In the books (*) and (**), the theory of electromagnetism has been reviewed by introducing a suitable nonlinear set of equations, allowing for a very extended space of solutions. The new formulation provides a far more accurate description of wave phenomena. We try here to explain the reasons for this choice, without discussing any technical detail. We also suggest a series of illuminating exercises aimed to explain the limits of the classical Maxwell’s setting and the advantages of the new approach.

A plane wave of infinite extent satisfies both the classical Maxwell’s equations in vacuum and the new model equations. There are however no other solutions of Maxwell’s equations, based on plane fronts and transversally evolving according to the rules of geometrical optics.Here below is an example of a portion of plane sinusoidal wave, suitably polarized. Each propagating front is an entire plane; the electromagnetic fields belong to it and are everywhere constant.The following animation shows the evolution of the electric field E (red) and the corresponding Poynting vectors (black). The magnetic field B is orthogonal to the page.

One can easily check that plane fronts, carrying (almost) any type of vector information, are always admissible by the new set of equations. Here below (left) is a train of wave-packets with bounded support, modulated in time by a sinus function. The transversal electric field belongs to each propagating front and is zero outside a prescribed region. The magnetic field is orthogonal to the page. We can obtain bounded solutions by rotating the picture about the propagation axis, so that the magnetic component lies on circles (right) and the electric one is radial. The body shifts without dissipation at the speed of light along the direction determined by V. There is no hope instead to get this result within the framework of the standard Maxwell’s model. In this way we get electromagnetic objects that travel undisturbed at the speed of light. At the same time, these can be viewed as particles (in principle of any size). It should be clear at this point that we are modeling photons in the classical fashion. The divergence of the electric field div(E) is now not allowed to be different from zero, however its average evaluated on the entire body is zero.

We provide further examples. Perfect spherical fronts, carrying any type of vector information on the local tangent planes, also satisfy the new set of equations. Here, the direction of the energy flow (associated to the Poynting vector) is the same as the one of the evolving fronts. The evolution is in perfect agreement with the rules of geometrical optics. For example, in spherical coordinates, a classical setting is the following one:

where only the azimuthal component of the electric field is different from zero. Despite common belief, Maxwell’s equations are not compatible with such a kind of wave displacement. Here below is the corresponding wave evolution. The energy diminishes when leaving the source, since it is distributed on spheres of increasing radius. The displacement agrees very well with what is observed in practice and these solutions are reported in many engineering books. Nevertheless, it is a straightforward calculation to check that the divergence of E is not zero even in vacuum. Here below is the vector field E (red) in the case of the above spherical example. On the right there is the vector V (normalized Poynting vectors) which is stationary. The information escapes radially at constant speed. The fronts evolve according to the rules of geometrical optics; hence they exactly satisfy the Huygens principle.

We could enforce the div(E)=0 condition, coming out with the Hertz solution that will be discussed later on. However, we lose the orthogonality of the Poynting vector with the propagating fronts. In this way the fronts do not develop according to the Huygens principle (see later).

A fragment of a perfect spherical wave also satisfies the whole set of equations. For example, E can be chosen according to the following expression:

for an arbitrary function f. The fronts now evolve along a narrower path. As in the previous case, the corresponding vector field V is constant, radial and stationary. Concerning Maxwell’s equations such a wave cannot be modelled. The situation is even worse than the case, previously considered, of a global spherical front. In fact, we cannot cut a piece of wave without altering the magnitude of div(E).

For the above reasons, the space of solutions obtainable with the classical Maxwell’s setting turns out to be extremely small. Removing the condition div(E)=0 allows for the inclusion of new families of electromagnetic emissions. In particular, photons can be incorporated in a classical field theory, with important consequences on the understanding of quantum phenomena.

Finally, Hertzian sinusoidal waves satisfy the relation div(E)=0, hence they are solution of the whole set of Maxwell’s equations. However, the Poynting vectors evolve in a very strange manner and the rules of geometrical optics are totally disregarded. The last two movies show the evolution of the electric field E and the velocity vector field V (normalized Poynting vectors) in the case of the Hertzian wave. The information spreads along concentric spheres, but the behavior of the Poynting vectors is quite uncontrolled, especially in proximity of the vertical axis. By better examining these solutions, one discovers that the propagation fronts are topologically equivalent to toroids, hence they are not spheres.

Conclusions – There are very few wave phenomena that can be modelled by Maxwell’s equations in vacuum. They are not certainly representative of what is actually seen in real world. The alternative system of equations allows instead for a very large space of solutions, containing, for example, solitary transverse waves with bounded support. The main achievement is the possibility of introducing photons through classical arguments, a subject that has been unsuccessfully investigated for more than a century.