Torna alla pagina principale Sigillo di Ateneo

Non Maxwellian examples

The following animations show the evolution of the electric field E (red) and the corresponding Poynting vectors (black). The magnetic field B is orthogonal to the screen.

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 a portion of a plane sinusoidal wave. Each propagating front is an entire plane;  the electromagnetic fields belong to it and are everywhere constant .


One can easily check that plane fronts, carrying (almost) any type of vector information, are admissible by the new set of equations. There is no hope instead to get this result  by the standard Maxwell’s equations. Here below is a train of wave-packets with compact support, modulated by a sinus function.  The transversal electromagnetic fields belong to  each propagating front and are zero outside a bounded region.


Perfect spherical fronts, carrying any type of vector information on the local tangent planes, also satisfy the new set of equations. Nevertheless, Maxwell’s equations are  not compatible with waves such that, in vacuum, the direction of the energy flow (associated to the Poynting vector) is the same as the one of the evolving fronts. 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. Here below is the corresponding wave evolution. The energy diminishes when leaving the source, since it is distributed on spheres of increasing radius. Such a solution agrees very well with what is observed in practice (for instance in the far field of dipole antennas). Nevertheless, it is a straightforward calculation to check that the divergence of E is not zero.wfr3

Here  below is the vector field V (normalized Poynting vectors) in the case of the above spherical example. The situation 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 cassical 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 Poyinting vectors is quite uncontrolled, especially in proximity of the vertical axis.

wfr5 wfr6

Conclusions – There are very few wave phenomena that can be modelled by Maxwell’s equations. 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 compact support. The main achievement is the possibility of introducing photons through classical arguments, a subject that has been unsucessfully investigated for more than a century.