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

surveying the region of electromagnetics, and working out the distinctive
consequences of the various theories, in order, wherever that was possible, to
decide between them by suitable experiments." 382.
In fact, in 1870, he started publishing a series of papers that constituted a
comprehensive study of electrodynamics. "With the object of clearing up this
confusion", he gave a theory, whose mathematical version included Weber, F.
Neumann and Maxwell’s theory as limiting cases. Helmholtz’s starting point is
relevant: it is the assumption of the existence not only of a force between current
elements but of a force and a torque. Both these assumptions contrast with
Ampére’s theory. Ampére admitted a potential only for closed currents and only
a central force between current elements. Helmholtz’s starting point was a
generalisation of F. Neumann’s theory: he gave the most general expression for
the energy of two current elements, consistent with the condition that the force
between two closed circuits should be the one given by Ampere:
where A is a constant depending on the unit of current, r the distance between the
elements of circuit Ds and Dσ traversed by the currents i and j. K too is a
constant, for K=+1 Helmholtz’s potential reduces to F. Neumann’s; for K=-1,
Weber’s potential is obtained; for K=0 plus a dielectric medium, Maxwell’s
theory can be obtained 383. An expression giving both F. Neumann’s and Weber’s
electrodynamic energies is:
where h is an arbitrary constant. The first term is Neumann’s value. The second
term is the difference between Weber’s and Neumann’s values. The second term
vanishes for closed currents, the only case analysed by F. Neumann 384.
In the fourth paragraph Helmholtz analyses the different theories with respect to
energy values 385. The whole energy Φ is the sum of the electrodynamic
382 Ibidem
383 Helmholtz, Hermann. "Ueber die Bewegungsgleichungen der Elektricität für
ruhende leitende Körper." In Borchardt's Journ 72 (1870): 57-129; repr. in WA 1 pp.545-628.
Pp. 549 and 567.
384 Ibidem pp.565-6.
385 Ibidem pp.579-85.