The Conservation Of Dynamical Energy

The correct theory of the energy of magnetic and electromagnetic fields is due mainly to W. Thomson (Lord Kelvin). Thomson’s researches on this subject commenced with one or two short investigations regarding the ponderomotive forces which act on temporary magnets.

In 1847, he discussed[40] the case of a small iron sphere placed in a magnetic field, showing that it is acted on by a ponderomotive force represented by - grad cR2, where c denotes a constant, and R denotes the magnetic force of the field; such a sphere must evidently tend to move towards the places where R2 is greatest.

The same analysis may be applied to explain why diamagnetic bodies tend to move, as in Faraday’s experiments, from the stronger to the weaker parts of the field.

Two years later, Thomson presented to the Royal Society a memoir[41] in which the results of Poisson’s theory of magnetism were derived from experimental data, without making use of the hypothesis of magnetic fluids.

This was followed in 1850 by a second memoir,[42] in which Thomson drew attention to the fact previously noticed by Poisson,[43] that the magnetic intensity at a point within a magnetized body depends on the shape of the small cavity in which the exploring magnet is placed.

Thomson distinguished two vectors;[44] one of these, by later writers generally denoted by B, represents the magnetic intensity at a point situated in a small crevice in the magnetized body, when the faces of the crevice are at right angles to the direction of magnetization; the vector B is always circuital.

The other vector, generally denoted by H, represents the magnetic intensity in a narrow tubular cavity tangential to the direction of magnetization; it is an irrotational vector, The magnetic potential tends at any point to a limit which is independent of the shape of the cavity in which the point is situated; and the space-gradient of this limit is identical with H. Thomson called B the “magnetic force according to the electro-magnetic definition,” and H the “magnetic force according to the polar definition”; but the names magnetic induction and magnetic force, proposed by Maxwell, have been generally used by later writers.

It may be remarked that the vector to which Faraday applied the term “magnetic force,” and which he represented by lines of force, is not H, but B; for the number of unit lines of force passing through any gap must depend only on the gap, and not on the particular diaphragm filling up the gap, across which the flux is estimated; and this can be the case only if the vector which is represented by the lines of force is a circuital vector.

Thomson introduced a number of new terms into magnetic science—as indeed he did into every science in which he was interested. The ratio of the measure of the induced magnetization Ii, in a temporary magnet, to the magnetizing force H, he named the susceptibility; it is positive for paramagnetic and negative for diamagnetic bodies, and is connected with Poisson’s constant kp[45] by the relation

where κ denotes the susceptibility. By an easy extension of Poisson’s analysis it is seen that the magnetic induction and magnetic force are connected by the equation

where I denotes the total intensity of magnetization: so if I0, denote the permanent magnetization, we have

where μ denotes (1 + 4πκ): μ was called by Thomson the permeability.

In 1851 Thomson extended his magnetic theory so as to include magnecrystallic phenomena. The mathematical foundations of the theory of magnecrystallic action had been laid by anticipation, long before the experimental discovery of the phenomenon, in a memoir read by Poisson to the Academy in February, 1824.

Poisson, as will be remembered, had supposed temporary magnetism to be due to “magnetic fields,” movable within the infinitely small “magnetic elements” of which he assumed magnetizable matter to be constituted. He had not overlooked the possibility that in crystals these magnetic elements might be non-spherical (e.g. ellipsoidal), and symmetrically arranged; and had remarked that a portion of such a crystal, when placed in a magnetic field, would act in a manner depending on its orientation. The relations connecting the induced magnetization I with the magnetizing force H he had given in a form equivalent to

Thomson now[46] showed that the nine coefficients a, b′, c′′ …, introduced by Poisson, are not independent of each other. kor a sphere composed of the magnecrystalline substance, iſ placed in a uniform field of force, would be acted on by a couple: and the work done by this couple when the sphere, supposed of unit volume, performs a complete revolution round the axis of x may be easily shown to be

But this work must be zero, since the system is restored to its primitive condition; and hence b′′ and c′ must be equal. Similarly c′′ =a′, and a′′ = b′. By change of axes three more coefficients may be removed, so that the equations may be brought to the form

where κ1, κ2, κ3 may be called the principal magnetic susceptibilities.

In the same year (1851) Thomson investigated the energy which, as was evident from Faraday’s work on self-induction, must be stored in connexion with every electric current. He showed that, in his own words,[47] “the value of a current in a closed conductor, left without electromotive force, is the quantity of work that would be got by letting all the infinitely small currents into which it may be divided along the lines of motion of the electricity come together from an infinite distance, and make it up.

Each of these ‘infinitely small currents’ is of course in a circuit which is generally of finite length; it is the section of each partial conductor and the strength of the current in it that must be infinitely small.”

Discussing next the mutual energy due to the approach of a permanent magnet and a circuit carrying a current, he arrived at the remarkable conclusion that in this case there is no electrokinetic energy which depends on the mutual action; the energy is simply the sum of that due to the permanent magnets and that due to the currents.

If a permanent magnet is caused to approach a circuit carrying a current, the electromotive force acting in the circuit is thereby temporarily increased; the amount of energy dissipated as Joulian heat, and the speed of the chemical reactions in the cells, are temporarily increased also.

But the increase in the Joulian heat is exactly equal to the increase in the energy derived from consumption of chemicals, together with the mechanical work done on the magnet by the operator who moves it; so that the balance of energy is perfect, and none needs to be added to or taken from the electrokinetic form.

Helmholtz escaped in this case the errors into which he was led in other cases by his neglect of electrokinetic energy; for in this case there was no electrokinetic energy to neglect.

Two years later, in 1853, Thomson[48] gave a new form to the expression for the energy of a system of permanent and temporary magnets.

The energy of such a system is represented by

where ρe denotes the density of Poisson’s equivalent magnetization for the permanent magnets, and φ denotes the magnetic potential, and where the integration may be extended over the whole of space. Substituting for ρ0 its value - div I0,[49] the expression may be written in the form

or, integrating by parts,

Since B = μH + 4πI0, this expression may be written in the form

but the former of these integrals is equivalent to

which vanishes, since B is a circuital vector. The energy of the field, therefore, reduces to

integrated over all space; which is equivalent to Thomson’s form.[50]

In the same memoir Thomson returned to the question of the energy which is possessed by a circuit in virtue of an electric current circulating in it. As he remarked, the energy may be determined by calculating the amount of work which must be done in and on the circuit in order to double the circuit on itself while the current is sustained in it with constant strength; for Faraday’s experiments show that a circuit doubled on itself has no stored energy. Thomson found that the amount of work required may be expressed in the form 1 2 Li2, where i denotes the current strength, and L, which is called the coefficient of self-induction, depends only on the form of the circuit.

It may be noticed that in the doubling process the inherent ​electrodynamic energy is being given up, and yet the operator is doing positive work. The explanation of this apparent paradox is that the energy derived from both these sources is being used to save the energy which would otherwise be furnished by the battery, and which is expended in Joulian heat.

Thomson next proceeded[51] to show that the energy which is stored in connexion with a circuit in which a current is flowing may be expressed as a volume-integral extended over the whole of space, similar to the integral by which he had already represented the energy of a system of permanent and temporary magnets. The theorem, as originally stated by its author, applied only to the case of a single circuit; but it may be established for a system formed by any number of circuits in the following way:—

If Ns denote the number of unit tubes of magnetic induction which are linked with the sth circuit, in which a current is is flowing, the electrokinetic energy of the system is

which may be written

I_{r}}, where Ir, denotes the total current flowing through the gap formed by the rth path unit tube of magnetic induction. But if H denote the (vector) magnetic force, and H its numerical magnitude, it is known that (1/4π)∫ Hds, integrated along a closed line of magnetic induction, measures the total current flowing through the gap formed by the line. The energy is therefore

, the summation being extended over all the unit tubes of magnetic induction, and the integration being taken along them. But if dS denote the cross-section of one of these tubes, we have BdS = 1, where B denotes the numerical magnitude of the magnetic induction B: so the energy is

, where the integration is extended over the whole of space; and since in the present case B = μH, the energy may also be represented by

But this is identical with the form which was obtained for a field due to permanent and temporary magnets. It thus appears that in all cases the stored energy of a system of electric currents and permanent and temporary magnets is

where the integration is extended over all space.

It must, however, be remembered that this represents only what in thermodynamics is called the “available energy”; and it must further be remembered that part even of this available energy may not be convertible into mechanical work within the limitations of the system: e.g., the electrokinetic energy of a current flowing in a single closed perfectly conducting circuit cannot be converted into any other for so long as the circuit is absolutely rigid. All that we can say is that the changes in this stored electrokinetic energy correspond to the work furnished by the system in any change.

The above form suggests that the energy may not be localized in the substance of the circuits and magnets, but may be distributed over the whole of space, an amount (μH2/8π) of energy being contained in each unit volume. This conception was afterwards adopted by Maxwell, in whose theory it is of fundamental importance.