Electromagnetism theory reached its pinnacle in the nineteenth century with
Maxwell's famous equations.
Let us take two examples.
The case of two parallel conductors carrying current.
1. Currents parallel.
The current is a slow movement of electrons with fixed positive charges. The electrons are moving parallel in the two wires so are stationary relative to each other. The positive charges are seen as moving. According to Einstein's theory of relativity the length of the positive charge is contracted as seen by the electrons and so the electrons see an increased charge density over the charge density of the electrons. This makes the force of attraction between unlike charges slightly greater than the force of repulsion between unlike charges. This means that there is a net force of attraction.
2. Currents anti-parallel.
The electrons are now moving antiparallel and so they see a length contraction of the other electron charge. The positive charges are also seen contracted but not as much. So the electrons are seen as having a greater charge density than the fixed positive charges. Thus the force of repulsion of like charges is greater than the force of attraction of unlike charges. This means there is a net force of repulsion.
Consider two conducting wires parallel.
One conductor has an alternating current flowing in it. This means that the electrons are accelerating and thus their electric field lines have a kink in them so there is a transverse component this field moves the electrons in the other wire. Thus producing an induced potential. The magnitude of the induced potential would be proportional to the rate of change of the current in the first conductor.
In all these cases no magnetic field was required to account for the phenomena involved. So invoking Occums razor, the magnetic field in not required and so does not exist.
Consider two long straight copper conductors each carrying the same current in the same direction. The electrons move together at the same velocity in each cable. They repel one another. The fixed positive charges repel each other. However the electrons wire A attract the fixed charges in wire B but because the electrons are moving relative to the fixed charges they will see a higher charge density than the charge density due to the electrons in B so the attraction of electrons to fixed positive charges is higher than the repulsion between the same number of electrons in B. Similarly for electrons in wire B and the fixed positive charges in wire A.
n=number of atoms per meter e=electronic charge
d=separation l=length P=permittivity
v=drift velocity of electrons c=velocity of light
The force of repulsion is P*2*((e*n)^2)*l /d (= F)
And the force of attraction is P*(2*((e*n)^2)*l/d)/sqr(1-(v/c)^2)
So the resultant force is
So P/c is the "permeability" and the force is proportional to the current squared and the length but inversely proportional to the separation.
There is no need for the idea of the magnetic field. There is no magnetic field
Karl Popper has explained the scientific method. Use the predictions of the theory and test experimentally when judging a philosophical idea.
Science in only science if you carry out experiments to test the theories of yourself or another. To play philosophical games with words and equations is no science.
Go get a bit of wire make a straight part 40 cm long and connect it to a constant current source of say 1 amp.
Get the old electrostatic kit out of the 18th century box and using a glass rod wiped with silk make a pith ball coated with gold leaf charged by bringing the glass rod close to the ball and touch the ball briefly with a finger. The ball will be repelled by the glass rod.
Put the pith ball (suspended by a silk thread from a curved glass stand) near the wire and note the deflection from the vertical when the current is switched on.
I predict it will be repelled if negatively charged and attracted if positively charged.
A bar magnet is in fact a group of spinning electrons which are coupled by spin-spin interactions (Quantum mechanics) that are mediated by virtual photons again a force that is electronic and connected to the Lorenz contraction.. The force on another bar magnet is actually due to spin-spin interactions (Quantum mechanics) which is electronic and related to the Lorenz contraction.
A loop of wire carrying a current is again interacting with another current carrying conductor by the electrostatic force (mediated by virtual photons) and is again a result of the Lorenz contraction.
The calculation to use is the biot-savart hypothesis integrated.
In view of this new insight we should re-name magnetism as the "Lorenz force". It also means that many of our reasoning based on observations made 300 years ago before Einstein developed his special theory of relativity are misconceived as they are based on the wrong model.
The Gaussian construction of magnetic shells does not come into it there is no magnetic flux or lines of force.
As a disclaimer I will point out that I am not the first or
only person to point this out.