Conservation and Innovation : Helmholtz's Struggle with Energy ...

solve the difficulty of the elastic-solid theory of reflection and refraction of light.
This footnote was the starting point of H. Lorentz’s 1875 doctoral thesis. A clear
schematization of the physical model of Helmholtz’s theory was given by Hertz
in 1892 388. Hertz distinguished two limiting cases, both refer to a conception of
action-at-a-distance starting from the charges on the plates that polarize the
dielectric between the plates. In the first case, the energy is supposed to be
concentrated on the plates and the actions (at a distance) produced by the
polarised dielectric are comparatively small. Ideally, removing B leaves some
action-at-adistance. In the second case, the energy is supposed to be concentrated
in the dielectric particles and so the dielectric action is the prevailing one. This
second case is mathematically equivalent to Maxwell’s. Helmholtz himself was
quite clear about the physical differences between his and Maxwell’s theory:
"The two theories are opposed to each other in a certain sense, since according to
the theory of magnetic induction originating with Poisson, which can be carried
through in a fully corresponding way for the theory of dielectric polarization of
insulators, the action-at-a-distance is diminished by the polarization, while
according to Maxwell’s theory on the other hand the action-at-a-distance is
exactly replaced by the polarization" 389
and
"It follows from these investigations that the remarkable analogy between the
motion of electricity in a dielectric and that of the light aether does not depend on
the particular form of Maxwell’s hypothesis, but results also in a basically similar
fashion if we maintain the older viewpoint about electrical action-at-adistance"
390.
Helmholtz’s comparison between the three theories had already led to
remarkable results: he judged Weber’s theory untenable (for the instability of
the equilibrium given by K < 0) and instead showed great interest in Maxwell’s
limit of his own theory. In addition, he considered this last approach suitable to
solve the difficulty of optical theory.
For my purposes, it is relevant to notice Helmholtz's shift of criticism towards
Weber’s law. Weber’s law is no longer considered inconsistent with the principle
388 Hertz, H. Untersuchungen über die Ausbreitung der elektrischen Kraft. Leipzig,
1892. Tr. by D.E.Jones Electric Waves. London: Mac Millan, 1892. Repr. New York: Dover,
1962. Pp.23-5.
389 Helmholtz "Beweg Elektr" pp.556-7.
390 Helmholtz "Beweg Elektr" p.558.