The edifice of electromagnetism has stood unassailable for 300 years since Gauss did his famous analysis.
However since Clerk Maxwell finally cracked it in an exam paper for a degree in the 1800's and they came to understand light and Hertzian waves there came Plank and his quantum law relating energy to frequency by plank's constant and Neils Bohr with his quantum mechanics.
Gravity and electromagnetism just did not fit. Maxwell's theory says the electromagnetic wave can have any energy but Plank showed that as a body gets hotter the spectrum does not behave as Maxwell predicted.
Is important. So to quantise radio waves the procedure is as follows: The energy of the photons that make up the wave (they come from accelerating electrons in the wire in the antenna) is given by hf where f is the frequency and h Plank's constant and the number of photons per unit time is ExH/hf.
In an antenna wire the electrons are continuously accelerating and this takes place in the manner of a "particle in a box" quantum mechanical result with each increase in energy taking place in a step and since the rate of change of energy (the force on the electrons is constant (simple harmonic motion) so the energy increment is equal) is related to the frequency of the exciter.
delta E=hf if the exciter has frequency f then an individual electron changes its energy by delta E as a series of increases in velocity. Each velocity has a wavelength linked to it, the de Broglie wavelength, so when the velocity changes the two frequencies interfere as the and the photon that is emitted has the frequency that is the difference between the two states. This is another example of a quantum mechanics.
So an electromagnetic wave is a superposition of many coherent photons added together each only about 30 cm long and having a fixed amplitude related to their energy this is their Pointing Vector x length of the train of waves in the photon.
I've done this in a microwave and solid state physics course but my understanding at the time was limited by medical intervention.
So that is done.
In the case of the electric field the force between two electrons is given by e.e/r^2 where e is the electronic charge, but that force is mediated by virtual photons exchanged between the two charged particles and is quantised. In other words only certain separations are stable. It is to do with quantum mechanics.
In the case of the magnetic field we don't need magnetism at all, it is merely negative charges moving relatively among the "fixed" positive charges of the wires.
Let us take two cases of parallel wires carrying a current i each.
1) currents parallel and 2) currents anti-parallel.
Case 1) Currents parallel;
The positive charges are fixed and charges in wire 1 repel those in wire 2 with a force per meter of +(Ne.Ne/r)e0 where N is the number of electrons in 1 meter and r the separation in meters, e0 is the permittivity of free space.
The electrons in wire 1 see the electrons in wire 2 as fixed as well so they see a force between them of +(Ne.Ne/r)e0
The electrons in wire 1 see a force of attraction between them and the fixed charges in the wire but because there is relative motion the force is -(Ne.Ne/r)e0/sqr(1-(v/c)^2).
The electrons in wire 2 see a force of attraction between them and the fixed charges in the wire and this again is -(Ne.Ne/r)e0/(sqr(1-(v/c)^2)
so the total force is +2(Ne.Ne/r)-2(Ne.Ne/re0/sqr(1-v/c)^2)
this is: 2(Ne.Ne/r)e0(1-1/sqr(1-v/c)^2).
1/sqr(1-(v/c)^2)= (1-(v/c)^2)^(-1/2) = (1+(1/2)(v/c)^2......) to a first term approximation for low velocities.
so the force is approximately: 2(Ne.Ne/r)e0(1-1+(1/2)(v/c)^2)
since Nve=i the current then the force is + (i^2/r)e0/c^2 since e0/c^2=u0 the permeability of free space then the force is u0(i.i/r) the normal answer found by experiment.
In the case of the currents anti parallel the analysis is similar but the result is a similar force of repulsion.
The 1/r force law is derived by integrating along the wire with wire elements repelling other wire elements by the inverse square law.
This is a well known result from electrostatics. So magnetism is actually the same as electrostatics and does away with any need for magnetism. This simplifies things but invalidates electromagnetism theory.
In the case of a curved wire as in a circular loop the force on an element carrying current is:
u0(idLidl/r^2)Sin(theta) where theta is the angle between them (biot-savart proved above).
To find the force on an element in a circular loop you integrate all round the loop.
The point at distance r from the axis at an angle psi from the vertical is a distance d from a point on the circular loop a distance R from the axis (the radius) and an angle chi from the vertical experiences a force between the two elements of (r angle psi - R angle chi) using r and psi fixed integrate for fixed R and chi going from 0 to 359 degrees using numerical methods.
That gives a net force towards the centre that increases from the axis towards the wire as the value of r is varied for each integration.
If you use a loop of wire inside carrying a current the force tends to push the loop to the centre.
These principles can be used to design machines that contain plasma for generating power using alternating current, transformers and motors using less iron than is usual and levitating ovens that can purify metal in a vacuum without needing a crucible.