A current at some point causes a vector potential at a point . A changing vector potential on the other hand causes an electromotive force on charges. The above equations are valid only if the permeablity is constant and homogeneous in the region of space we are looking at.
This makes total sense. We have a rather high vector potential (aether wind) close to the conductor. The wind gets smaller with increasing distance from the wire.
We calculate the magnetic field now. The rotation of a vector field in cylinder coordinates is given by
Since we have and in our special case this simplifies to
This gives us the magnetic field for the short wire:
For we get
which is the accepted expression for a very long straight wire found in many text books.
The integration path is given by
We look at a soleonoidal coil of length with its center in the origin of the coordinate system and
We then have
Let's revisit our expression Eq. 3 for the vector potential caused by a current carrying straight conductor
and assume that a test charge moves with velocity towards the center of the wire. The charge will experience an ever increasing vector potential (perpendicular to its movement). According to Eq. 2
this should cause a force
in accordance with conventional expectation (Lorentz Force).
A charge moving parallel to an (extremely long) current carrying wire would experience no change in at the first glance and thus no force, at least not according to Eq. 1 and Eq. 2. However, another effect is to be looked at here. Let's consider two charges and , each moving with velocity
but displaced by . Let's look at the scene at where
The vector potential along the x-axis is then
It can clearly be seen that the vector potential is larger between the two moving charges, assuming that we can add up the vector potential caused by the two charges which seems reasonable. According to Bernoullie this higher velocity of aether causes a region of less pressure between the two charges which eventually leads to an additional aether wind component directed inwards.
To get a reasonable prediction for interaction of matter in space we have to marry
What happens if we place a sheet of bismuth () over a magnet? The magnet rotates the aether. Since aether velocity has to be continuous due to viscosity of the aether, the aether winds in the air on top of the magnet are extremely high. Above the air gap in the bismuth structure, the winds have to be ...
The square brackets in this equation is the retardation symbol indicating that the quantities between the brackets are to be evaluated for the time .
Let be a vector field on a bounded domain , which is twice continuously differentiable, and let S be the surface that encloses the domain V. Then can be decomposed into a curl-free component and a divergence-free component (Helmholtz's Theorem):
This allows us to write as
To simplify things let's (for now) assume that space is void of matter (no aether consumtion) and that the aether is not compressed (moderate current changes). We then have and can then simplify to
We make the educated guess that a changing vector potential generates an electromagnetic force.
With the vector identity
this can be rewritten to
We (again) assume that is divergence free and thus simplify to
He uses the Helmholtz Theorem
to write like so
He then assumes charge free space and therefore simplifies to
He then substtutes Eq. 7 into Eq. 8 and gets
He now utitlizes the vector identity
He uses another vector identity
puts the surface far far away (where is zero) and thus simplifies to
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