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 it 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. 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

The Longitudinal E-field derived from the Liénard–Wiechert Potentials

Believe it or not, the electromagnetic potentials formulated independently by Alfred-Marie Liénard[7] and Emil Wiechert[8] anticipate an electric field with longitudinal components.

The Liénard–Wiechert potentials [math]\varphi[/math] (scalar potential field) and [math]\mathbf{A}[/math] (vector potential field) are for a source point charge [math]q_s[/math] at position [math]\mathbf{r}_s[/math] traveling with velocity [math]\mathbf{v}_s[/math]:

[math]\varphi(\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \left(\frac{q_s}{(1 - \mathbf{n} \cdot \boldsymbol{\beta}_s)|\mathbf{r} - \mathbf{r}_s|} \right)_{t_r}[/math]

and

[math]\mathbf{A}(\mathbf{r},t) = \frac{\mu_0c}{4 \pi} \left(\frac{q_s \boldsymbol{\beta}_s}{(1 - \mathbf{n} \cdot \boldsymbol{\beta}_s)|\mathbf{r} - \mathbf{r}_s|} \right)_{t_r} = \frac{\boldsymbol{\beta}_s(t_r)}{c} \varphi(\mathbf{r}, t)[/math]

where [math]\boldsymbol{\beta}_s(t) = \frac{\mathbf{v}_s(t)}{c}[/math] and [math]\mathbf{n} = \frac{\mathbf{r} - \mathbf{r}_s}{|\mathbf{r} - \mathbf{r}_s|}[/math].

The electric field derived from this is:

[math]\begin{align}\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2(1 - \mathbf{n} \cdot {\boldsymbol \beta}_s)^3} \left[\left(\mathbf{n} - {\boldsymbol \beta}_s\right)(1-{\beta_s}^2) + |\mathbf{r} - \mathbf{r}_s|(\mathbf{n} \cdot \dot{\boldsymbol \beta}_s/c) (\mathbf{n} - {\boldsymbol \beta}_s) - |\mathbf{r} - \mathbf{r}_s|\big(\mathbf{n} \cdot (\mathbf{n} - {\boldsymbol \beta}_s)\big) \dot{\boldsymbol \beta}_s/c \right]\end{align}[/math]

which is equal to [math]-{\boldsymbol \nabla}\varphi - \frac{d\mathbf{A}}{dt}[/math]

If acceleration [math]\dot{\boldsymbol \beta}_s[/math] is negligible, then we have:

[math]\begin{align}\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2(1 - \mathbf{n} \cdot {\boldsymbol \beta}_s)^3} \left(\mathbf{n} - \hat{{\boldsymbol \beta}}_s\right)(1-{\beta_s}^2)\end{align}[/math]

If [math]\mathbf{n}[/math] is parallel to [math]{\boldsymbol \beta}_s[/math], then:

[math]\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2(1 - \beta_s)^3} \left(\mathbf{n} - {\boldsymbol \beta}_s\right)(1 - {\beta_s}^2)[/math]
[math]\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2(1 - \beta_s)^3} \left(\mathbf{n} - {\boldsymbol \beta}_s\right)(1 - \beta_s)(1 + \beta_s)[/math]
[math]\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2(1 - \beta_s)^2} \left(\mathbf{n} - {\boldsymbol \beta}_s\right)(1 + \beta_s)[/math]

If in addition to the aforementioned condition, the condition [math]|{\boldsymbol \beta}_s| \ll |\mathbf{n}|[/math] is also satisfied, then we can approximate [math]\mathbf{\hat{n}} - \hat{{\boldsymbol \beta}}_s[/math] as [math]\mathbf{\hat{n}}[/math]. This affects only the direction of the calculated electric field, not its magnitude. Regarding the instance of [math]\left(\mathbf{n} - {\boldsymbol \beta}_s\right)[/math], what we can do is factor out the magnitude [math]|\mathbf{n} - {\boldsymbol \beta}_s|[/math] from the vector, and the use the approximation from the unit vector. Note, remember that [math]\mathbf{n}[/math] and [math]{\boldsymbol \beta}_s[/math] were taken to be co-linear. Therefore:

[math]\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2(1 - \beta_s)^2} \mathbf{\hat{n}} \left(1 - \beta_s\right)(1 + \beta_s)[/math]
[math]\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2(1 - \beta_s)^2} \mathbf{\hat{n}} (1 + \beta_s)[/math]
[math]\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2} \frac{1 + \beta_s}{1 - \beta_s} \mathbf{\hat{n}}[/math]

Recognizing that the above condition limits the situation to non-relativistic velocities, that is to say [math]{\boldsymbol \beta} \ll 1[/math] or [math]|\mathbf{v}_s| \ll c[/math], then we can approximate [math]1 - \beta_s[/math] as [math]1[/math] without significantly affecting the numerator [math]1 + \beta_s[/math]. Therefore:

[math]\mathbf{E}(\mathbf{r}, t) = \frac{q_s}{4 \pi \epsilon_0} \frac{1 + \beta_s}{|\mathbf{r} - \mathbf{r}_s|^2} \mathbf{\hat{n}}[/math]

The "rigid" contribution to the electric field is:

[math]\mathbf{E}(\mathbf{r}, t)_{rigid} = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2} \mathbf{\hat{n}}[/math]

If we consider an electromagnetic device composed of electrically neutral objects, what remains is a "quasistatic" contribution to the electric field.

[math]\mathbf{E}(\mathbf{r}, t)_{quasistatic} = \frac{q_s}{4 \pi \epsilon_0} \frac{\beta_s}{|\mathbf{r} - \mathbf{r}_s|^2} \mathbf{\hat{n}}[/math]

Given that:

[math]\mathbf{A}(\mathbf{r},t) = \frac{\boldsymbol{\beta}_s(t_r)}{c} \varphi(\mathbf{r}, t)[/math]

We can substitute for [math]\beta_s[/math]:

[math]|\mathbf{A}(\mathbf{r},t)|c = \beta_s \varphi(\mathbf{r}, t)[/math]
[math]\beta_s = \frac{|\mathbf{A}(\mathbf{r},t)|c}{\varphi(\mathbf{r}, t)}[/math]

Because this scenario considers non-relativistic velocities, that is to say [math]{\boldsymbol \beta} \ll 1[/math] or [math]|\mathbf{v}_s| \ll c[/math], we can again approximate the denominator. Therefore:

[math]\beta_s = \frac{|\mathbf{A}(\mathbf{r},t)|c}{\frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|}}[/math]

Therefore, substitution for [math]\beta_s[/math] gives:

[math]\mathbf{E}(\mathbf{r}, t)_{quasistatic} = \frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|^2} \frac{|\mathbf{A}(\mathbf{r},t)|c}{\frac{q_s}{4 \pi \epsilon_0} \frac{1}{|\mathbf{r} - \mathbf{r}_s|}} \mathbf{\hat{n}}[/math]
[math]\mathbf{E}(\mathbf{r}, t)_{quasistatic} = \frac{|\mathbf{A}(\mathbf{r},t)|c}{|\mathbf{r} - \mathbf{r}_s|} \mathbf{\hat{n}}[/math]

Next, substituting for [math]\mathbf{A}(\mathbf{r},t)[/math] gives:

[math]\mathbf{E}(\mathbf{r}, t)_{quasistatic} = \frac{\frac{\beta_s(t_r)}{c} \varphi(\mathbf{r}, t)c}{|\mathbf{r} - \mathbf{r}_s|} \mathbf{\hat{n}}[/math]
[math]\mathbf{E}(\mathbf{r}, t)_{quasistatic} = -\beta_s(t_r) \nabla\varphi(\mathbf{r}, t)[/math]

Force is four-dimensional, not three dimensional

At the beginning of physics studies, students are normally treated to the famous equation [math]\mathbf{F} = m \mathbf{a}[/math] where [math]\mathbf{F}[/math] is the force, [math]m[/math] is the mass, and [math]\mathbf{a}[/math] is the acceleration. However, in special relativity, force is four-dimensional, because momentum is four-dimensional, and force is generally the rate change of momentum over time.[9] Even light itself is known to have momentum[10].

There are three dimensions are associated with space, and one dimension is associated with time. Per Emmy Noether's theorem, momentum conservation is due to invariance due to spatial translations, while energy conservation is due to invariance due to time translations[11].

It is known that the norm of the spatial-symmetry contribution of the four-momentum is frame dependent, while the norm of the entire four-momentum is itself invariant[12]. This is a manifestation of the fact that a mass may radiate or absorb energy, and therefore the time-symmetry contribution of the four-momentum may change. The contribution to the four-force due to time-symmetry is essentially equal to velocity vector multiplied by power absorbed divided by the speed of light squared, which essentially gives velocity times the rate change of mass over time[13].

The Lorentz force gives us the rate change of the momentum per charge with respect to time. However, because the relationship between momentum [math]\mathbf{p}[/math] and energy [math]\mathcal{E}[/math] is[14]:

[math]\mathbf{p} = \frac{\mathbf{v}}{c^2}\mathcal{E}[/math]

The acceleration [math]\mathbf{a}[/math] of a target point charge [math]q_t[/math] due to the Lorentz force it receives is[14]:

[math]\mathbf{a} = \frac{q_t}{m\gamma}\left[\mathbf{E} + \boldsymbol{\beta}_t \times \mathbf{B} - \boldsymbol{\beta}_t \left(\boldsymbol{\beta}_t\ \dot\ \mathbf{E}\right)\right][/math]

I ( S.H.O. talk) hereby make the suggestion that the first instance of [math]\mathbf{E}[/math] is different than the second instance of [math]\mathbf{E}[/math]. The first instance corresponds to the electric field according to the Lorentz-Maxwell model of the electric field. Under that model of the electric field, there is no effect on the longitudinal components of the electric field dependent on the chosen inertial frame. Therefore, as per the Joules-Bernoulli equation[15] describing the transformation of the electric field between differing inertial frames, the following is true for the component of the electric field co-linear with the velocity:

[math]\mathbf {{E}_{\parallel}}' = \mathbf {{E}_{\parallel}}[/math]

This is in contrast to the electric field model of Liénard–Wiechert, whose components co-linear with the velocity are dependent on the magnitude velocity, as is evident in the prior subsection titled "The Longitudinal E-field derived from the Liénard–Wiechert Potentials". I ( S.H.O. talk) am making a suggestion today (Wednesday March 15, 2017) that the second instance for the above Lorentz acceleration equation applies for the electric field as derived from the potentials of Alfred-Marie Liénard[7] and Emil Wiechert[8].

To distinguish between these two electric fields, [math]\mathbf{E}_{LM}[/math] will denote the electric field used in Lorentz force equation + Maxwell equations, while [math]\mathbf{E}_{LW}[/math] will denote the electric fields as derived by the Liénard–Wiechert potentials. Therefore the acceleration [math]\mathbf{a}[/math] becomes:

[math]\mathbf{a} = \frac{q_t}{m\gamma}\left[\mathbf{E}_{LM} + \boldsymbol{\beta}_t \times \mathbf{B} - \boldsymbol{\beta}_t \left(\boldsymbol{\beta}_t\ \dot\ \mathbf{E}_{LW}\right)\right][/math]

The extra acceleration term [math]\mathbf{a}_{LW}[/math] is therefore:

[math]\mathbf{a}_{LW} = - \boldsymbol{\beta}_t \left(\boldsymbol{\beta}_t\ \dot\ \mathbf{E}\right)[/math]

Deriving an effective longitudinal force on a current element

Per the section titled "The Longitudinal E-field derived from the Liénard–Wiechert Potentials" there is a "quasistatic" contribution to the electric field based upon the Liénard–Wiechert potentials.

[math]\mathbf{E}(\mathbf{r}, t)_{quasistatic} = \frac{|\mathbf{A}(\mathbf{r},t)|c}{|\mathbf{r} - \mathbf{r}_s|} \mathbf{\hat{n}}[/math]

Then extra acceleration term [math]\mathbf{a}_{LW}[/math] from the previous section titled "Force is four-dimensional, not three dimensional" becomes:

[math]\mathbf{a}_{LW} = - \frac{q_t}{m\gamma} \boldsymbol{\beta}_t \left(\boldsymbol{\beta}_t\ \dot\ \mathbf{\hat{n}} \frac{|\mathbf{A}(\mathbf{r},t)|c}{|\mathbf{r} - \mathbf{r}_s|} \right)[/math]

Because this scenario considers non-relativistic velocities, that is to say [math]{\boldsymbol \beta} \ll 1[/math] or [math]|\mathbf{v}_s| \ll c[/math], we can approximate the Lorentz factor [math]\gamma[/math] as [math]1[/math]. Therefore:

[math]\mathbf{a}_{LW} = - \frac{q_t}{m} \boldsymbol{\beta}_t \left(\boldsymbol{\beta}_t\ \dot\ \mathbf{\hat{n}} \frac{|\mathbf{A}(\mathbf{r},t)|c}{|\mathbf{r} - \mathbf{r}_s|} \right)[/math]

If acceleration [math]\dot{\boldsymbol \beta}_s[/math] of the source point charge [math]q_s[/math] at position [math]\mathbf{r}_s[/math] is negligible, then:

[math]\mathbf{a}_{LW} = - \frac{q_t}{m} \boldsymbol{\beta}_t \left(\boldsymbol{\beta}_t\ \dot\ \nabla\right) \mathbf{A}(\mathbf{r},t)c[/math]
[math]\mathbf{a}_{LW} = - \frac{q_t}{m} \boldsymbol{\beta}_t \left(\frac{\mathbf{v}_t}{c} \dot\ \nabla\right) \mathbf{A}(\mathbf{r},t)c[/math]
[math]\mathbf{a}_{LW} = - \frac{q_t}{m} \boldsymbol{\beta}_t \left(\mathbf{v}_t \dot\ \nabla\right) \mathbf{A}(\mathbf{r},t)[/math]

The effective longitudinal force on charge [math]+q[/math] of velocity [math]\mathbf{v}_{+q}[/math] is:

[math]\mathbf{F_{+q}}_{LW} = - (+q) \boldsymbol{\beta}_{+q} \left(\mathbf{v}_{+q} \dot\ \nabla\right) \mathbf{A}(\mathbf{r},t)[/math]

Likewise, the effective longitudinal force on charge [math]-q[/math] of velocity [math]\mathbf{v}_{-q}[/math] is:

[math]\mathbf{F_{-q}}_{LW} = - (-q) \boldsymbol{\beta}_{-q} \left(\mathbf{v}_{-q} \dot\ \nabla\right) \mathbf{A}(\mathbf{r},t)[/math]

Per the above results, the effective longitudinal force is proportional to the square of the velocity of the charge [math]q[/math]. In the case where [math]\mathbf{v}_{-q} \gt \mathbf{v}_{+q}[/math] The total force on a pair of equal and opposite charges [math]+q[/math] and [math]-q[/math] is varies as:

[math]\mathbf{F_{-q}}_{LW} \propto v_{-q}^2 - v_{+q}^2[/math]

[math]\mathbf{F_{-q}}_{LW} \propto (v_{avg}+v_{dev})^2 - (v_{avg}-v_{dev})^2[/math]

[math]\mathbf{F_{-q}}_{LW} \propto v_{avg} v_{dev}[/math]

Where:

  • [math]v_{avg}[/math] represents the "center of velocity" of the current element, considering both positive and negative charges.
  • [math]v_{dev}[/math] is represents one-half of the drift velocity of the negative charges relative to the positive charges.

This result is consistent with the results of the Marinov Generator experiment conducted by Cyril W. Smith (See figure 9 of [3]). Sincerely, S.H.O. talk 23:41, 15 March 2017 (PDT)

To finalize the above results, let's consider the specific weighted average squared velocity:

[math]v_{weighted}^2 = (50%) v_{-q}^2 - (50%) v_{+q}^2[/math] [math]v_{weighted}^2 = (1/2) v_{-q}^2 - (1/2) v_{+q}^2[/math]

In terms of [math]v_{avg}[/math] and [math]v_{dev}[/math], the result is:

[math]v_{weighted}^2 = (1/2) (v_{avg}+v_{dev})^2 - (1/2) (v_{avg}-v_{dev})^2[/math] [math]v_{weighted}^2 = 2 v_{avg}v_{dev}[/math]

Since the deviation velocity [math]v_{dev}[/math] is one-half the drift velocity, let [math]v_{drift}[/math] be the drift velocity. In the case that the drift velocity is co-linear with average velocity, the force on a current element [math]I[/math] due to the aforementioned longitudinal force is:

[math]\mathbf{F}_{LW} = - (-q) \boldsymbol{\beta}_{avg} \left(\mathbf{v}_{drift} \dot\ \nabla\right) \mathbf{A}(\mathbf{r},t)[/math]

In the case of the Marinov Generator, as the conductive slip ring increases speed, the charge carriers conveying the current are thinned out due to the sliding motion of the contacts, and so their quantity per length varies inversely to the speed. The drift velocity is determined by the voltage applied to the slip ring divided by the slip ring resistance and is not a function of externally driven shaft's rotational rate. As a result, the induced voltage increase directly with the increase of the velocity of the slip ring [math]v_{+q}[/math] and therefore [math]\boldsymbol{\beta}_{+q}[/math]. Note comparatively minute drift velocities of the current-carrying loose electrons [math]-q[/math] against the remaining charge of the slip ring [math]+q[/math].

This further confirms the above prior results.

I am more certain now than before that the M.A.K.E.R.A.R.C will work. Sincerely, S.H.O. talk 00:29, 16 March 2017 (PDT)

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 "Altered-Lenz" Devices

It is anticipated that the longitudinal force described in the previous section may explain some types of 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 mostly outside the cylindrical boundary of the permanent magnet:

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. 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://redshift.vif.com/JournalFiles/Pre2001/V05NO3PDF/v05n3phi.pdf
  2. ↑ http://www.distinti.com/docs/pdx/paradox2.pdf
  3. ↑ 3.0 3.1 3.2 3.3 http://overunity.com/14691/the-marinov-generator/
  4. ↑ 4.0 4.1 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. ↑ 7.0 7.1 https://docs.google.com/file/d/0B817m31MAj0wZTZjZmMwMjgtY2Y5YS00YTQ5LThjM2EtNzhjYTYzNzFlZDY0/edit?hl=en_GB&pli=1
  8. ↑ 8.0 8.1 https://docs.google.com/file/d/0B817m31MAj0wMDI1YjllYjctY2NhOS00M2M2LWFlMTUtYjVmYTkyZmVlY2M2/edit?hl=en_GB
  9. ↑ https://en.wikipedia.org/wiki/Four-force
  10. ↑ https://en.wikipedia.org/wiki/Photon#Physical_properties
  11. ↑ https://en.wikipedia.org/wiki/Noether's_theorem#Examples_2
  12. ↑ https://en.wikipedia.org/wiki/Four-momentum
  13. ↑ https://www.physicsforums.com/threads/force-on-a-spherically-uniform-radiator-moving-through-space.887479/
  14. ↑ 14.0 14.1 http://physics.usask.ca/~hirose/p812/notes/Ch10.pdf
  15. ↑ https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Transformation_of_the_fields_between_inertial_frames

See also

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