On the Definition of Radiation by a System of Charges

In a broad sense, radiation has come to mean a flow of energy through some medium, possibly vacuum. In a classical view, the energy can be carried by both particles (α and β particle radiation, etc.) and by waves (acoustic radiation, electromagnetic radiation, etc.). An early view (see, for example, Newton [1]) of optical radiation was that it consists of “rays” which emanate in straight lines from a source. Then, the number of rays crossing any surface enclosing the source is the same, and the number of rays crossing a area element normal to the rays falls off as the square of the distance from the source. A simple model is that optical rays are particles that move with some constant velocity along the path of the ray. The energy carried by the ray is the kinetic energy of the particles. In the early 1800’s Young [2] and Fresnel [3] argued that the optical phenomena of interference and stellar aberration imply that optical rays are actually an aspect of (transverse) waves in an æther. The energy carried by these rays was imagined to be that of the kinetic and potential energy of the undulations of the æther. Maxwell [4] identified optical rays with electromagnetic waves, whose energy is now ascribed to that of the electric and magnetic fields, rather than to a mechanical æther. The concept of rays for waves is only defined on scales larger than a wavelength. A challenge addressed in the present note is to provide an understanding of what can be meant by radiation of electromagnetic energy close to its source(s). In the quantum theory of electromagnetic fields, they can also be regarded as consisting of particles (photons), and whether their field or particle character is more prominent depends on details of the experiments devised to ascertain that character. This raises the question as to whether Maxwell’s equations for electromagnetic fields can lead to results of a particle-like character when considering electromagnetic radiation. Also, in the quantum view, photons have an extent at least that of a characteristic wavelength, and the flow of energy of these photons is not as highly localized as is assumed to be possible in a classical description. Hence, a classical description of the flow of energy close to charges and currents is expected to have finer detail than that possible in the quantum view. For example, lines of classical energy flow in Young’s double-slit experiment pass through only one slit or the other, while the quantum view of the resulting interference pattern for a single photon is that the photon has a probability amplitude to pass through both slits. A possible lesson is that one should