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

was interpreted by Helmholtz as a new electromotive force: the force of the
induced current. The law is similar to Neumann's, but more precise : Helmholtz
gave the value 1/a for what in Neumann was an indetermined constant 217.
This "demonstration" became very famous as an indication of the heuristic power
of the principle 218. Instead the fourth case 219 discussed in this section of the
Erhaltung , concerning the interactions between two currents, became famous for
the opposite reason: the lack of correct deductions from the principle.
Here Helmholtz simply extended the previous formulation: tension forces
provided by the batteries of the two circuits are
A 1 J 1 and A 2 J 2,
living forces identified with the heat produced by the current in the two circuits
are
J 2
1 W 1
and J21
W2 plus 1
a J1J dV
2 dt
and 1/a J1J2 dV/dt is interpreted as the living force of a circuit under the effect of
the current circulating in the other one
His claim that the results so obtained were in agreement with Weber's does not
imply that his result was correct; Helmholtz in fact dismissed two kinds of
potentials that, this time, really exist : the mutual potential of the two currents
(electrokinetic energy) and the potential of a current on itself (selfinduction) 220.
The conflicting outcome of the two cases discussed point to a serious problem: to
what extent Helmholtz's principle was a useful heuristic device? In fact the first,
correct, deduction was in a way a reinterpretation of already existing knowledge.
Helmholtz did not include terms that could have been reasonably included not to
violate Neumann's law. What about, for instance, the tension forces given by the
potential JV of the current on the magnet? Have they to be taken in account or
not? From a theoretical point of view we could easily give a formulation of the
217 Helmholtz Erhaltung P.65.
218 In 1873 it was still quoted and praised in Maxwell's Treatise Maxwell, James
Clerk. A Treatise on Electricity and Magnetism. 2 Vols. 1873; Thomson J.J. ed of 3rd edition
1891; repr New York: Dover,1954; Pp.190-3;but J.J.Thomson remarked in a note to third
edition that the law of induction cannot be deduced through the principle of conservation of
energy alone: another equation is needed. Ibidem P.192.
219 Helmholtz Erhaltung Pp.67-8.
220 see Planck Princip P.47