# Maxwell on the Electromagnetic Field

### Maxwell on the Electromagnetic Field

### follows the course of Maxwell’s thinking as he searches to give form to an idea that is new to mathematical physics–the idea of the field.

This is by no means the story of a smooth progression, but rather the record of a series of impasses, each followed by a radical turn of thought. Maxwell makes this journey in a series of three dramatic steps, each marked by its own distinctive style. These are first “Faraday’s Lines of Force”, then “A Physical Theory of the Electromagnetic Field”, and finally, “A Dynamical Theory of the Electromagnetic Field.”

Because each of these steps moves through the darkness of a specific form of difficulty into a new form of light, they together trace a truly dialectical progression of thought. A remarkable fruit of this odyssey is the discovery of the electromagnetic theory of light: Maxwell’s realization that disturbances of the electromagnetic field must propagate with a velocity which appears to be the very velocity of light We trace the exacting strategy by which turns this exciting surmise into an exact prediction, and thereby arrives at the full electromagnetic theory of light.

Only after completion of this creative journey is Maxwell in a position to write his “Treatise on Electricity and Magnetism”, in which the entire theory of the electromagnetic field is set out in extensive form.

Maxwell unavailingly gives credit to Michael Faraday for the underlying concept of the field, and the whole series of papers can best be viewed as an effort to give mathematical form to the very essence of Faraday’s idea. Faraday himself had no formal mathematics, though Maxwell believed that Faraday was in some deeper sense a true mathematician (see the lecture, “Faraday’s Mathematics”.) The first paper is in effect addressed to Faraday, and carries the idea of the magnetic field as far as possible by means of an analogy, simple but effective, to fluid flow. This method has limited power, however, and to carry the idea forward Maxwell takes a completely different approach, envisioning an imaginative physical mechanism which would do in a realm of wheels and gears, what Faraday has shown electricity and magnetism actually do. But this, for all its success, is a project of thought, of wit and fantasy, and by no means the science Maxwell is seeking

Between the second and third papers, Maxwell has encountered the mathematical theory of Lagrange, which for the first time opens the way to a mathematical form which will fully capture the essence of Faraday’s field concept; this is a theory of a connected system, characterized by its overall energy, in which the whole is primary, while the parts – if any such exist at all – play only a secondary role. Here, in this dyanamical theory, Maxwell has found in Lagrange a formulation that expresses beautifully the wholeness of the field as the primary reality, and leaves behind physical hypothesis and conjecture.