## Vector Potential

We have found the following set of equations to describe electromagnetic phenomena:

 (1)

 (2)

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.

## Current carrying straight conductor

We look at a short straight conductor of length and determine the vector potential at distance from the wire (at the center of the conductor).

 (3)

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:

 (4)

For we get

 (5)

which is the accepted expression for a very long straight wire found in many text books.

## Current carrying soleonoid

We try to determine the vector potential outside of a soleonoidal coil.

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

## Moving charges in the aether

Let's consider a charge moving with velocity through the aether. We assume the majority of aether particles to be at rest. Very few are moving slowly towards a center of mass (far away) to feed its matter. This gives us a small and mainly constant vector potetial (aether wind) . The moving charge therefore experiences and thus no accelerating or decelerating force. The charge moves with (almost) constant velocity. We say almost because the behaviour of planets suggest that there is a minor friction component involved in celestial mechanics causing all orbits to eventually end up in circles. However, this friction is extremely weak.

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

and

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

with

and

A r

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 ...

## New Aether Model

A current through a wire causes (with time retardation) a vector potential at some point .

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

## Jefimenko

Jefimenko reflects over Maxwells equations

 (6) (7)

He uses the Helmholtz Theorem

to write like so

He then assumes charge free space and therefore simplifies to

 (8)

He then substtutes Eq. 7 into Eq. 8 and gets

He now utitlizes the vector identity

to get

 (9)

He uses another vector identity

to transform Eq. 9 into

puts the surface far far away (where is zero) and thus simplifies to

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