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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:

• 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.

1. W. Weber, Elektrodynamische Maassbestimmungen [Determinations of Electrodynamic Measure], Leipzig, 1846: p. 142 ff. of Weber's Werke vol. 3 or p. 81 (§18 or PDF p. 82) 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:

   • Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 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. (also here)

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)].

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:

• 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.

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:

   • Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 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. (also here)

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)].

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:

• 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.

1. W. Weber, Elektrodynamische Maassbestimmungen [Determinations of Electrodynamic Measure], Leipzig, 1846: p. 142 ff. of Weber's Werke vol. 3 or p. 81 (§18 or PDF p. 82) 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:

   • Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 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. (also here)

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)].

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:

• 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.

1. W. Weber, Elektrodynamische Maassbestimmungen [Determinations of Electrodynamic Measure], Leipzig, 1846: p. 142 ff. of Weber's Werke vol. 3 or p. 81 (§18 or PDF p. 82) 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:

   • Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 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. (also here)

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)].

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:

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

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.

1. W. Weber, Elektrodynamische Maassbestimmungen [Determinations of Electrodynamic Measure], Leipzig, 1846: p. 142 ff. of Weber's Werke vol. 3 or p. 81 (§18 or PDF p. 82) 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:

   • Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 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. (also here)

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)].

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:

• 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.

1. W. Weber, Elektrodynamische Maassbestimmungen [Determinations of Electrodynamic Measure], Leipzig, 1846: p. 142 ff. of Weber's Werke vol. 3 or p. 81 (§18 or PDF p. 82) 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:

   • Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 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. (also here)

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)].