zerotensor's coil project

[zerotensor]

I am planning to build a special coil of my own design, one that I believe will have some remarkable properties.  I am calling on the community here to come along for the ride, and I would welcome your participation at any level.

This work is a spin-off of my theoretical work regarding the structure of the electron.
In my model, the electron is a tiny black hole, with a toroidal event horizon.  It has two distinct angular momenta, which combine to impart a helical frame-drag on the surrounding space.  For you relativity buffs out there who say that a toroidal event horizon is forbidden, I can assure you that this is a special case which elegantly evades this restriction.  The electromagnetic field is identified with the helical frame-drag.  Lightlike geodesics on the horizon twist once around, once through the torus, closing on themselves to form perfect circles.

It occurred to me that a conducting coil whose turns match the lightlike geodesics on the electron event horizon might have some interesting properties.  It turns out that there is a fascinating mathematical object which embodies this geometry:  The Hopf fibration.

I decided it might be helpful to model the magnetostatic field of my theoretical coil before beginning construction.  Using the software package Radia, which was used to design magnets at the European Synchrotron Radiation Facility, I have accomplished this and done some basic R&D regarding the coil parameters.  I found that when the ratio of (the distance from the center of the torus to the center of the circular loops) to (the radius of the loops) is exactly pi (!), the magnetic field inside the tube of the resulting toroid has nearly equal components in the axial and the toroidal directions.  The magnetic field inside the "donut hole" is purely axial at the midplane, and forms a potential well at the center.

Attached are some images.  The first is a basic visualization of the coil surface.  The second is a plot of surfaces of constant magnetic vector potential; the intersecting contours represent the toroidal and axial components of the field.  The third is a plot of the axial and toroidal field components taken along the x-axis,  notice the conjunction of the axial and toroidal components within the tube.

[Earl]

Good work ZT.

Here are some of my first thoughts.

I would be interested in seeing graphic outputs of your simulation involving the golden mean, phi and Phi, in addition to PI.
A very interesting site with many amazing photos is:
http://www.goldenmeangauge.co.uk/
http://www.goldenmeangauge.co.uk/cropcircles.htm
http://www.goldenmeangauge.co.uk/nature.htm

As the coil makes one revolution around the circumference of the toroid, it could make one turn around the torus - or two - or three.

If there was a two strand winding around the torus, whereby one strand completed one turn during one circumference while the second strand completed two turns during one circumference, then

one of these windings could be excited with sine wave of frequency f1 and
the other winding could be excited with sine wave of frequency f2
whereby f1 and f2 could be exact multiples/sub-multiples of each other.

OR

one of these windings could be excited with pulses of duration t1 and
the other winding could be excited with pulses of duration t2

whereby it is understood that the pulses have durations in the nanoseconds
and transitions in the pico- to nano-second range.  t1 and t2 could be
related such that they are exact multiples/sub-multiples of each other.

OR

one of these windings could be excited with pulses of duration t1 and
repetition rate of pps1 and

the other winding could be excited with pulses of duration t1 and
repetition rate of pps2

pps1 and pps2 could be related such that they are exact multiples/
sub-multiples of each other.


Another question is where to locate the electronics?  It looks like for
the simulated case as shown, maybe the middle of the torus,
inside the winding?  Your simulation shows a B field minimum there,
however there is an artifact which I do not understand.

As far as the A field is concerned, the center of the toroid appears
to have the minimum A field.

Earl

[zerotensor]

I would be interested in seeing graphic outputs of your simulation involving the golden mean, phi and Phi, in addition to PI.

The pi ratio was discovered by accident, while attempting to find some limiting behavior of the magnetic field for different tori.  I have done some simulations in which the golden mean enters as an underlying ratio;  I'll try to generate a few graphics from these for you.

As the coil makes one revolution around the circumference of the toroid, it could make one turn around the torus - or two - or three.

Different winding ratios are intriguing, but for the purposes of this project, I am sticking to a 1:1 winding ratio.  In the far limits, other ratios will degenerate into the common toroidal solenoid, or the common current loop.  I am splitting the difference, half-way in between these two extremes.  This ensures that the windings are perfect circles, and the current will emulate the spin structure of the electron, with each winding threading all the other windings.

There are, however, two ways to combine the rotations, producing a right-handed or left-handed twist.  Coils with opposite handedness could be wound on the same core, producing (or reacting to) added dipole fields and canceled toroidal fields, or vice-versa.  Also, multiple coils of varying toroidal parameter could be nested one inside the other, while preserving the winding ratio.  Differently scaled versions of the same coil could be brought into close proximity (e.g. stacked atop one another), to create a venturi-like effect.

Also, it occurs to me that there might be some benefit to situating an untwisted circular coil inside the tube.  So many possibilities!

...one of these windings could be excited with sine wave of frequency f1 and
the other winding could be excited with sine wave of frequency f2
whereby f1 and f2 could be exact multiples/sub-multiples of each other....

Yes, the coils will be pulsed.  When I get to the experimental phase of the project, I'll try out a whole range of pulsing methods.  I agree that pulses with short duration and even shorter rise-time are the most likely to produce interesting effects, especially with a multi-stranded coil.

Another question is where to locate the electronics?  It looks like for
the simulated case as shown, maybe the middle of the torus,
inside the winding?

Good question.  I imagine that a parallel-plate capacitor in the dipole field might display some interesting behavior.  I've got a long way to go before I can start to answer that question.

there is an artifact which I do not understand.

The B-field graph is a plot of the z-component (axial), and the y-component of the field, taken along the x-axis.  The y- component is positive on one side and negative on the other, because the B-field circulates inside the tube.  I'll post some vector field plots to illustrate this.

The second plot is in the x-z plane, and the first one is in a plane slightly above the equator, parallel to the the x-y plane.  We find that the field has a dipole characteristic outside the tube, and a mixed toroidal and dipole  field inside the tube.

[Earl]

ZT,

here is another image concerning an artifact.

Is the blue trace or the red trace an extraneous artifact,
which should not be there?

It disturbs me to not see perfect symmetry when everything else
is perfectly symmetrical.

Which should be removed, the red trace or the blue trace?

Earl

[zerotensor]

ZT,

here is another image concerning an artifact.

Is the blue trace or the red trace an extraneous artifact,
which should not be there?

It disturbs me to not see perfect symmetry when everything else
is perfectly symmetrical.

Which should be removed, the red trace or the blue trace?

Earl

Earl,

Both traces should be there.  The field is sampled along the x-axis, which is the abscissa in this graph.

In your markup, you colored the red trace blue and the blue trace red, which is a bit confusing, but no matter, just follow along closely:  In my original graph, the red trace is the y-component of the magnetic field.  The blue trace is the z-component.  You can think about it like this;  on the right, the y-component of the field inside the tube is coming toward you, out of the screen, and on the left, it is going away, into the screen.  The z- and y- components have nearly the same magnitude inside the tube, so on the left in this graph their individual traces appear to merge.  Here's a close-up of the field on the left, showing the individual traces.

It is precisely this (near) convergence of the field intensity profile within the tube that led me to believe that this particular "pi-ratio" geometry is "special".  Other tori do not display this close agreement of the field intensities, instead either the axial or the toroidal field begins to dominate within the tube. The slight non-uniformity within the tube I think is due to the asymmetry of the fringe fields, which can exit the coil more easily on the outer perimeter of the torus, where the windings are farther apart.    When the ratio of the radius of the individual rings to the displacement of their individual centers from the origin is 3.141592653589793..., we get the closest agreement of the two traces.  This was discovered empirically, and it still baffles me as to "why pi?".