Physics: Electromagnetic Waves Field Theory

The greatest change in the axiomatic basis of physics - in other words, of our conception of the structure of reality - since Newton laid the foundation of theoretical physics was brought about by Faraday's and Maxwell's work on electromagnetic field phenomena. (Albert Einstein, 1931)

 Michael Faraday's Electromagnetic Force Field, Particle/Field Duality (1832)

Faraday's Continuous Electromagnetic Force Field is a Mathematical Approximation of Many Discrete Standing Wave Interactions

Faraday (1832) developed the mathematical concept of the 'electro-magnetic force field' as a way of mathematically describing action-at-a-distance for charged particles (i.e. electrons and protons). This is a continuous mathematical 'plotting' of the effects (forces and thus accelerated motions) that matter has on other matter in the Space around it, thus it is a description of effects rather than causes (Inductive / a posteriori rather than deductive / a priori. And this becomes important when you read Hume and Kant, for they explain that the ultimate Principles of Physics must be a priori, not a posteriori!).
This field concept replaced Newton's instant action-at-a-distance between discrete particles. Importantly, the electromagnetic (e-m) field is a vector (directional) quantity that defines force anddirection of acceleration of many charged particles upon one another. It is continuous in the sense that the distance and force between particles can vary by infinitely small amounts.
For example, electrons near one another in Space experience a mutual force of repulsion and this behaviour can be mathematically described using Faraday's e-m field which quantifies this force and describes how it varies with distance and direction. As Albert Einstein explains;

Faraday must have grasped with unerring instinct the artificial nature of all attempts to refer electromagnetic phenomena to actions-at-a-distance between electric particles reacting on each other. How was each single iron filing among a lot scattered on a piece of paper to know of the single electric particles running round in a nearby conductor? All these electric particles together seemed to create in the surrounding space a condition which in turn produced a certain order in the filings. These spatial states, today called fields, would, he was convinced, furnish the clue to the mysterious electromagnetic interactions.
He conceived these fields as states of mechanical stress in an elastically distended body (ether/space). For at that time this was the only way one could conceive of states that were apparently continuously distributed in space. The peculiar type of mechanical interpretation of these fields remained in the background - a sort of placation of the scientific conscience in view of the mechanical (Newtonian) tradition of Faraday's time. (Albert Einstein, 1940)

It seems that the 'electromagnetic force field' is a poorly understood concept which causes considerable confusion. It is quite basic though, as it is nothing more than a mathematical description of how matter affects and moves other matter in the Space around it. This mathematical 'force field' is a very powerful tool for mathematical physicists (as is the particle) and as a consequence many physicists (including Faraday, Maxwell, and Lorentz) imagined this 'field' to be real and therefore assumed that an 'Aether' (made up of many smaller particles!) must exist in Space as the medium for this 'field'. Born describes the ether as follows;

The undulatory, or wave theory, on the other hand, sets up an analogy between the propagation of light and the motion of waves on the surface of water or sound waves in air. For this purpose it has to assume the existence of an elastic medium that permeates all transparent bodies; this is the luminiferous ether. The individual particles of this substance merely oscillate about their positions of equilibrium. That which moves on as the light wave is the state of motion of the particles and not the particles themselves. (Born, 1924)

In fact there is no 'ether' simply because there are no 'force fields'. Both are mathematical constructions (rather clumsy and confusing ones at that) to try to explain how matter 'particles' interacted with other 'particles' in the space around them.
Once we understand the Spherical Wave Structure of Matter in Space though, then we no longer need these mathematical ideas, instead we realise that Space itself is a continuous wave medium (which necessarily connects all things) and there are no such things as discrete particles.

 James Clerk Maxwell's Equations
The Finite Velocity of Light Waves (1876)

When Maxwell (1876) used this field theory to assume that light was an Electromagnetic Wave, and then correctly deduced the finite velocity of light, it was a powerful logical argument for the existence of the electromagnetic force field, and that light was a wave like change in the field (electromagnetic radiation) that propagated with the velocity of light c through the ether.
In fact Maxwell was simply confirming that all Wave-Center to Wave-Center (particle) interactions are not instantaneous as Newton assumed, but are limited by the velocity of the In-Waves which is the Velocity of Light c.
So while Maxwell misunderstood the true nature of the waves (which are physical waves in Space rather than mathematical vector e-m waves), he is largely correct. This new knowledge was significant as it established the importance of the finite velocity of light c and further enhanced the field theory, thus rejecting Newton's theory of particles and instant action-at-a-distance.

The precise formulation of the time space laws of those fields was the work of Maxwell (1870s). Imagine his feelings when the differential equations he had formulated proved to him that the electromagnetic fields spread in the form of polarized waves and with the speed of light! To few men in the world has such an experience been vouchsafed.
Only after Hertz (1888) had demonstrated experimentally the existence of Maxwell's electromagnetic waves did resistance to the new theory break down. And what was true for electrical action could not be denied for gravitation. Everywhere Newton's (instant) actions-at-a-distance gave way to fields spreading with finite velocity.
At that thrilling moment he surely never guessed that the riddling nature of light, apparently so completely solved, would continue to baffle succeeding generations. (Albert Einstein, 1954)

And this is true. Because they were using a mathematical construction of a continuous e-m wave, rather than the true Spherical Standing Wave, they were in for a rather disturbing discovery not long thereafter. For standing wave interactions only occur at discrete frequencies, like notes on the string of a guitar, thus while the true Spherical Standing Wave Structure of Matter predicts that wave interactions will be discrete, the continuous e-m wave does not anticipate this.
Thus when Max Planck (1900) discovered that there are only certain allowed discrete energy states for electrons in molecules and atoms, and that light is only ever emitted and absorbed by electrons in discrete amounts or 'quanta', contrary to Maxwell's formulation that light is a continuous electromagnetic wave, then this caused a fundamental problem for the field theory that was never resolved. It is only now, with knowledge of the true foundations of physics and reality, that we can understand, and thus anticipate and correct, the errors of contemporary modern physics.