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What can we anticipate for the lines of equipotential in the current flowing through a resistor case?
Using the magnetoelectrostatic case i would expect the lines of equipotential to be the limit case of magnetoelectrostatic dipoles. I would expect these dipoles to reflect the differences in the substances in which they are formed. I would expect some additional dynamical components to have effevts so that torque and translation would be dynamic torque systems and dynamic translation systems, and thus i would expect these stationary equipotential lines to represent a dynamic interchange in which interference patterning both constructive and desttructive would play a significant part.

The empirical evidence supports at least this minimal description. I would further add that Fractal scaling effects or free scaling effects would be identifiable.

Remembering that the role of model making is not to exposit the ontology of these behaviours but to model them as is as precisely as possible.

I stated that the two element description for these magnetoelectro effects is not sufficient, and that is the case, In fact , theoretically a binary scale of distinctions is used. This means that though it always sounds like we ae discussing only 2 distinctions we are in fact discussing 2 n distinctions depending on the structure we need. This is a modulo arithmetic structure, a factorisation into 2 prime , a biadic numeral structure etc. Simply to call it fractal on the scale of 2 is therefore an additional terminological simplification, that may confuse, however, if you understand Mandelbrot fractal geometry it helps to keep things in perspective. In addition the pictorial or iconic representation of the Mandelbrot set based on the complex plane, that is a binary disposition of the magnitudes of space. SET ORTHOGANALLY to each other may give meditiative insight.

In the current across a resistor case, the typical arrangement i a current wire connected to both sides of a flat resistor which may be of any dielectric(now diamagnetoelectric). Typically a diamagnetoelectrolyte is used or a semi conductive material.

Materials that offer resistance to the diamagnetic spark effect thereby classify themselves and grade themselves by the complex diamagnetic array, which requires at least 3 dimensions to characterise the dipole differences in terms of orientation(orthogonality), current flow(force effect), and these 3 can quickly become complexified by dual distinctions within them, let alone the material and environmental distinctions. To make out that this i anything but a complex phenomenon is to do a disservice to many observers who still find new things to explore!

What we know is that if the conducting wires are placed in the presence of a magneto electric dipole that is permanent( a permanent magnet) then this dipole responds by pointing. That is it torques to a position that reflects the current carrying status of the wire. These dipoles do not point at the wire, and so are not like our grass seed compasses that pointed toward the "source of disturbance in the torque". These dipoles circle around the current carrying wire as i to protect it.

Now when we move these dipoles to the current carrying resitor we find that they deflect but this time in a way effected by the layout or spaciometry of the resistor. We find that the dipoles torque to the equipotential lines in the resistor. Thus they clear;y identify current tracks in the resistor which must be the limit cases of the dipole tracks in the magnetoelectrostaitc case.
http://www.allianceorg.com/pdfs/MMPA_Magnet_Design_Guide.pdf

The torque disturbance created by a current flow represents the bounding or regionalising effect of the magnetoelectric force. That this force structures space in this way is explainable in terms of vortex theory.
For a force in a substance to act, whether that substance be space or some other form of space, the law of action and reaction must hold.

Given that motion in general is arbitrary, the best effective model of the constraints is a vortex, which is regional and fractal. If the arbitrary force was to be opposed by an arbitrary reaction then there would necessarily be levels to that interaction in which certain directions would be completely blocked in motion while others would be free to move. as this is entirely arbitrary in the model all motions we know of are possibl at a point. However, should those motions colect into a regional behaviour them boundaries would form, and the simples boundary is that of the cone or vortex, simplest only in the sense of the count of surfaces in classification. In fact the cone represents a sophisticated solution, and in the case of a bilateral flow a conej joined by its base to a second cone represents a minimal solution. This already has the attributes of a dipole.
Bearing in mind the mechanical pressure such a system would exert pole wise, providing they were lined up correctly , wuth input at the commom base radially, a kind of venturi like effect is envisaged in the dipole state, the axial pulse being the familiar magnetoelectric spark while the radial base absorption, absorbing two sorts of substance supplies the torque adjustment.
The 2 substances spinning into this common base create a gyroscopic phenomenon which therefore tends to rotate the region into a stable force position vis a vis its neighbours. The points of these vortices aim at the higher regions of tension facilitating a transferrence of the high tension force to the low tension region respective to the substance.

The proposing of such regions was in fact put forward by Weiss as a postulant, who begging, secured the interest of his colleagues who found the idea very satisfactory, explaining much satisfactorily in permanent magnets.

The case in electromagnets is less satisfacory, because in adapting Weiss formulation, a physical boundary was accepted as being visible and demonstrable in permanent magnetics. Not so in space.

The assumption of empty or free space has to be dropped to make progress here. The vorticular boundaries themselves have to be accepted as the boundary effect of the magnetoelectic effect, and that the limit of the magnetoelectrostatic case has to be employed. In addition the bilateral flow of substances must not be ignored, which it routinely is, thus making it a puzzle as to how viable cavities of vortices may be created , sustained and indeed destroyed.

The equipotential "symmetry" for magnetoelectric effects indicates that an orthogonal plane or surface effect is demonstrable in space and also as space from which we may measure those phenomenon of magnetoelectricity. In time i hope they will come to be seen as bilateral plasma effects in a vorticularly structured dynamism, plasma being a fundamental substance in at least 2 forms that constitutes dynamic space.http://www.lintronicstech.com/index%20pdf/Hall_intro01.pdf
http://physics.stackexchange.com/a/22782http://physics.stackexchange.com/questions/22776/in-electrostatics-why-the-conductor-is-an-equipotential-surfacehttp://www.ornl.gov/~jzh/JHolmes/USPASAccFund/2_Lecture2.pdf
http://en.wikisource.org/wiki/A_Treatise_on_Electricity_and_Magnetism/Part_I/Chapter_VII

Some mathematical descriptions seem to give impossible results. but this is an indication that the model is not correct, and the principle cause of error is the particle electron, proton etc. A vorticular wave model is well able to lay within a surface. Also a dynamic transport across the surface of equipotentiality is allowed in the form of a standing wave solution, if 2 substances are included.
Ivor Catt Electromagnetic theory
.http://video.mit.edu/watch/5-e-grad-v-more-on-equipotential-surfaces-conductors-electrostatic-shielding-faraday-cage-12585/

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