Eqs. (4) to (6) are derived in UFT381 and they are valid for any omega sub 0 and A. There is an equation:

curl (omega sub 0 A) = 0

which is combined with

E = – omega sub 0 A

So if the dipole electric field E is known, and omega sub 0 is known, then A can be calculated directly. The standard model calculates the electric dipole field as described in chapter four of Jackson, third edition, which is in S. I. units. The usual (Jackson standard model) method is E = – del phi, where phi is given by a multipole expansion, Jackson’s eq. (4.1). The methodology of UFT384 and Notes 385(1) and 385(2) is to accept E (which is well tested experimentally) but to reexpress it as equation (1) of the note, the first antisymmetry law. Then A is found from E assuming the existence of the universal timelike spin connection, a vacuum frequency in hertz:

omega sub = = – c / r

The vector potential A given by wikipedia is just the usual standard model one for a magnetic dipole field, and not an electric dipole field. It is seen that Jackson’s chapter four is developed entirely in terms of phi, and the chapter does not contain a vector potential. As you know, a static electric field is not associated with a vector potential in the standard model. The vector potential given by wiki is of course dogmatically standard model, and it is probably the same as Jackson’s (5.32), calculated from the standard B = curl A and the standard curl B = mu0 J, the Ampere law. This is not the entirely new electric vector potential of Note 385(2). I have assumed that omega sub 0 is universal, as in UFT384, and related to the mass of the vacuum particle introduced in earlier UFT papers as you know, and written up in PECE2 (UFT366). So omega sub 0 does not depend on anything except the vacuum structure.

It is possible to express the vector antisymmetry laws in any coordinate system, and I can write them out in cylindrical and spherical polars. The spherical polar system is used in Eqs. (7) and (11) of the note, following Jackson’s chapter four. I agree about the symmetry of the electric dipole field.

Finally I can set up simultaneous equations for the electric dipole problem which can be solved for all the relevant unknowns: omega sub 0, bold omega, bold A and phi, given that E is the electric dipole field, or indeed any static electric field and given that B is zero and that A is independent of time. If that can be done by computer then there will be no need to assume a model for omega 0. I will think about this in the next note. The main purpose of Note 385(2) is to show that there is conservation of antisymmetry for any static electric field. That is a significant advance.

To: EMyrone@aol.com

Sent: 04/08/2017 11:58:50 GMT Daylight Time

Subj: Re: 385(2): Conservation of Antisymmetry for the Electric Dipole Field StrengthTo my understanding eqs. (4-6) have been derived for a point charge. I wonder if these relations are valid (or can be assumed concerning omega_0) for a dipole. I guess that omega_0 must be different from a simple 1/r dependence because of the dipole geometry.

According to https://en.wikipedia.org/wiki/Dipole

The vector potential of a (magnetic) dipole is

The dipole fied is rotationally symmetric, therefore we could restrict to the plane X=0 or Y=0 a priori.

Can the antisymmetry eqs.(10) be rewritten to spherical coordinates? Then we could use the electric dipole field directly in this form. The phi dependence cancels out. Re-transforming to cartesian coordinates requires the right sign handling when taking the square roots which could become complicated for a graphical representation.Horst

Am 03.08.2017 um 11:15 schrieb EMyrone:

The methodology developed in Note 385(1) is applied to the electric dipole field strength (7) in electrostatics. The spin connections can be worked out and graphed with computer algebra using Eqs. (9) and (10), and in the absence of a magnetic flux density B, automatically conserve antisymmetry. In general, the latter is conserved for any electric field strength E in ECE2 electrostatics in the absence of B, and with the assumptions (2) and (4) that A has no time dependence and that the scalar spin connection is universal:

omega sub 0 = – c / r.

In the special case where the electric dipole moment p is aligned in the Z axis, the electric field strength is given in Cartesian components by Eq. (26), and the vector potential by Eq. (27). So the spin connections can be worked out and graphed from Eqs. (10) and (27). They will have a very interesting graphical structure and will all conserve ECE2 antisymmetry. The calculation can be repeated for the electric n pole field, e.g. quadrupole, octopole, hexadecapole and so on. Any electric field strength of electrostatics will conserve antisymmetry under the conditions (2) and (4). The next not will apply this same methodology to magnetostatics.