Maxwell's equations

“ Other essential features of Maxwell's view are that electric, magnetic, and electromagnetic action is transmitted by means of stresses in a medium which possesses some sort of elasticity and inertia not exactly of an ordinary mechanical kind, and that the energy of all such action resides in the medium. 'Maxwell's equations', especially as modified by H. A. Lorentz so as to take account of the atomic structure of electricity, are fundamental in modern electrodynamics and the electron theory of matter. ”

“ After I had recognized that the existing theories, including mine, do not meet the requirements, I have asked me the question, if it is possible to be consistent with experience while keeping Maxwell's equations and the principle of equality of action and reaction. This is possible when we rely on the following principle: the ponderomotive force between two systems in relative uniform motion to each other, by consideration of the sign, is the force calculated from the Maxwell-Lorentz equations acting upon a system which is arbitrarily regarded as at rest. ”

“ It is without doubt one of the most important tasks of theoretical physics, to connect with each other the, at the beginning, completely isolated areas of mechanical and electromagnetic phenomena, and to derive the differential equations applying to any of them from a common foundation. Maxwell and Thomson and subsequently Boltzmann and Hertz, have taken the, at the beginning, surely natural way to chose mechanics as the foundation and to derive Maxwell's equations from it. ”

“ A greater and really serious danger is threatening from the side of the quantum theory, for the existence of amounts of energy that remain concentrated in small spaces during their propagation, to which several phenomena seem to point, is in absolute contradiction to Maxwell's equations. ”

“ A consideration of Einstein's theory shows, that if one nevertheless wants (with keeping this principle) to obtain agreement with my results, already Maxwell's equations of resting bodies must be modified, a step to which hardly anyone will decide himself for the time being.
We will rather remain at the assumption for the time being, that the physical phenomena depend on the relative motion to a very special coordinate system, which we call the absolutely resting aether. ”

“ Moreover, Maxwell's equations are after all only empirical laws, with a mystery of their own; it was a notable advance to connect the change of mass at high speeds with other phenomena whose strangeness has disappeared by long familiarity, but there is still scope for a more far-reaching explanation. Einstein takes us straight to the root of the mystery, and he clears up one point which was misleading, if not actually wrong, in the older explanation. The change of mass does not in any way depend on whether the mass is of electrical origin or not ”

“ Although the Gaussian aspect of the subject, which would simply assert that the primary atoms of matter exert actions on each other which are transmitted in time across space in accordance with Maxwell's equations, is a formally sufficient basis on which to construct physical theory, yet the question whether we can form a valid conception of a medium which is the seat of this transmission is of fundamental philosophical interest, quite independently of the fact that in default of the analogy at any rate of such a medium this theory would be too difficult for development. ”

“ The discussion was suggested by the fact that it has been proved that Maxwell's equations represent equally well the sequence of electromagnetic phenomena relative to a set of axes moving relative to the aether, as relative to a set of axes fixed in the aether. ”

“ The metrical, gravitational and electromagnetic fields are all included. We build the quantities enumerated above; and they obey the great laws of field-physics in virtue of the way in which they have been built. That is the special feature; the field laws—conservation of energy, mass, momentum and of electric [Pg 237] charge, the law of gravitation, Maxwell’s equations—are not controlling laws. [38] They are truisms. Not truisms when approached in the way the mind looks out on the world, but truisms when we encounter them in a building up of the world from a basal structure. ”

“ The general equations are next applied to the case of a magnetic disturbance propagated through a non-conductive field, and it is shown that the only disturbances which can be so propagated are those which are transverse to the direction of propagation, and that the velocity of propagation is the velocity v, found from experiments such as those of Weber, which expresses the number of electrostatic units of electricity which are contained in one electromagnetic unit.
This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself ”

“ In his image the friction wheels represent what we call electricity; in a conductor we must conceive them to be freely movable, whereas the centres of the larger wheels or balls have fixed positions. It is easily seen how a motion of translation imparted to the friction wheels can give rise to rotations of the larger balls; this is how a current produces a magnetic field. Other phenomena can be explained on the same lines; it is sufficient for this that Maxwell's equations can be deduced by means of the model. ”

“ Before proceeding to prove this theorem analytically we may observe, that if be taken to represent the components of the velocity of a homogeneous incompressible fluid of density unity, and if be taken to represent the potential at any point of space of forces acting on the fluid, then the general and superficial equations of continuity ( (1) and (2) ) indicate that every part of the space is, and continues to be, full of the fluid, and equation (4) is the condition to be fulfilled at a surface through which the fluid does not pass. ”

“ These are the analytical equations derived by Maxwell in his mathematical development of Faraday's views as to an electric medium: and they are the same as the equations arrived at by MacCullagh a quarter of a century earlier in his formulation of the dynamics of optical media. ”

“ While the manner in which Maxwell obtained his equations is all his own, he was not alone in stating and discussing general equations of the electro-magnetic field. The next steps which we are about to consider are, however, in a special manner due to him. An electrical or magnetic system is the seat of energy; this energy is partly electrical, partly magnetic, and various expressions can be found for it. In Maxwell’s theory it is a fundamental assumption that energy has position. ”

“ But when Maxwell had proved that his equations of electromagnetics could be thrown into a form compatible with the principle of action, and when he had succeeded in amalgamating electricity, magnetism and optics into one science, the universal validity of the principle was accepted. Inasmuch as this principle includes that of the conservation of energy, we can understand why the principle of action was often referred to as the supreme principle of physical science. ”

“ Since more recent observation has shown that the numerical results of Maxwell’s work connecting viscosity and temperature are erroneous, this last135 deduction does not hold; the inverse fifth power law of force will not give the correct relation between viscosity and temperature. Maxwell himself at a later date, “On the Stresses in Rarefied Gases,” Phil. Trans., 1879, realised this; but even in this last paper he adhered to the fifth power law because it leads to an important simplification in the equations to be dealt with. ”

“ Hence comes Maxwell’s electro-magnetic theory of light.
It is only in dielectric media that electric force is propagated by wave motion. In conductors, although the third and fourth of Maxwell’s principles given on page 185 still are true, the relation between the electric force and the electric current differs from that which holds in a dielectric. Hence the equations satisfied by the force are different. The laws of its propagation resemble those of the conduction of heat rather than those of the transmission of light. ”

“ Let the forces in the field be those due to the circuits A and B, then the electromagnetic momentum of A is
where and are the currents in A and B, and
Hence, if there is no motion of the circuit A,
where is a function of , and , which is indeterminate as far as regards the solution of the above equations, because the terms depending on it will disappear on integrating round the circuit. The quantity can always, however, be determined in any particular case when we know the actual conditions of the question. ”

“ As we know, subsequent experiment has justified Maxwell’s deep vision. It is much the same with the theory of relativity. One of the reasons for its great appeal to mathematicians is the extreme harmony, beauty and simplicity which it permits us to bestow on many of the equations of physics.
However, if we leave these aesthetic urges aside, we may say that the methodology that has yielded us the theory of relativity is the same methodology that has yielded all the great scientific discoveries. It is the methodology of Galileo, Newton and Maxwell: First, ascertain experimental facts ”

“ Such a description of the methods of pure dynamics is not unnecessary, because Lagrange and most of his followers, to whom we are indebted for these methods, have in general confined themselves to a demonstration of them, and, in order to devote their attention to the symbols before them, they have endeavoured to banish all ideas except those of pure quantity, so as not only to dispense with diagrams, but even to get rid of the ideas of velocity, momentum, and energy, after they have been once for all supplanted by symbols in the original equations. ”

“ The above equations, together with those which express the proportionality of E to D in insulators, and to ι in conductors, constituted Maxwell's system for a field formed by isotropic bodies which are not in motion. When the magnetic field is due entirely to currents (including both conduction-currents and displacement-currents) , so that there is no magnetization, we have
so that the vector-potential is connected with the total current by an equation of the same form as that which connects the scalar potential with the density of electric charge. ”

“ Accordingly, we find Euler and D’Alembert devoting their talent and their patience to the establishment of the laws of the rotation of solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligible propositions supersede equations. ”

“ If we write we obtain from these equations Hence, either , in which case the wave is not propagated at all; or, , which leads to the equation for given by Fresnel; or the quantities within brackets vanish, in which case the vector whose components are , , is normal to the wave-front and proportional to the electric volume-density. Since the medium is a non-conductor, the electric density at any given point is constant, and therefore the disturbance indicated by these equations is not periodic, and cannot constitute a wave. ”

“ The resultant of all the vectors thus found is the potential of the whole current. Since the current is a vector quantity, its potential is also a vector. See Art. 422.
When the distribution of electric currents is given, there is one, and only one, distribution of the values of , such that , is every where finite and continuous, and satisfies the equations
and vanishes at an infinite distance from the electric system. This value is that given by equations (5) , which may be written
Quaternion Expressions for the Electromagnetic Equations. ”

“ While Hertz's mechanics is obviously aimed at presenting the electromagnetic equations as consequences, the relation is directly reversed here. With respect to logical form, the electromagnetically founded mechanics of course cannot compare itself to Hertz's one, just because the system of Maxwell's differential equations has found no precise-critical treatment at all, but it has, as it seems to me, a very remarkable advantage, namely (as it was shown) that it transcends ordinary mechanics, which therefore is only to be described as a first approximation. ”

“ The line integral,
,
is zero, because the curve , is reduced to a point at which there is but one value of and of .
The two line integrals,
+ ,
destroy each other, because the point is identical with the point .
The expressions (8) is therefore reduced to
Since the curve is identical with the closed curve s, we may write this expression
p. 80, in equations (3) , (4) , (6) , (e) , (17) , (18) , (19) , (20) , (21) , (22) , for read . ”

“ If we treat as covariant, then this is evidently somewhat arbitrary in terms of Cartesian orthogonal coordinates, since covariants and contravariantes coincide here. If we would require (following Abraham) that the transform like the products of the corresponding components of the radius vector, then we would have to postulate the as contravariants. By passage to the reciprocal system we can, however, represent them as convariants. For the sake of being easier related to the integral forms, we stick to the covariant description. ”