Function Conjunction → Electromagnetic Potentials

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The basic idea here is that the electromagnetic potentials [math]\varphi[/math] and [math]A[/math] may be the underlying key to several inventions related to electromagnetic forces.

Introduction

From December 2016 to March 2017, I (S.H.O. talk) have been conducting electromagnetic simulations using JavaScript and the THREE.js script library (http://threejs.org). Based on these results, I have determined that the magnetic component of the Lorentz force:

[math]\mathbf{F_{mag}} = q\ \mathbf{v} \times \mathbf{B} = q\ \left[ \nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A}) - (\mathbf{v}\cdot\nabla)\mathbf{A} \right] [/math]

causes transfers of energy within the magnetic rotor assembly (such as electron kinetic energy and inductive storage energy into wire kinetic energy (wired rotor example), or atomic electron kinetic energy to magnetic domain kinetic energy (permanent magnet rotor example)) that, in the low-frequency approximation, pretty much matches the amount of energy transfer between fields and the driving (stator) coils which is due to the transformer induction electric field [math]- \frac{\partial \mathbf{A}}{\partial t}[/math] acting on currents in the coils from the relative motion of the magnetic rotor assembly.

As a result, the only way using conventional physics to explain various devices, such as the Marinov Motor[1], the Distinti Paradox2[2], and, especially, the Marinov Generator[3][4] is to more accurately define the electric field [math]\mathbf{E}[/math], which is part of the full Lorentz Force equation, and then see if the leads to predictions that confirm observations, especially those found figure 9 of Cyril Smith's 2009 paper on the "Marinov Generator".[4] S.H.O. talk 22:09, 5 March 2017 (PST)

Prior content in the "Comment Record" section:

Prior content in the "Background" section:

http://www.sho.wiki/index.php?title=Electromagnetic_Potentials&diff=next&oldid=1162

Novel Force laws proposed by various researchers

Stefan Marinov's proposal

Stefan Marinov proposed adding the "motional-transformer induction" on charge q:[5]

[math](\mathbf{v_{wire}}\cdot\nabla)\mathbf{A}[/math]

or rather:

[math]-(\mathbf{v_{charge}}\cdot\nabla)\mathbf{A}[/math]

to the Lorentz force.

The extra term is equivalent to:[6]

[math]-(\mathbf{v}\cdot\nabla)\mathbf{A} = \begin{matrix} - \left[ v_x \frac{∂A_x}{∂x} + v_y \frac{∂A_x}{∂y} + v_z \frac{∂A_x}{∂z} \right] \mathbf{e}_x \\ - \left[ v_x \frac{∂A_y}{∂x} + v_y \frac{∂A_y}{∂y} + v_z \frac{∂A_y}{∂z} \right] \mathbf{e}_y \\ - \left[ v_x \frac{∂A_z}{∂x} + v_y \frac{∂A_z}{∂y} + v_z \frac{∂A_z}{∂z} \right] \mathbf{e}_z \end{matrix}[/math]

Where [math]\mathbf{v}[/math] is the velocity of the charge.

The Lorentz force is:

[math]\mathbf{F} = q\left[-\nabla \varphi- \frac{\partial \mathbf{A}}{\partial t}+ \nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A})-(\mathbf{v}\cdot\nabla)\mathbf{A} \right][/math]

Therefore, adding the extra term proposed by Stefan Marinov results in:

[math]\mathbf{F} = q\left[-\nabla \varphi- \frac{\partial \mathbf{A}}{\partial t}+ \nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A})-2(\mathbf{v}\cdot\nabla)\mathbf{A} \right][/math]

The problem with this modification:

In the case of an electrical charge approaching a wire, this additional term proposed by Marinov doubles the force of deflection. This is not observed.

Consider the vector potential due a current-carrying wire on the x-axis. Both the current and the vector potential of this current point in the +x direction. Now have a charge approaching this wire perpendicularly. Both the magnetic Lorentz force and the additional Marinov term predict the same force. Now consider the case where the wire is moving toward the charge. In this case, both the transformer induction [math]- \frac{\partial \mathbf{A}}{\partial t}[/math] and the additional Marinov term predict the same force acting on the charge directed parallel to the vector potential of the current. The term proposed by Marinov doubles the observed force in these cases.

Cyril Smith's proposal

Cyril Smith proposed adding the following gradient to the Lorentz force:[3]

[math]- \nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A})[/math]

This is equal to:

[math]- \nabla_\mathbf{A}(\mathbf{v} \cdot \mathbf{A}) = \begin{matrix} - \left[ v_x \frac{∂A_x}{∂x} + v_y \frac{∂A_y}{∂x} + v_z \frac{∂A_z}{∂x} \right] \mathbf{e}_x \\ - \left[ v_x \frac{∂A_x}{∂y} + v_y \frac{∂A_y}{∂y} + v_z \frac{∂A_z}{∂y} \right] \mathbf{e}_y \\ - \left[ v_x \frac{∂A_x}{∂z} + v_y \frac{∂A_y}{∂z} + v_z \frac{∂A_z}{∂z} \right] \mathbf{e}_z \end{matrix}[/math]

The idea behind this was to explain an observation in an experiment involving a "Marinov Generator"[3] in which longitudinal induction forces were produced.

The Lorentz force is:

[math]\mathbf{F} = q \left[ -\nabla \varphi - \frac{\partial \mathbf{A}}{\partial t} + \nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A}) - (\mathbf{v}\cdot\nabla)\mathbf{A} \right][/math]

Therefore, adding the extra term proposed by Cyril Smith results in:

[math]\mathbf{F} = q \left[ -\nabla \varphi - \frac{\partial \mathbf{A}}{\partial t} + 0\nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A}) - (\mathbf{v}\cdot\nabla)\mathbf{A} \right][/math]

The problem with this modification:

In the case of two parallel current-carrying wires, this additional term proposed by Cyril Smith negates the magnetic forces between the currents. It turns out that the extra term may yield forces perpendicular to the velocity. These are:

[math]- \nabla_\mathbf{A}(\mathbf{v} \cdot \mathbf{A})_\bot = \begin{matrix} - \left[ v_y \frac{∂A_y}{∂x} + v_z \frac{∂A_z}{∂x} \right] \mathbf{e}_x \\ - \left[ v_x \frac{∂A_x}{∂y} + v_z \frac{∂A_z}{∂y} \right] \mathbf{e}_y \\ - \left[ v_x \frac{∂A_x}{∂z} + v_y \frac{∂A_y}{∂z} \right] \mathbf{e}_z \end{matrix}[/math]

Explaining Cyril Smith's "Marinov Generator" using standard Lorentz Transforms

As it turns out, under both cases, the role of the electric scalar potential [math]\varphi[/math] was ignored. Under a rather simple restriction however, the longitudinal force observed in the Marinov Generator experiment can emerge without any additional perpendicular components. The term we are looking for corresponds to [math]-v_x \frac{∂A_x}{∂x} \mathbf{e}_x[/math] in the case that the Cartesian x-axis is defined to be aligned with the velocity [math]\mathbf{v}[/math].

The solution is to simply require that the gradient of the electric scalar potential (in the rest frame of the charge) [math]\varphi'[/math] is (statistically) zero.

The Lorentz transforms of the lab frame electric scalar potential [math]\varphi[/math] and magnetic vector potential [math]\mathbf{A}[/math] are as follows:[7][8]

[math]\varphi' = \gamma (\varphi - v A_\parallel)[/math]
[math]A_\parallel' = \gamma (A_\parallel - v \varphi /c^2)[/math]
[math]A_\bot' = A_\bot[/math]

By setting the gradient of the scalar potential in the charge's rest frame to zero, the result is:

[math]\nabla \varphi' = 0 = \gamma \nabla (\varphi - v A_\parallel)[/math]
[math]- \nabla \varphi = - v \nabla A_\parallel[/math]

This can also be expressed as:

[math] - |\mathbf{v}| \nabla A_\parallel [/math]

Substituting this result in the Lorentz force yields:

[math]\mathbf{F} = q \left[ - |\mathbf{v}| \nabla A_\parallel - \frac{\partial \mathbf{A}}{\partial t} + \nabla_{A}(\mathbf{v}\cdot\mathbf{A}) - (\mathbf{v}\cdot\nabla)\mathbf{A} \right][/math]

In the case that the magnetic field is steady [math] - \frac{\partial \mathbf{A}}{\partial t} = 0[/math] and magnetic Lorentz force on the wire is opposed by the physical restriction to the apparatus, the observed longitudinal induction field is:

[math]\mathbf{E} = - |\mathbf{v}| \nabla A_\parallel[/math]

Integrated over the path taken by the charge yields the same voltages that were predicted and observed by Cyril Smith (see Table 1 and Figure 9 at [4]), but here we use a correct expression and without modifying Einstein's theory of Special Relativity.

A New Idea: The Makerarc

The above result supports a development beyond the S.H.O. Drive. A much more powerful and compact permanent magnet system is now envisioned, the viability of which rests on the condition that [math]\langle \nabla \varphi' \rangle = 0[/math] on each charge [math]q[/math]. The preliminary name is Makerarc (MAY-KERR-ARK), which stands for:

  • Magnetic
  • Atom
  • Kinetic
  • Energy
  • Reservoir
  • And
  • Resource
  • Channel

Details pending. Stay tuned. Sincerely, S.H.O. talk 22:09, 5 March 2017 (PST)

Explaining "Reduced-Lenz" Devices

It is anticipated that the above stated condition [math]\langle \nabla \varphi' \rangle = 0[/math] which leads to the electric field [math]- |\mathbf{v}| \nabla A_\parallel[/math] may explain some types of Reduced-Lenz devices. A good example can be found in the video below ("Ray's No Back EMF Generator"):

Simulations in JavaScript and THREE.js have determined that in many cases the magnetic Lorentz force [math]q\ \mathbf{v} \times \mathbf{B}[/math] will be opposed in part by the additional force [math]q \left[ - |\mathbf{v}| \nabla A_\parallel \right][/math]. This makes certain types of magnetic circuit arrangements more efficient at electrical power generation (in terms of output power vs. input power) but less efficient as an electrical motor. Simulations of the S.H.O. Drive indicated that the additional force may vary between assisting or opposing the Lorentz force, which is helpful depending on whether the goal is to develop an efficient motor or an efficient generator, respectively. The simulated S.H.O. Drive designs appeared to switch between these two extremes every quarter cycle. The only way to make it work would have been to increase the inductive reactance (so that the current would be delayed by about a quarter cycle), and even then, simulations indicated that the extra term was typically insufficient to completely reverse the net force. However simulations of various rotor and S.H.O. coil curve modifications for the S.H.O. Drive showed that it was possible for the additional force to be a significant percentage of the Lorentz force. Per more recent simulations (early March), the Makerarc design (above) will improve upon this many fold, unless if permanent magnets do not permit the [math]\langle \nabla \varphi' \rangle = 0[/math] condition. S.H.O. talk 23:39, 5 March 2017 (PST)

References

  1. ↑ http://redshift.vif.com/JournalFiles/Pre2001/V05NO3PDF/v05n3phi.pdf
  2. ↑ http://www.distinti.com/docs/pdx/paradox2.pdf
  3. ↑ 3.0 3.1 3.2 http://overunity.com/14691/the-marinov-generator/
  4. ↑ 4.0 4.1 4.2 http://www.overunityresearch.com/index.php?action=dlattach;topic=2470.0;attach=13897
  5. ↑ https://archive.org/stream/thornywayoftruthpart4maririch#page/104/mode/2up/search/motional-transformer+induction
  6. ↑ http://www.overunityresearch.com/index.php?action=dlattach;topic=2470.0;attach=13908
  7. ↑ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
  8. ↑ https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#The_.CF.86_and_A_fields

See also

Site map

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