The Conservation Of Dynamical Energy

Shortly before the mid-19th century, the transformation of energy from one form to another was studied.

It had long been known:

• that the energy of motion and the energy of position of a dynamical system are convertible into each other, and
• that their sum remains invariable when the system is self-contained.

This principle of conservation of dynamical energy had been extended to optics by Fresnel. He had assumed[26] that the energy brought to an interface by incident light is equal to the energy carried away from the interface by the reflected and refracted beams.

A similar conception was involved in Roget’s and Faraday’s defence[27] of the chemical theory of the voltaic cell.

They argued that the work done by the current in the outer circuit must be provided at the expense of the chemical energy stored in the cell, and showed that the quantity of electricity sent round the circuit is proportional to the quantity of chemicals consumed, while its tension is proportional to the strength of the chemical affinities concerned in the reaction.

This theory was extended and completed by James Prescott Joule, of Manchester, in 1841.

Joule believed[28] that heat is producible from mechanical work and convertible into it, measured[29] the amount of heat evolved in unit time in a metallic wire, through which a current of known strength was passed.

He found the amount to be proportional to the resistance of the wire multiplied by the square of the current-strength; or (as follows from Ohm’s law) to the current-strength multiplied by the difference of electric tensions at the extremities of the wire.

The quantity of energy yielded ap as heat in the outer circuit being thus known, it became possible to consider the transference of energy in the circuit as a whole.

Joule writes:

In the same year, he[30] showed that the quantities of heat which are evolved by the combustion of the equivalents of bodies are proportional to the intensities of their affinities for oxygen, as measured by the electromotive force of a battery required to decompose the oxide electrolytically.

The theory of Roget and Faraday was thus perfected by Joule. It enables us to trace quantitatively the transformations of energy in the voltaic cell and circuit.

The primary source of energy is the chemical reaction. In a Daniell cell Zn|Zn SO4|Cu SO4|Cu for instance, it is the substitution of zinc for copper as the partner of the sulphion.

The strength of the chemical affinities concerned is in this case measured by the difference of the heats of formation of zinc sulphate and copper sulphate.

This determines the electromotive force of the cell.[31]

The amount of energy which is changed from the chemical to the electrical form in a given interval of time is measured by the product of the strength of the chemical affinity into the quantity of chemicals decomposed in that time, or (what is the same thing) by the product of the electromotive force of the cell into the quantity of electricity which is circulated. This energy may be either dissipated as heat in conformity to Joule’s law, or otherwise utilized in the outer circuit.

The importance of these principles was emphasized by:

• Hermann von Helmholtz (b. 1821, d. 1894), in an 1847 memoir and
• W. Thomson (Lord Kelvin) in 1851[32].

The equations have subsequently received only one important modification from Helmholtz.[33]

He pointed out that the electrical energy furnished by a voltaic cell need not be derived exclusively from the energy of the chemical reactions: for the cell may also operate by abstracting heat-energy from neighbouring bodies, and converting this into electrical energy. The extent to which this takes place is determined by a law which was discovered in 1855 by Thomson.[34]

Thomson showed that if E denotes the “available energy,” i.e., possible output of mechanical work, of a system maintained at the absolute temperature T, then a fraction of this work is obtained. This is not at the expense of the thermal or chemical energy of the system itself, but at the expense of the thermal energy of neighbouring bodies.

In the case of the voltaic cell, the principle of Roget, Faraday, and Joule is expressed by the equation:

where E denotes the available or electrical energy, which is measured by the electromotive force of the cell, and where λ denotes the heat of the chemical reaction which supplies this energy. In accordance with Thomson’s principle, we must replace this equation by

which is the correct relation between the electromotive force of a cell and the energy of the chemical reactions which occur in it.

In general, the term λ is much larger than the term TdE/dT; but in certain classes of cells—e.g., concentration-cells—λ is zero; in which case the whole of the electrical energy is procured at the expense of the thermal energy of the cells’ surroundings.

Helmholtz’s memoir of 1847, to which reference has already been made, bore the title, “On the Conservation of Force.”

It was originally read to the Physical Society of Berlin[35]. The younger physicists of the Society received it enthusiastically. But the prejudices of the older generation prevented its acceptance for the Annalen der Physik.

It was eventually published as a separate treatise.[36] This memoir asserted[37] that:

• the conservation of energy is a universal principle of nature
• the kinetic and potential energy of dynamical systems may be converted into heat according to definite quantitative laws, as taught by Rumford, Joule, and Robert Mayer[38].

Any of these forms of energy may be converted into the chemical, electrostatic, voltaic, and magnetic forces. The latter Helmholtz examined systematically.

Consider first the energy of an electrostatic field.

It will be convenient to suppose that the system has been formed by continually bringing from a very great distance infinitesimal quantities of electricity, proportional to the quantities already present at the various points of the system; so that the charge is always distributed proportionally to the final distribution.

Let e typify the final charge at any point of space, and V the final potential at this point.

Then at any stage of the process the charge and potential at this point will have the values λc and λV, where λ denotes a proper fraction. At this stage let charges edλ be brought from a great distance and added to the charges λe. The work required for this is

so the total work required in order to bring the system from infinite dispersion to its final state is

By reasoning similar to that used in the case of electrostatic distributions, it may be shown that the energy of a magnetic field, which is due to permanent magnets and which also contains bodies susceptible to magnetic induction, is

where ρ0 denotes the density of Poisson’s equivalent magnetization, for the permanent magnets only, and φ denotes the magnetic potential.[39]

Helmholtz, moreover, applied the principle of energy to systems containing electric currents.

For instance, when a magnet is moved in the vicinity of a current, the energy taken from the battery may be equated to the sum of that expended as Joulian heat, and that communicated to the magnet by the electromagnetic force:

This equation shows that the current is not proportional to the electromotive force of the battery, i.e. it reveals the existence of Faraday’s magneto-electric induction, As, however, Helmholtz was at the time unacquainted with the conception of the electrokinetic energy stored in connexion with a current, his equations were for the most part defective.

But in the case of the mutual action of a current and a permanent magnet, he obtained the correct result that the time-integral of the induced electromotive force in the circuit is equal to the increase which takes place in the potential of the magnet towards a current of a certain strength in the circuit.