sophistication to properly understand. I have seen a rather glib demonstration of the conserved quantities like the poynting vector in Maxwell's equations and thought I had seen the last of it. I think I shall devote quite a bit more time to truly mastering electrodynamics. – Rascalniikov Jan 2 '17 at 15:08
  • @Rascalniikov Maxwell called Ampère the "Newton of electricity" and said Ampère's force law "must always remain the cardinal formula of electro-dynamics." (Treatise on E&M vol. 2, §528). – Geremia Jan 2 '17 at 22:52
  • @Rascalniikov Unfortunately, modern E&M textbooks (e.g., Jackson's) don't start with Ampère's force law that he ingeniously derived from 4 null-experiments, although they do discuss the related (but not equivalent) Biot-Savart Law. See also this good intro. article on Ampère's force law. – Geremia Jan 2 '17 at 22:59
  • @Rascalniikov Also, Peter Graneau did many experiments (beginning in circa 1980s) related to Ampère's force law (e.g., this Nature article by him). – Geremia Jan 2 '17 at 22:59
  • @Rascalniikov See the article I added: Rosenfeld, L. “The Velocity of Light and the Evolution of Electrodynamics.” Il Nuovo Cimento 4, no. 5 (September 1, 1956): 1630–69. doi:10.1007/BF02745315. – Geremia Jan 5 '17 at 15:32

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