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Reference: Evolution of Physics

This paper presents Chapter III, section 3 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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The Reality of the Field

The quantitative, mathematical description of the laws of the field is summed up in what are called Maxwell’s equations. The facts mentioned so far led to the formulation of these equations, but their content is much richer than we have been able to indicate. Their simple form conceals a depth revealed only by careful study.

The formulation of these equations is the most important event in physics since Newton’s time, not only because of their wealth of content, but also because they form a pattern for a new type of law.

The characteristic features of Maxwell’s equations, appearing in all other equations of modern physics, are summarized in one sentence. Maxwell’s equations are laws representing the structure of the field.

Maxwell’s equations are laws representing the structure of the field.

Why do Maxwell’s equations differ in form and character from the equations of classical mechanics? What does it mean that these equations describe the structure of the field? How is it possible that, from the results of Oersted’s and Faraday’s experiments, we can form a new type of law, which proves so important for the further development of physics?

We have already seen, from Oersted’s experiment, how a magnetic field coils itself around a changing electric field. We have seen, from Faraday’s experiment, how an electric field coils itself around a changing magnetic field. To outline some of the characteristic features of Maxwell’s theory, let us, for the moment, focus all our attention on one of these experiments, say, on that of Faraday. We repeat the drawing in which an electric current is induced by a changing magnetic field. We already know that an induced current appears if the number of lines of force, passing the surface bounded by the wire, changes. Then the current will appear if the magnetic field changes or the circuit is deformed or moved: if the number of magnetic lines passing through the surface is changed, no matter how this change is caused. To take into account all these various possibilities, to discuss their particular influences, would necessarily lead to a very complicated theory. But can we not simplify our problem? Let us try to eliminate from our considerations everything which refers to the shape of the circuit, to its length, to the surface enclosed by the wire. Let us imagine that the circuit in our last drawing becomes smaller and smaller, shrinking gradually to a very small circuit enclosing a certain point in space. Then everything concerning shape and size is quite irrelevant. In this limiting process where the closed curve shrinks to a point, size and shape automatically vanish from our considerations and we obtain laws connecting changes of magnetic and electric field at an arbitrary point in space at an arbitrary instant.

Thus, this is one of the principal steps leading to Maxwell’s equations. It is again an idealized experiment performed in imagination by repeating Faraday’s experiment with a circuit shrinking to a point.

We should really call it half a step rather than a whole one. So far our attention has been focused on Faraday’s experiment. But the other pillar of the field theory, based on Oersted’s experiment, must be considered just as carefully and in a similar manner. In this experiment the magnetic lines of force coil themselves around the current. By shrinking the circular magnetic lines of force to a point, the second half-step is performed and the whole step yields a connection between the changes of the magnetic and electric fields at an arbitrary point in space and at an arbitrary instant.

The Maxwell equations are obtained by shrinking Oersted’s and Faraday’s circuits to an idealized point, such that everything concerning shape and size of the circuits become quite irrelevant.  We then obtain laws connecting the changes of the magnetic and electric fields at an arbitrary point in space and at an arbitrary instant.

But still another essential step is necessary. According to Faraday’s experiment, there must be a wire testing the existence of the electric field, just as there must be a magnetic pole, or needle, testing the existence of a magnetic field in Oersted’s experiment. But Maxwell’s new theoretical idea goes beyond these experimental facts. The electric and magnetic field or, in short, the electromagnetic field is, in Maxwell’s theory, something real. The electric field is produced by a changing magnetic field, quite independently, whether or not there is a wire to test its existence; a magnetic field is produced by a changing electric field, whether or not there is a magnetic pole to test its existence.

Maxwell’s theoretical ideas goes beyond the experimental fact that there must be a wire testing the existence of the electric field, just as there must be a magnetic pole, or needle, testing the existence of a magnetic field. The electromagnetic field is something quite independently real.

Thus two essential steps led to Maxwell’s equations. The first: in considering Oersted’s and Rowland’s experiments, the circular line of the magnetic field coiling itself around the current and the changing electric field had to be shrunk to a point; in considering Faraday’s experiment, the circular line of the electric field coiling itself around the changing magnetic field had to be shrunk to a point. The second step consists of the realization of the field as something real; the electromagnetic field once created exists, acts, and changes according to Maxwell’s laws.

Maxwell’s equations describe the structure of the electromagnetic field. All space is the scene of these laws and not, as for mechanical laws, only points in which matter or charges are present.

According to Maxwell, all space is the scene of these laws and not only the points in which matter or charges are present. Therefore, the duality of matter and void, as in the mechanical view, is eliminated.

We remember how it was in mechanics. By knowing the position and velocity of a particle at one single instant, by knowing the acting forces, the whole future path of the particle could be foreseen. In Maxwell’s theory, if we know the field at one instant only, we can deduce from the equations of the theory how the whole field will change in space and time. Maxwell’s equations enable us to follow the history of the field, just as the mechanical equations enabled us to follow the history of material particles.

Maxwell’s equations enable us to follow the history of the field, just as the mechanical equations enabled us to follow the history of material particles.

But there is still one essential difference between mechanical laws and Maxwell’s laws. A comparison of Newton’s gravitational laws and Maxwell’s field laws will emphasize some of the characteristic features expressed by these equations.

With the help of Newton’s laws we can deduce the motion of the earth from the force acting between the sun and the earth. The laws connect the motion of the earth with the action of the far-off sun. The earth and the sun, though so far apart, are both actors in the play of forces.

In Maxwell’s theory there are no material actors. The mathematical equations of this theory express the laws governing the electromagnetic field. They do not, as in Newton’s laws, connect two widely separated events; they do not connect the happenings here with the conditions there. The field here and now depends on the field in the immediate neighbourhood at a time just past. The equations allow us to predict what will happen a little farther in space and a little later in time, if we know what happens here and now. They allow us to increase our knowledge of the field by small steps. We can deduce what happens here from that which happened far away by the summation of these very small steps. In Newton’s theory, on the contrary, only big steps connecting distant events are permissible. The experiments of Oersted and Faraday can be regained from Maxwell’s theory, but only by the summation of small steps each of which is governed by Maxwell’s equations.

In Newton’s theory only big steps connecting distant events are permissible. In Maxwell’s theory, the field here and now depends on the field in the immediate neighborhood at a time just past. We can deduce what happens here from that which happened far away by the summation of these very small steps.

A more thorough mathematical study of Maxwell’s equations shows that new and really unexpected conclusions can be drawn and the whole theory submitted to a test on a much higher level, because the theoretical consequences are now of a quantitative character and are revealed by a whole chain of logical arguments.

Let us again imagine an idealized experiment. A small sphere with an electric charge is forced, by some external influence, to oscillate rapidly and in a rhythmical way, like a pendulum. With the knowledge we already have of the changes of the field, how shall we describe everything that is going on here, in the field language?

The oscillation of the charge produces a changing electric field. This is always accompanied by a changing magnetic field. If a wire forming a closed circuit is placed in the vicinity, then again the changing magnetic field will be accompanied by an electric current in the circuit. All this is merely a repetition of known facts, but the study of Maxwell’s equations gives a much deeper insight into the problem of the oscillating electric charge. By mathematical deduction from Maxwell’s equations we can detect the character of the field surrounding an oscillating charge, its structure near and far from the source and its change with time. The outcome of such deduction is the electromagnetic wave. Energy radiates from the oscillating charge travelling with a definite speed through space; but a transference of energy, the motion of a state, is characteristic of all wave phenomena.

The Maxwell’s equations provide the insight that energy radiates from the oscillating charge, traveling with a definite speed through space; but a transference of energy, the motion of a state, is characteristic of all wave phenomena.

Different types of waves have already been considered. There was the longitudinal wave caused by the pulsating sphere, where the changes of density were propagated through the medium. There was the jellylike medium in which the transverse wave spread. A deformation of the jelly, caused by the rotation of the sphere, moved through the medium. What kind of changes are now spreading in the case of an electromagnetic wave? Just the changes of an electromagnetic field! Every change of an electric field produces a magnetic field; every change of this magnetic field produces an electric field; every change of…, and so on. As field represents energy, all these changes spreading out in space, with a definite velocity, produce a wave. The electric and magnetic lines of force always lie, as deduced from the theory, on planes perpendicular to the direction of propagation. The wave produced is, therefore, transverse. The original features of the picture of the field we formed from Oersted’s and Faraday’s experiments are still preserved, but we now recognize that it has a deeper meaning.

Energy is diluted substance—force that is spread out in space. This is field, which is maintaining a balance between motion and inertia dynamically at every point. This balance gives it a certain density and velocity.

The electromagnetic wave spreads in empty space. This, again, is a consequence of the theory. If the oscillating charge suddenly ceases to move, then its field becomes electrostatic. But the series of waves created by the oscillation continues to spread. The waves lead an independent existence and the history of their changes can be followed just as that of any other material object.

There are electromagnetic waves of different densities (frequencies). Those densities are maintained. Therefore, we have many different densities spreading through the same region independent of each other. But they may mix in some manner without losing individual identities.

We understand that our picture of an electromagnetic wave, spreading with a certain velocity in space and changing in time, follows from Maxwell’s equations only because they describe the structure of the electromagnetic field at any point in space and for any instant.

There is another very important question. With what speed does the electromagnetic wave spread in empty space? The theory, with the support of some data from simple experiments having nothing to do with the actual propagation of waves, gives a clear answer: the velocity of an electromagnetic wave is equal to the velocity of light.

There are two distinct velocities: The velocity in material domain, and the velocity in radiation domain. The inertia of these two domains is very far apart, and so are their velocities. All the velocities in the radiation domain appear to be the same from the perspective of material domain. So, it is no surprise that the velocity of electromagnetic wave is same as the velocity of light.

Oersted’s and Faraday’s experiments formed the basis on which Maxwell’s laws were built. All our results so far have come from a careful study of these laws, expressed in the field language. The theoretical discovery of an electromagnetic wave spreading with the speed of light is one of the greatest achievements in the history of science.

Experiment has confirmed the prediction of theory. Fifty years ago, Hertz proved, for the first time, the existence of electromagnetic waves and confirmed experimentally that their velocity is equal to that of light. Nowadays, millions of people demonstrate that electromagnetic waves are sent and received. Their apparatus is far more complicated than that used by Hertz and detects the presence of waves thousands of miles from their sources instead of only a few yards.

The electromagnetic wave produced by an oscillating charge is of a different substantiality then that of the electromagnetic field existing among the nuclei of the atoms.

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Final Comment

Maxwell’s equations provide laws for the structure of the electromagnetic field. At every point in this field there is a magnetic (spinning) component that contributes to its inertia, and an electrical (linear) component that contributes to its motion. These two components maintain a balance. The electromagnetic field related to a certain frequency of spin seems to confine itself to the surface of a sphere of certain radius. This field is continuous throughout the spherical surface.

Different frequencies have different spherical surfaces that share the same center, and are stacked on top of each other. Frequency increases and velocity decreases as the radius of the surface decreases. There is a constant relationship among the frequency, wavelength (radius) and the velocity (motion on the spherical surface). This can be worked out mathematically from this model.

The surface of the sphere represents a merging of force and distance, which is energy. The frequency represents the density of this energy layer. The wavelength represents the radius of the spherical surface. These spherical surfaces seems to stack up on top of the spherical atom. We may visualize the atom consisting of a “solid” nucleus, a “liquid” electronic region, and a “gaseous” envelop of radiation.

Newton’s laws of motion seem to address the “solid” core of the atom. Maxwell’s equations seem to address the “liquid” region of the atom; and Einstein’s equation may address the “gaseous” envelop of the atom.

Thus, the “thickness” of substance decreases from the center of the atom towards its periphery. 

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