Electromagnetism theory reached its pinnacle in the nineteenth century with Maxwell's famous equations, but are they necessary for developing a full theory of electrical phenomena.

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 antiparallel.

          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.

Induction.

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.

Maxwell's theory is not a good representation of electromagnetism.

Chris.