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<ref="Marinov Motor">http://redshift.vif.com/JournalFiles/Pre2001/V05NO3PDF/v05n3phi.pdf</ref>, the Distinti Paradox2[1], and, especially, the Marinov Generator[2][3] 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 it leads to predictions that confirm observations, especially those found figure 9 of Cyril Smith's 2009 paper on the "Marinov Generator".[3] 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

James Paul Wesley's proposal

James Paul Wesley proposed adding the "motional induction" on charge q:[4]

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

to the Lorentz force.

The idea behind this was to explain an observation in an experiment involving a "Marinov Motor"[4] in which longitudinal induction forces were produced.

The extra term is equivalent to:[5]

[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 William Smith's proposal

Cyril Smith proposed adding the following gradient to the Lorentz force:[2]

[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"[2] 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. The relevant field components 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 the Marinov Motor and Cyril Smith's "Marinov Generator" using Conventional Physics

In the sub-sections below, CGS units are used unless otherwise noted.

The E-field derived from the Liénard–Wiechert Potentials

From the paper titled "Onoochin's Paradox" by Kirk T. McDonald[6][7], we have following statement:

For calculations of the Lorentz force to be accurate to order [math]\frac{1}{c^2}[/math], it suffices to use eq. (4) for the magnetic field. However, to maintain the desired accuracy the electric field of a moving charge must also include effects of retardation, as can be obtained from an expansion of the Liénard–Wiechert fields [8][9] (for details, see the appendix of [10]),
[math]\mathbf{E} \approx q\frac{\mathbf{\hat{r}}}{r^2} \left(1 + \frac{v^2}{2c^2} - 3\frac{\left(\mathbf{\hat{r}} \cdot \mathbf{v}\right)^2}{2c^2} \right) - \frac{q}{2c^2r}\left[\mathbf{a}+\left(\mathbf{a} \cdot \mathbf{\hat{r}} \right)\mathbf{\hat{r}}\right][/math]
where [math]\mathbf{a}[/math] is the acceleration [math]\mathbf{a}[/math] of the charge [math]q[/math] at the present time.

Let's consider the situation where the acceleration [math]\mathbf{a}[/math] of charge [math]q[/math] is negligible:

[math]\mathbf{E} = q\frac{\mathbf{\hat{r}}}{r^2} \left(1 + \frac{v^2}{2c^2} - 3\frac{\left(\mathbf{\hat{r}} \cdot \mathbf{v}\right)^2}{2c^2} \right)[/math]

Next, let's clarify that [math]q[/math] is the source of E field by attaching a subscript s to it:

[math]\mathbf{E} = q_s\frac{\mathbf{\hat{r}}}{r^2} \left(1 + \frac{v^2}{2c^2} - 3\frac{\left(\mathbf{\hat{r}} \cdot \mathbf{v}\right)^2}{2c^2} \right)[/math]

In the Coulomb gauge, the first term in the parentheses comes from the electric scalar potential of a charge at rest in the observer's inertial frame. In event that the charge is contained within an electrically-neutral body, the electric field reduces to:

[math]\mathbf{E} = q_s\frac{\mathbf{\hat{r}}}{r^2} \left(\frac{v^2}{2c^2} - 3\frac{\left(\mathbf{\hat{r}} \cdot \mathbf{v}\right)^2}{2c^2} \right)[/math]

Let's consider a target charge [math]q_t[/math] located at [math]\mathbf{r}[/math] at rest in the observer's inertial frame. The inertial observer and the charge [math]q_t[/math] agree on what the electric field [math]E[/math] is, they agree that that there is no magnetic force on [math]q_t[/math], and finally, they agree on the acceleration of [math]q_t[/math].

The above equation can be broken up into two parts, one based on the relative azimuthal velocity of the source [math]\mathbf{v}_{s_\theta}[/math], and one based on the relative radial velocity of the source [math]\mathbf{v}_{s_r}[/math]:

[math]\mathbf{E} = q_s\frac{\mathbf{\hat{r}}}{r^2} \left(\frac{\mathbf{v}_{s_\theta}^2}{2c^2} - \frac{\mathbf{v}_{s_r}^2}{c^2} \right)[/math]

Scenario 1: All charges confined to a straight line

We can consider in the surrounding environment a system of charges with arbitrary position and velocity which observe different electric fields [math]\mathbf{E}[/math] that are unique to each charge's rest frame.

However, let's start off by considering the simplest case where all the charges are distributed along a straight line and all of their velocities are tangent to this line. Therefore, the velocity-dependent electric field experienced by a charge [math]q_t[/math] in its rest frame due a charge [math]q_s[/math] is:

[math]\mathbf{E} = - q_s\frac{\mathbf{\hat{r}}}{r^2} \frac{\left(\mathbf{v}_s-\mathbf{v}_t\right)^2}{c^2}[/math]
[math]\mathbf{E} = - q_s\frac{\mathbf{\hat{r}}}{r^2} \frac{\left(\mathbf{v}_{s \to r}\right)^2}{c^2}[/math]

Special Scenario 1.1: A current element and a charge confined to a straight line

Let's consider that along this line there is an element of a wire at velocity [math]\mathbf{v}_{wire}[/math] relative to charge [math]q_t[/math] carrying loose electrons with drift velocity [math]\mathbf{v}_{drift}[/math] relative to that wire. And let's say the total electrical charge of that wire [math]Q_{wire}=+Q[/math] + loose electrons [math]Q_{drift}=-Q[/math] is zero. What is the electric field created by this current element according to a target charge [math]q_r[/math] moving at velocity [math]\mathbf{v}_t[/math]?

[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{Q \mathbf{v}_{wire \to r}^2}{c^2} + \frac{-Q \left(\mathbf{v}_{wire \to r}+\mathbf{v}_{drift}\right)^2}{c^2}\right)[/math]
[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{Q \mathbf{v}_{wire \to r}^2}{c^2} - \frac{Q \left(\mathbf{v}_{wire \to r}+\mathbf{v}_{drift}\right)^2}{c^2}\right)[/math]
[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{2 Q \mathbf{v}_{wire \to r}\mathbf{v}_{drift} + \mathbf{v}_{drift}^2}{c^2}\right)[/math]

To simplify this result, let's consider the average velocity of charges comprising the wire+current:

[math]\mathbf{v}_{rel} = \mathbf{v}_{wire \to r} + \mathbf{v}_{drift}/2[/math]

This is effectively the relative velocity of a current element [math]\mathbf{I}_s dl = Q_{drift} \mathbf{v}_{drift}[/math] with respect to receiver charge [math]q_t[/math]

From here, we can find an alternate expression for the electric field at charge [math]q_t[/math] due to the current element [math]\mathbf{I}_s dl[/math]:

[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{Q \left(\mathbf{v}_{avg} - \mathbf{v}_{drift}/2\right)^2}{c^2} + \frac{-Q \left(\mathbf{v}_{avg} + \mathbf{v}_{drift}/2\right)^2}{c^2}\right)[/math]
[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{Q \left(\mathbf{v}_{avg} - \mathbf{v}_{drift}/2\right)^2}{c^2} - \frac{Q \left(\mathbf{v}_{avg} + \mathbf{v}_{drift}/2\right)^2}{c^2}\right)[/math]
[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{-Q v_{avg} v_{drift}}{c^2}\right)[/math]
[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{Q_{drift} v_{avg} v_{drift}}{c^2}\right)[/math]
[math]\mathbf{E} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{I_s v_{avg}}{c^2}\right)[/math]

Special Scenario 1.2: Forces between co-linear current elements confined to a straight line

If we wish to consider the force on charge [math]q_s[/math] due to the electric field of this current element, we simply multiply by target charge [math]q_t[/math].

[math]\mathbf{F} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{I_s q_t v_{avg}}{c^2}\right)[/math]

If [math]q_t[/math] was a loose electron embedded in another electrically-neutral wire element, let [math]\mathbf{I}_t dl[/math] stand for this target current element, analogous to the source current element [math]\mathbf{I}_s dl[/math]. Therefore the force on current element [math]\mathbf{I}_t dl[/math] due to current element [math]\mathbf{I}_s dl[/math] is:

[math]\mathbf{F} = - \frac{\mathbf{\hat{r}}}{r^2} \left( \frac{I_s I_t}{c^2}\right)[/math]

In SI units, this becomes:

[math]\mathbf{F} = - \frac{\mathbf{\hat{r}}}{4 \pi \epsilon r^2} \left( \frac{I_s I_t}{c^2}\right)[/math]

This is equivalent to:

[math]\mathbf{F} = - \left( \mathbf{I}_t \dot\ \nabla \right)\mathbf{A_s}[/math]

This is essentially the "Marinov Force", but with restrictions. Like the Marinov Force, it exists without necessitating a magnetic field [math]\mathbf{B}[/math] and so may exist in regions where the curl of [math]\mathbf{A_s}[/math] is zero. However, this scenario is limited to the case that "all charges are confined to a line" with the restriction that the current elements are electrically neutral.

Scenario 2: Large drift velocities and negligible wire velocities

The magnetization currents embodied inside permanent magnets consist of electrons with very high effective velocities with respect to atomic nuclei per the Virial theorem which relates the kinetic energy to the potential energy of the system[11]. Therefore, in the case of permanent magnets interacting with other permanent magnets, one can simply consider the effective velocity of the charges comprising the effective magnetization currents and disregard the velocities of the atoms which confine these currents in place.

Let [math]\mathbf{v}_{rel}[/math] be the relative velocity of charge [math]q_s[/math] respect to [math]q_t[/math]. In order to account for the forces due to the relative motion with respect to the positive charges which screen the Coulomb electric field of [math]q_s[/math], the velocity [math]\mathbf{v}_t[/math] of charge [math]q_t[/math] is added to [math]\mathbf{v}_{rel}[/math]. Therefore the effective relative velocity [math]v_{net}[/math] is equal to:

[math]\mathbf{v}_{net} = \mathbf{v}_{rel} + \mathbf{v}_t = \mathbf{v}_s - \mathbf{v}_t + \mathbf{v}_t = \mathbf{v}_s[/math]

As indicated in the parent section, the electric field due to charge [math]q_s[/math] moving at velocity [math]\mathbf{v}_s[/math] is equal to:

[math]\mathbf{E} = q_s\frac{\mathbf{\hat{r}}}{r^2} \left(\frac{\mathbf{v}_{s_\theta}^2}{2c^2} - \frac{\mathbf{v}_{s_r}^2}{c^2} \right)[/math]

The electric force on [math]q_t[/math] by [math]q_s[/math] is therefore:

[math]\mathbf{F} = q_t q_s\frac{\mathbf{\hat{r}}}{r^2} \left(\frac{\mathbf{v}_{s_\theta}^2}{2c^2} - \frac{\mathbf{v}_{s_r}^2}{c^2} \right)[/math]

Special Scenario 2.1: Forces between magnetization currents from magnets of the same grade

In the case where the magnets are of the same grade (e.g. N52), the effective drift speed of the magnetization currents are identical. This makes it possible to calculate the above using current elements [math]\mathbf{I}_s[/math] and [math]\mathbf{I}_t[/math].

The differential version of this is:

[math]d^2\mathbf{F} = \frac{\mathbf{\hat{r}}}{r^2} \left(\frac{\mathbf{I}_{s_\theta}^2}{2c^2} - \frac{\mathbf{I}_{s_r}^2}{c^2} \right)[/math]

In SI units, this becomes:

[math]d^2\mathbf{F} = \frac{\mathbf{\hat{r}}}{4 \pi \epsilon r^2} \left(\frac{\mathbf{I}_{s_\theta}^2}{2c^2} - \frac{\mathbf{I}_{s_r}^2}{c^2} \right)[/math]

In this special scenario:

  • [math]\mathbf{I}_{s_\theta}[/math] is the component of [math]\mathbf{I}_s[/math] which is perpendicular to [math]\mathbf{r}[/math].
  • [math]\mathbf{I}_{s_r}[/math] is the component of [math]\mathbf{I}_s[/math] which is parallel to [math]\mathbf{r}[/math].

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 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 "Altered" Lenz' Law Devices

It is anticipated that the longitudinal force described in the previous section may explain some types of purported Reduced-Lenz devices. A good example can be found in the video below ("Ray's No Back EMF Generator"), although in this example, the longitudinal force increases the drag, mainly in positions where pancake generator coil is mostly outside the cylindrical boundary of the permanent magnet. This creates an illusion of a "Reduced" Lenz' Law effect when the magnet is mostly within the cylindrical boundary of the permanent magnet:

Simulations in JavaScript and THREE.js have determined that in many other configurations of currents and magnets, the magnetic Lorentz forces [math]q\ \mathbf{v} \times \mathbf{B}[/math] will be opposed in part by the additional force. 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 magnetic 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 magnetic Lorentz force. Per more recent simulations (early March), the Makerarc design (previous section) will improve upon this many fold. S.H.O. talk 23:39, 5 March 2017 (PST)

Explaining the Newman Motor

A Newman Motor-style coil and magnet arrangement, like that shown in the video below, have been simulated by me using JavaScript and THREE.js.

The extra electric field term predicts a significant opposition to the magnetic Lorentz force at angles slightly straying from the "top-dead-vertical" position, making it a better generator than a motor. However, when energy is discharged from the "generator coil" to the "motor coil" of Newman's motor, the rotor will have often changed position to the point where the magnetic Lorentz force becomes increasingly significant, helpful for motive purposes. Newman's motor operated at a high Q, which facilitated energy recovery. S.H.O. talk 00:00, 6 March 2017 (PST)

References

  1. ↑ http://www.distinti.com/docs/pdx/paradox2.pdf
  2. ↑ 2.0 2.1 2.2 http://overunity.com/14691/the-marinov-generator/
  3. ↑ 3.0 3.1 http://www.overunityresearch.com/index.php?action=dlattach;topic=2470.0;attach=13897
  4. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Marinov_Motor
  5. ↑ http://www.overunityresearch.com/index.php?action=dlattach;topic=2470.0;attach=13908
  6. ↑ http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.299.8534
  7. ↑ freeweb.siol.net/markoor/onoochin.pdf
  8. ↑ https://docs.google.com/file/d/0B817m31MAj0wZTZjZmMwMjgtY2Y5YS00YTQ5LThjM2EtNzhjYTYzNzFlZDY0/edit?hl=en_GB&pli=1
  9. ↑ https://docs.google.com/file/d/0B817m31MAj0wMDI1YjllYjctY2NhOS00M2M2LWFlMTUtYjVmYTkyZmVlY2M2/edit?hl=en_GB
  10. ↑ http://web.archive.org/web/20170318210550/http://puhep1.princeton.edu/~mcdonald/examples/ph501/ph501lecture24.pdf
  11. ↑ https://en.wikipedia.org/wiki/Virial_theorem

See also

Site map

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