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| | |titlemode=replace | | |titlemode=replace |
| | |keywords=electricity,magnetism,motor,generator | | |keywords=electricity,magnetism,motor,generator |
| − | |description=The basic idea here is that the electromagnetic potentials <math>\phi</math> and <math>A</math> and their derivatives can be used to derive all electromagnetism. | + | |description=The basic idea here is that the electromagnetic potentials φ and A may be the underlying key to several inventions related to electromagnetic forces. |
| | }} | | }} |
| | {{DISPLAYTITLE:Function Conjunction → Electromagnetic Potentials}} | | {{DISPLAYTITLE:Function Conjunction → Electromagnetic Potentials}} |
| − | The basic idea here is that the electromagnetic potentials <math>\phi</math> and <math>A</math> and their derivatives can be used to derive all electromagnetism. | + | 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. |
| | | | |
| − | ==Comment Record== | + | ==Introduction== |
| | | | |
| − | Beginning with the velocity-dependent electromagnetic potential, one normally may derive the Lorentz force from it. This occurs by substituting a gradient term via an approximate, rather than exact, vector identity <math>\nabla (\mathbf{A} \cdot \mathbf{v}) = \nabla_\mathbf{A}(\mathbf{A} \cdot \mathbf{v})</math>. This is "allowed" normally because velocities in Special Relativity are not an explicit function of the coordinates, a matter simply assumed to be fact.<ref>http://www.nhn.ou.edu/~gut/notes/cm/lect_09.pdf ("The triple cross product can be written in a more compact form where we use the fact that the velocity is not an explicit function of the coordinates.")</ref> However, by taking velocity to be an explicit function of the coordinates, as per the S.R.-like Lorentz Ether theory, the extra force term not seen in the Lorentz force appears, which is <math>\nabla_\mathbf{v}(\mathbf{A} \cdot \mathbf{v})</math>. ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'' 03:18, 12 September 2016 (PDT)
| + | From December 2016 to March 2017, I ( ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'') 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: |
| | | | |
| − | So herein, the actual vector identity for the gradient of a dot is employed, resulting in mathematical consistency, as opposed to "magically" waving away the velocity-gradient terms as is usually done to impose consistency of the Electromagnetic Lagrangian with the Lorentz Force. These extra terms are gauge-dependent, and so an appropriate gauge must be selected (by Nature itself) to render these (heretical) gauge-dependent forces meaningful. Applying the Lorenz gauge would make it consistent with the finite speed of light, while applying the Coulomb gauge would imply dependence of the force on the instaneous position of the sources of electromagnetic potential. ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'' 13:37, 28 August 2016 (PDT)
| + | : <math>\mathbf{F_{mag}} = q \mathbf{v} \times \mathbf{B} = \nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A}) - (\mathbf{v}\cdot\nabla)\mathbf{A}</math> |
| | | | |
| − | The extra force term <math>\nabla_\mathbf{v}(\mathbf{A} \cdot \mathbf{v})</math> consists of a changing effective mass-correction term (in units of kg/s) multiplied by a velocity. Such a force is not preserved under Galilean transformations, but neither is the derivative <math>\frac{d(m\mathbf{v})}{dt} = m \frac{d\mathbf{v}}{dt} + \mathbf{v} \frac{dm}{dt}</math>. The extra force term is essentially a <math>\mathbf{v} \frac{dm}{dt}</math> term and could conceivably substitute for it, unless other terms may also assume a similar role (more on this below). Such forces change the time-like component of the relativistic 4-momentum, and therefore they are related to changes of rest energy of a particle (charge) subject to potentials, as viewed by an arbitrary inertial observer. Whatever other such terms may be, after cancelling the terms on both sides, what remains on the left hand side is equal to <math>m \frac{d\mathbf{v}}{dt}</math> of the particle (charge) and the right hand side yields the standard Lorentz force on the particle (charge). Unlike the 4-momentum, which is Lorentz invariant, the 3-momentum (i.e. the set of the 3 space-like components of the 4-momentum) is not. The Lorentz force, as normally expressed, only deals with changes of the 3-momentum over time. An observer may observe the time-like component of its 4-momentum transform as it changes velocity resulting in an unaccounted for "thrust", particularly if the time-like component of the momentum <math>E/c</math> in some way depends on the ''velocity-dependent electromagnetic potentials'', which in turn would yield, in effect, a ''velocity-dependent rest mass''.
| + | causes changes in kinetic energy that, in the low-frequency approximation, pretty much matches the amount of energy induced into surrounding coils. |
| | | | |
| − | To allude this possibility, consider that in cases where there are changing charge densities due to divergent/convergent electrical currents, and yet where current densities are constant, the kinetic forces <math>m \frac{d\mathbf{v}}{dt}</math> between charges do not sum to zero. A key such example of another non-Galilean invariant terms involving the variation mass over time at a given velocity, is the hypothetical Longitudinal force density conceived by Koen J. van Vlaenderen in his 2015 paper "General Classical Electrodynamics"<ref>http://vixra.org/abs/1512.0297</ref>, which from equations (2.8) and (2.9) can be written as <math>\mathbf{f}_L(\mathbf{x}) = \mathbf{J} (- \nabla \cdot \mathbf{A})</math>. The extra force term, adapted to help generalize Whittaker's force law to cover field force densities, was intended to preserve Newton's Third Law of Motion ("For every action there is an equal and opposite reaction"). Under the Lorenz gauge condition, this equals <math>\mathbf{f}_L(\mathbf{x}) = \frac{dq}{dV} \mathbf{v} \left( \epsilon_0 \mu_0 \frac{∂\varphi}{∂t} \right) = \frac{d(E_{coulomb})}{c^2dtdV} \mathbf{v}</math>, which when integrated over volume elements <math>dV</math> gives <math>\frac{d(m_{coulomb})}{dt} \mathbf{v} = \frac{d(E_{coulomb})}{c^2dt} \mathbf{v}</math>. When this <math>\mathbf{v} \frac{dm}{dt}</math> type term is subtracted from both sides of the full force equation (a <math>\frac{d(m \mathbf{v})}{dt}</math> type equation), we once again return to the Lorentz 3-force, which is the standard electromagnetic force of type <math>\mathbf{F} = m \frac{d\mathbf{v}}{dt} = m\mathbf{a}</math>. ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'' 03:18, 12 September 2016 (PDT)
| + | As a result, the only way using conventional physics to explain various devices, such as the Marinov Motor,<ref>http://redshift.vif.com/JournalFiles/Pre2001/V05NO3PDF/v05n3phi.pdf</ref> the Distinti Paradox2<ref>http://www.distinti.com/docs/pdx/paradox2.pdf</ref>, and, especially, the Marinov Generator<ref name="Marinov Generator">http://overunity.com/14691/the-marinov-generator/</ref><ref name="Marinov Generator (paper)">http://www.overunityresearch.com/index.php?action=dlattach;topic=2470.0;attach=13897</ref> 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".<ref name="Marinov Generator (paper)"/> ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'' 22:09, 5 March 2017 (PST) |
| | | | |
| − | ==Background==
| + | '''Prior content in the "Comment Record" section:''' |
| | | | |
| − | According Emil John Konopinski, a nuclear scientist<ref name="nyt obit">[http://query.nytimes.com/gst/fullpage.html?res=9C0CE6D81739F93BA15756C0A966958260 Emil Konopinski, 78, Atomic Bomb Scientist], New York Times</ref> and professor of Mathematics<ref>[http://www.genealogy.ams.org/id.php?id=104307 Eugene Greuling] at the Mathematics Genealogy Project</ref> who worked on the Manhattan Project<ref>{{cite journal|title=Obituary: Emil J. Konopinski|journal=Physics Today|date=October 1991|volume=44|issue=10|pages=144|url=http://www.physicstoday.org/resource/1/phtoad/v44/i10/p144_s1?bypassSSO=1|doi=10.1063/1.2810306|bibcode = 1991PhT....44j.144E }}</ref>, the electromagnetic fields <math>\mathbf{E}</math> and <math>\mathbf{B}</math> can be re-expressed in terms of the electromagnetic potentials <math>\varphi</math> and <math>A</math> through substitutions. From his article on "What the electromagnetic vector potential describes"<ref>http://exvacuo.free.fr/div/Sciences/Dossiers/EM/ScalarEM/J%20Konopinski%20-%20What%20the%20Electromagnetic%20Vector%20Potential%20Describes%20-%20ajp_46_499_78.pdf</ref>, he presents the equation of motion for a localized point charge:
| + | * http://www.sho.wiki/index.php?title=Electromagnetic_Potentials&diff=1036&oldid=951 |
| | + | * http://www.sho.wiki/index.php?title=Electromagnetic_Potentials&diff=next&oldid=1165 |
| | | | |
| − | : <math>d(M\mathbf{v})/dt = q \left[ \mathbf{E} + \mathbf{v} \times \mathbf{B}/c \right]\ (Gaussian\ units)</math>: | + | '''Prior content in the "Background" section:''' |
| − | : <math>d(M\mathbf{v})/dt = q \left[ \mathbf{E} + \mathbf{v} \times \mathbf{B} \right]\ (SI\ units)</math>
| + | |
| | | | |
| − | With the standard substitutions for the fields in terms of the potentials, which were taken to be:
| + | http://www.sho.wiki/index.php?title=Electromagnetic_Potentials&diff=next&oldid=1162 |
| | | | |
| − | : <math>\mathbf{E} = - \nabla \varphi - \frac{∂\mathbf{A}}{c∂t}\ (Gaussian\ units)</math>
| + | ==Novel Force laws proposed by various researchers== |
| − | : <math>\mathbf{E} = - \nabla \varphi - \frac{∂\mathbf{A}}{∂t}\ (SI\ units)</math>
| + | |
| − | : <math>\mathbf{B} = \nabla \times \mathbf{A}\ </math>
| + | |
| | | | |
| − | Konopinski determined the function relating the time derivative of the "total" momentum with the velocity-dependent potential, as evaluated by an inertial observer who sees the localized charge <math>q</math> of mass <math>m</math> travelling with velocity <math>\mathbf{v}</math> which is subject to a magnetic vector potential <math>\mathbf{A}</math> and an electric scalar potential <math>\varphi</math>.
| + | ===Stefan Marinov's proposal=== |
| | | | |
| − | : <math>\frac{d}{dt} \left[ M\mathbf{v} + (q/c)\mathbf{A} \right] = - q \left[ \nabla \varphi - (\mathbf{v}/c) \cdot \mathbf{A} \right]\ (Gaussian\ units)</math> | + | Stefan Marinov proposed adding the "motional-transformer induction" on charge q:<ref>https://archive.org/stream/thornywayoftruthpart4maririch#page/104/mode/2up/search/motional-transformer+induction</ref> |
| − | : <math>\frac{d}{dt} \left[ M\mathbf{v} + q\mathbf{A} \right] = - q \left[ \nabla \varphi - \mathbf{v} \cdot \mathbf{A} \right]\ (SI\ units)</math>
| + | |
| | | | |
| − | The terms on the right can be separated as follows:
| + | : <math>(\mathbf{v_{wire}}\cdot\nabla)\mathbf{A}</math> |
| | | | |
| − | : <math>\frac{d}{dt} \left[ M\mathbf{v} + q\mathbf{A} \right] = - q \nabla \varphi + q \nabla (\mathbf{v} \cdot \mathbf{A})\ (SI\ units)</math> | + | or rather: |
| | | | |
| − | Using Feynman subscript notation, we can separate the last term on the right into two separate terms:
| + | : <math>-(\mathbf{v_{charge}}\cdot\nabla)\mathbf{A}</math> |
| | | | |
| − | : <math>q\nabla (\mathbf{v} \cdot \mathbf{A}) = q\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) + q\nabla_\mathbf{A}(\mathbf{v} \cdot \mathbf{A})</math>
| + | to the Lorentz force. |
| | | | |
| − | In an article titled "A Discussion on the Magnetic Vector Potential"<ref>http://www.overunityresearch.com/index.php?action=dlattach;topic=2470.0;attach=13908</ref>, Cyril W. Smith (Professor Ph.D of Electronic and Electrical Engineering from 1964-1989<ref>http://www.positivehealth.com/author/cyril-smith-ph-d</ref>), the last term (without the charge <math>q</math>) can be expressed as:
| + | The extra term is equivalent to:<ref>http://www.overunityresearch.com/index.php?action=dlattach;topic=2470.0;attach=13908</ref> |
| | | | |
| − | : <math>\nabla_\mathbf{A}(\mathbf{v} \cdot \mathbf{A}) = \begin{matrix} | + | : <math>-(\mathbf{v}\cdot\nabla)\mathbf{A} = \begin{matrix} |
| − | \mathbf{a}_x \left[ v_x \frac{∂A_x}{∂x} + v_y \frac{∂A_y}{∂x} + v_z \frac{∂A_z}{∂x} \right] \\
| + | - \left[ v_x \frac{∂A_x}{∂x} + v_y \frac{∂A_x}{∂y} + v_z \frac{∂A_x}{∂z} \right] \mathbf{e}_x \\ |
| − | \mathbf{a}_y \left[ v_x \frac{∂A_x}{∂y} + v_y \frac{∂A_y}{∂y} + v_z \frac{∂A_z}{∂y} \right] \\ | + | - \left[ v_x \frac{∂A_y}{∂x} + v_y \frac{∂A_y}{∂y} + v_z \frac{∂A_y}{∂z} \right] \mathbf{e}_y \\ |
| − | \mathbf{a}_z \left[ v_x \frac{∂A_x}{∂z} + v_y \frac{∂A_y}{∂z} + v_z \frac{∂A_z}{∂z} \right] | + | - \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> | | \end{matrix}</math> |
| | | | |
| − | Written in this form, <math>\mathbf{a}_x</math>, <math>\mathbf{a}_y</math>, and <math>\mathbf{a}_z</math> are the unit basis vectors for x, y, and z, respectively. However, for the purposes of the S.H.O. Drive Wiki Site, <math>\mathbf{a}</math> will stand for the vector for acceleration, so it doesn't hurt that the standard variables for the basis vectors are really <math>\mathbf{e}_x</math>, <math>\mathbf{e}_y</math>, and <math>\mathbf{e}_z</math>.
| + | Where <math>\mathbf{v}</math> is the velocity of the charge. |
| | | | |
| − | Predictably, the values for first term on the right (without the charge <math>q</math>) can be expressed as:
| + | The Lorentz force is: |
| | | | |
| − | : <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) = \begin{matrix}
| + | <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> |
| − | \left[ A_x \frac{∂v_x}{∂x} + A_y \frac{∂v_y}{∂x} + A_z \frac{∂v_z}{∂x} \right] \mathbf{e}_x \\ | + | |
| − | \left[ A_x \frac{∂v_x}{∂y} + A_y \frac{∂v_y}{∂y} + A_z \frac{∂v_z}{∂y} \right] \mathbf{e}_y \\
| + | |
| − | \left[ A_x \frac{∂v_x}{∂z} + A_y \frac{∂v_y}{∂z} + A_z \frac{∂v_z}{∂z} \right] \mathbf{e}_z
| + | |
| − | \end{matrix}</math> | + | |
| | | | |
| − | The sum of the first and last terms gives:
| + | Therefore, adding the extra term proposed by Stefan Marinov results in: |
| | | | |
| − | :<math>\nabla (\mathbf{v} \cdot \mathbf{A}) = \begin{matrix}
| + | <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> |
| − | \left[ v_x \frac{∂A_x}{∂x} + v_y \frac{∂A_y}{∂x} + v_z \frac{∂A_z}{∂x} + A_x \frac{∂v_x}{∂x} + A_y \frac{∂v_y}{∂x} + A_z \frac{∂v_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} + A_x \frac{∂v_x}{∂y} + A_y \frac{∂v_y}{∂y} + A_z \frac{∂v_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} + A_x \frac{∂v_x}{∂z} + A_y \frac{∂v_y}{∂z} + A_z \frac{∂v_z}{∂z} \right] \mathbf{e}_z
| + | |
| − | \end{matrix}</math>
| + | |
| | | | |
| − | By choosing the <math>x</math> Cartesian axis so that it is aligned with the velocity <math>\mathbf{v}</math> of charge <math>q</math> such that <math>\mathbf{v}/|\mathbf{v}| = \mathbf{\hat{v}} = \mathbf{e}_x</math>, the above reduces to:
| + | The problem with this modification: |
| | | | |
| − | :<math>\nabla (\mathbf{v} \cdot \mathbf{A}) = \begin{matrix}
| + | 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. |
| − | \left[ v_x \frac{∂A_x}{∂x} + A_x \frac{∂v_x}{∂x} + A_y \frac{∂v_y}{∂x} + A_z \frac{∂v_z}{∂x} \right] \mathbf{e}_x \\
| + | |
| − | \left[ v_x \frac{∂A_x}{∂y} \right] \mathbf{e}_y \\
| + | |
| − | \left[ v_x \frac{∂A_x}{∂z} \right] \mathbf{e}_z
| + | |
| − | \end{matrix}</math>
| + | |
| | | | |
| − | <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A})</math> simplifies to: | + | 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. |
| | | | |
| − | :<math>\left[ \mathbf{A} \cdot \mathbf{a}/|\mathbf{v}| \right] \mathbf{\hat{v}} = \left[ A_x \frac{∂v_x}{∂x} + A_y \frac{∂v_y}{∂x} + A_z \frac{∂v_z}{∂x} \right] \mathbf{e}_x</math>
| + | ===Cyril Smith's proposal=== |
| | | | |
| − | The above may also be expressed as:
| + | Cyril Smith proposed adding the following gradient to the Lorentz force:<ref name="Marinov Generator"/> |
| | | | |
| − | :<math>\nabla_\mathbf{v}(\mathbf{A} \cdot \mathbf{v}) = (\mathbf{A} \cdot \mathbf{a})\mathbf{v} / |\mathbf{v}|^2</math>
| + | <math>- \nabla_\mathbf{A}(\mathbf{v}\cdot\mathbf{A})</math> |
| | | | |
| − | ==Draft==
| + | This is equal to: |
| | | | |
| − | The field experienced by a charge <math>q</math> viewed at rest in a static electromagnetic field is:
| + | : <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> |
| | | | |
| − | : <math>\mathbf{F}_{rest,static}/q = - \nabla \varphi</math>
| + | The idea behind this was to explain an observation in an experiment involving a "Marinov Generator"<ref name="Marinov Generator"/> in which longitudinal induction forces were produced. |
| | | | |
| − | The field experienced by a charge <math>q</math> viewed at rest in a dynamic electromagnetic field is: | + | The Lorentz force is: |
| | | | |
| − | : <math>\mathbf{F}_{rest,dynamic}/q = - \nabla \varphi - ∂\mathbf{A}/∂t</math>
| + | <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> |
| | | | |
| − | The field experienced by a moving charge <math>q</math> in a dynamic electromagnetic field (ignoring dilation of proper time relative to coordinate time) is:
| + | Therefore, adding the extra term proposed by Cyril Smith results in: |
| | | | |
| − | : <math>\mathbf{F}_{moving,dynamic}/q = - \nabla \varphi_q - ∂\mathbf{A}_q/∂t</math>
| + | <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> |
| | | | |
| − | Where:
| + | The problem with this modification: |
| | | | |
| − | * <math>\varphi_q = \varphi - \mathbf{v} \cdot A</math> is the scalar potential experienced by the moving charge.<ref>http://exvacuo.free.fr/div/Sciences/Dossiers/EM/ScalarEM/J%20Konopinski%20-%20What%20the%20Electromagnetic%20Vector%20Potential%20Describes%20-%20ajp_46_499_78.pdf</ref>
| + | 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>∂ \mathbf{A}_q/∂t = ∂\mathbf{A}/∂t + (\mathbf{v} \cdot \nabla)\mathbf{A}</math> is the partial time derivative of the magnetic vector potential experienced by the moving charge.
| + | : <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 \\ |
| − | Substituting per the above, the field experienced by the moving charge <math>q</math> is:
| + | - \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 |
| − | : <math>\mathbf{F}/q = - \nabla (\varphi-\mathbf{v} \cdot \mathbf{A}) - ∂\mathbf{A}/∂t - (\mathbf{v} \cdot \nabla)\mathbf{A}</math> | + | \end{matrix}</math> |
| − | : <math>\mathbf{F}/q = - \nabla \varphi + \nabla (\mathbf{v} \cdot \mathbf{A}) - ∂\mathbf{A}/∂t - (\mathbf{v} \cdot \nabla)\mathbf{A}</math>
| + | |
| − | | + | |
| − | Using Feynman subscript notation:
| + | |
| − | | + | |
| − | : <math>\nabla (\mathbf{v} \cdot \mathbf{A}) = \nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) + \nabla_\mathbf{A}(\mathbf{v} \cdot \mathbf{A})</math>
| + | |
| − | : <math>\nabla_\mathbf{A}(\mathbf{v} \cdot \mathbf{A}) = \mathbf{v} \times \nabla \times \mathbf{A} + (\mathbf{v} \cdot \nabla)\mathbf{A}</math>
| + | |
| − | : <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) = \mathbf{A} \times \nabla \times \mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v}</math>
| + | |
| − | | + | |
| − | Substuting for the curl of the vector potential and the curl of the immediate velocity field for a moving charge, we have:
| + | |
| − | | + | |
| − | : <math>\nabla_\mathbf{A}(\mathbf{v} \cdot \mathbf{A}) = \mathbf{v} \times \mathbf{B} + (\mathbf{v} \cdot \nabla)\mathbf{A}</math>
| + | |
| − | : <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) = \mathbf{A} \times \mathbf{ω}_\mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v}</math>
| + | |
| − | | + | |
| − | Where:
| + | |
| − | | + | |
| − | : <math>\mathbf{B} = \nabla \times \mathbf{A}</math> is the magnetic field.
| + | |
| − | : <math>\mathbf{ω}_\mathbf{v} = \nabla \times \mathbf{v}</math> is the angular rate of deflection.
| + | |
| − | | + | |
| − | Substituting per the above, the field experienced by the moving charge is:
| + | |
| − | | + | |
| − | : <math>\mathbf{F}/q = - \nabla \varphi - ∂\mathbf{A}/∂t + \mathbf{v} \times \mathbf{B} + (\mathbf{v} \cdot \nabla)\mathbf{A} + \mathbf{A} \times \mathbf{ω}_\mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v} - (\mathbf{v} \cdot \nabla)\mathbf{A}</math>
| + | |
| − | : <math>\mathbf{F}/q = - \nabla \varphi - ∂\mathbf{A}/∂t + \mathbf{v} \times \mathbf{B} + \mathbf{A} \times \mathbf{ω}_\mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v}</math>
| + | |
| − | | + | |
| − | This field includes the field from Lorentz plus two additional terms:
| + | |
| − | | + | |
| − | : <math>\mathbf{F}/q = \mathbf{F}_{Lorentz}/q + \mathbf{A} \times \mathbf{ω}_\mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v}</math>
| + | |
| − | : <math>(\mathbf{A} \cdot \nabla)\mathbf{v}</math> is the dot product of the magnetic vector potential with the gradient of the velocity field.
| + | |
| − | | + | |
| − | For a velocity field defined in the immediate neighborhood of a moving charge <math>q</math> at point <math>p</math>, where the local <math>(\nabla_\mathbf{v} \mathbf{A})_p</math> is a tangent vector on <math>\mathbf{A}</math> (the Lie derivative of <math>\mathbf{v}</math> along <math>\mathbf{A}</math>), the above is equivalent to:
| + | |
| − | | + | |
| − | : <math>(\mathbf{A} \cdot \nabla)\mathbf{v} = |\mathbf{A}|(\nabla_\mathbf{v} \mathbf{A})_p = (\mathbf{A} \cdot \mathbf{v})\mathbf{a}/|\mathbf{v}|^2</math>
| + | |
| − | | + | |
| − | Where <math>\mathbf{a}</math> is the convective acceleration of the charge, which equals:
| + | |
| − | | + | |
| − | : <math>\mathbf{a} = (∂\mathbf{v}/∂x)|(∂\mathbf{x}/∂t)|</math>
| + | |
| − | | + | |
| − | If the charge is taken as a point particle, the convective acceleration is the same as the acceleration.
| + | |
| − | | + | |
| − | : <math>\mathbf{a} = ∂²\mathbf{x}/∂t²</math>
| + | |
| − | : <math>\mathbf{A} \times \mathbf{ω}_\mathbf{v}</math> is the cross product of the magnetic vector potential and the angular rate of deflection.
| + | |
| − | : <math>\mathbf{ω}_\mathbf{v} = (\mathbf{v} \times \mathbf{a})/|\mathbf{v}|^2</math>
| + | |
| − | | + | |
| − | When fields are static, the field experienced by a moving charge is:
| + | |
| − | | + | |
| − | : <math>\mathbf{F}_{moving,static}/q = - \nabla \varphi + \mathbf{v} \times \mathbf{B} + \mathbf{A} \times \mathbf{ω}_\mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v}</math>
| + | |
| − | | + | |
| − | So for the case of static fields, the field on an accelerating charge is:
| + | |
| − | | + | |
| − | : <math>\mathbf{F}_{moving,static}/q = - \nabla \varphi + \mathbf{v} \times \mathbf{B} + \mathbf{A} \times (\mathbf{v} \times \mathbf{a})/|\mathbf{v}|^2 + (\mathbf{A} \cdot \mathbf{v})\mathbf{a}/|\mathbf{v}|^2</math>
| + | |
| − | | + | |
| − | While the power on an accelerating charge q subject to a static field is:
| + | |
| | | | |
| − | : <math>P_{moving,static} = q \left[\ -\nabla \varphi \cdot \mathbf{v} + (\mathbf{v} \times B) \cdot \mathbf{v} + \mathbf{A} \times (\mathbf{v} \times \mathbf{a}) \cdot \mathbf{v}/|\mathbf{v}|^2 + (\mathbf{A} \cdot \mathbf{v})(\mathbf{a} \cdot \mathbf{v})/|\mathbf{v}|^2\ \right]</math>
| + | ==Explaining Cyril Smith's "Marinov Generator" using standard Lorentz Transforms== |
| − | : <math>P_{moving,static} = q \left[\ -\nabla \varphi \cdot \mathbf{v} + \mathbf{A} \times (\hat{\mathbf{v}} \times \mathbf{a}) \cdot \hat{\mathbf{v}} + (\mathbf{A} \cdot \mathbf{\hat{v}})(\mathbf{a} \cdot \mathbf{\hat{v}})\ \right]</math>
| + | |
| | | | |
| − | The field on a moving charge in a changing electromagnetic field becomes:
| + | 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>. |
| | | | |
| − | : <math>\mathbf{F}_{moving,dynamic}/q = - \nabla \varphi - ∂\mathbf{A}/∂t + \mathbf{v} \times \mathbf{B} + \mathbf{A} \times (\mathbf{v} \times \mathbf{a})/|\mathbf{v}|^2 + (\mathbf{A} \cdot \mathbf{v})\mathbf{a}/|\mathbf{v}|^2</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 zero. |
| | | | |
| − | Summarizing the derivation of the last two terms above, we have:
| + | The Lorentz transforms of the electric scalar potential <math>\varphi</math> and magnetic vector potential <math>\mathbf{A}</math> are as follows:<ref name="Physics Formulas 2010">The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.</ref> |
| | | | |
| − | : <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) = \mathbf{A} \times \nabla \times \mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v}</math> | + | : <math>\varphi' = \gamma (\varphi - v A_\parallel)</math> |
| − | : <math>\mathbf{ω}_\mathbf{v} = \nabla \times \mathbf{v}</math> is the angular rate of deflection.
| + | : <math>A_\parallel' = \gamma (A_\parallel - v \varphi /c^2)</math> |
| − | : <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) = \mathbf{A} \times \mathbf{ω}_\mathbf{v} + (\mathbf{A} \cdot \nabla)\mathbf{v}</math>
| + | : <math>A_\bot' = A_\bot</math> |
| − | : <math>\mathbf{A} \times \mathbf{ω}_\mathbf{v} = \mathbf{A} \times (\mathbf{v} \times \mathbf{a})/|\mathbf{v}|^2</math> | + | |
| − | : <math>(\mathbf{A} \cdot \nabla)\mathbf{v} = (\mathbf{A} \cdot \mathbf{v})\mathbf{a}/|\mathbf{v}|^2</math>
| + | |
| − | : <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) = \mathbf{A} \times (\mathbf{v} \times \mathbf{a})/|\mathbf{v}|^2 + (\mathbf{A} \cdot \mathbf{v})\mathbf{a}/|\mathbf{v}|^2</math> | + | |
| | | | |
| − | A concise alternative to the above is:
| + | By setting the gradient of the scalar potential in the charge's rest frame to zero, the result is: |
| | | | |
| − | : <math>\nabla_\mathbf{v}(\mathbf{v} \cdot \mathbf{A}) = (\mathbf{A} \cdot \mathbf{a})\mathbf{v}/|\mathbf{v}|^2 </math> | + | : <math>\nabla \varphi' = 0 = \gamma \nabla (\varphi - v A_\parallel)</math> |
| | + | : <math>- \nabla \varphi = - v \nabla A_\parallel</math> |
| | | | |
| − | The field on a moving charge <math>q</math> in a changing electromagnetic field becomes:
| + | This can also be expressed as: |
| | | | |
| − | : <math>\mathbf{F}_{moving,dynamic}/q = - \nabla \varphi - ∂\mathbf{A}/∂t + \mathbf{v} \times \mathbf{B} + (\mathbf{A} \cdot \mathbf{a})\mathbf{v}/|\mathbf{v}|^2</math>
| + | <math> - |\mathbf{v}| \nabla A_\parallel </math> |
| | | | |
| − | The power on a moving charge <math>q</math> in a changing electromagnetic field becomes:
| + | Substituting this result in the Lorentz force yields: |
| | | | |
| − | : <math>P_{moving,dynamic} = q \left[\ \left(-\nabla \varphi - ∂\mathbf{A}/∂t\right) \cdot \mathbf{v} + (\mathbf{A} \cdot \mathbf{a})(\mathbf{v} \cdot \mathbf{v})/|\mathbf{v}|^2\ \right]</math>
| + | <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> |
| − | : <math>P_{moving,dynamic} = q \left[\ \left(-\nabla \varphi - ∂\mathbf{A}/∂t\right) \cdot \mathbf{v} + \mathbf{A} \cdot \mathbf{a}\ \right]</math>
| + | |
| | | | |
| − | ===Caveat=== | + | 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: |
| | | | |
| − | If the parent section to this subsection is correct, that the field on a moving charge <math>q</math> in a changing electromagnetic field becomes:
| + | <math>\mathbf{E} = - |\mathbf{v}| \nabla A_\parallel</math> |
| | | | |
| − | : <math>\mathbf{F}_{moving,dynamic}/q = - \nabla \varphi - ∂\mathbf{A}/∂t + \mathbf{v} \times \mathbf{B} + (\mathbf{A} \cdot \mathbf{a})\mathbf{v}/|\mathbf{v}|^2</math>
| + | Integrated over the path taken by the charge yields same voltages that were predicted by Cyril Smith,<ref name="Marinov Generator (paper)"/> but here we use a correct expression and without modifying Einstein's theory of Special Relativity. |
| | | | |
| − | The next question one may have is if the force generated by multiplying the field by the charge value (i.e. <math>\mathbf{F}_{moving,dynamic}</math>) is a <math>\frac{d(m \mathbf{v})}{dt}</math> type force or a <math>m \frac{d\mathbf{v}}{dt} = m\mathbf{a}</math> type force. If the latter is true, then the presence of a permanent magnet having a magnetic moment <math>\mathbf{m}</math> in the vicinity of a copper loop, which provides the vector potential <math>\mathbf{A}</math>, would cause the effective inductance to change via the last term on the right <math>(\mathbf{A} \cdot \mathbf{a})\mathbf{v}/|\mathbf{v}|^2</math>. However, the last term on the right is a <math>\mathbf{v} \frac{dm}{dt}</math> type term. If this is indeed the case, then such a term cannot contribute to acceleration and therefore may not contribute to any changes to velocity. Rather this force would be seen as changing the "effective-inertia" of the charge, implying that the term in essence carries with it, in part, the mass of other objects which provide the vector potential <math>\mathbf{A}</math>, in addition to the mass of the charge <math>q</math>. It could very be argued that mass is not a property of a particle-in-itself but rather a property of a particle coupled to the potentials and/or fields of other particles, such that the mass is the result of exchange forces between particles. If that is true, then to derive the force in terms of the common definition of <math>\mathbf{F} = m \frac{d\mathbf{v}}{dt} = m\mathbf{a}</math>, the last term should be subtracted from both sides. The result of doing so is the usual Lorentz Force:
| + | ===The New Idea=== |
| | | | |
| − | : <math>\mathbf{F}_{Lorentz} = m \mathbf{a} = q \left[\ - \nabla \varphi - ∂\mathbf{A}/∂t + \mathbf{v} \times \mathbf{B}\ \right]</math>
| + | This new result has inspired a development beyond the S.H.O. Drive. A much more powerful and compact permanent magnet system is now envisioned, the viability of which is rests on the condition that <math>\langle \nabla \varphi' \rangle = 0</math> each charge <math>q</math>. The preliminary name is Makerarc (MAY-KERR-ARK), which stands for: |
| | | | |
| − | According to Kirk T. McDonald, professor of High-Energy Experimental Physics at Princeton,<ref>http://www.physics.princeton.edu/~mcdonald/</ref> the presence of a stationary permanent magnet does not affect the inductance of a stationary solenoid.<ref>Self Inductance of a Solenoid
| + | * '''M'''agnetic |
| − | with a Permanent-Magnet Core http://www.physics.princeton.edu/~mcdonald/examples/magsol.pdf</ref> This indicates that the extra term above cannot be allowed to affect in anyway the acceleration of a charge moving through the field of a magnet. This is great news for the S.H.O. Drive project because it means that it will not take additional energy to drive current to repel the central permanent magnet. The above predicted additional force term (depending on the gauge-dependent Magnetic Vector Potential) may be used to help enforce the rule of conservation of momentum (in 3-space) in the case of time-varying charge densities produced by non-radiating, discontinuous, yet constant current densities. We can add to this contribution the force term suggested by Koen J. van Vlaenderen in his 2015 paper "General Classical Electrodynamics"<ref>http://vixra.org/abs/1512.0297</ref> (as mentioned in the Comment Record section above) which is dependent on the gauge-dependent Electric Scalar Potential. Perhaps both of theses "forces" only change "mass" not velocity, or perhaps more accurately, they change the inertia that a charge carries by virtue of interactions via the Magnetic Vector Potential and Electric Scalar Potential.
| + | * '''A'''tom |
| | + | * '''K'''inetic |
| | + | * '''E'''nergy |
| | + | * '''R'''eservoir |
| | + | * '''A'''nd |
| | + | * '''R'''esource |
| | + | * '''C'''hannel |
| | | | |
| − | So in the end, the actual accelerations of charges in 3-space would appear to follow exactly from what you would expect Special Relativity / Lorentz force to predict, but perhaps there are indeed underlying and non-apparent energy and force exchanges that may prove more apparent in future discoveries. Sincerely, ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'' 14:53, 15 September 2016 (PDT)
| + | Details pending. Stay turned. Sincerely, ''[[User:S.H.O.|S.H.O.]] <sup>[[User_talk:S.H.O.|talk]]</sup>'' 22:09, 5 March 2017 (PST) |
| | | | |
| − | ===References===
| + | ==References== |
| | <references /> | | <references /> |
| | | | |
causes changes in kinetic energy that, in the low-frequency approximation, pretty much matches the amount of energy induced into surrounding coils.
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)
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.
The idea behind this was to explain an observation in an experiment involving a "Marinov Generator"[3] in which longitudinal induction forces were produced.
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:
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 zero.
By setting the gradient of the scalar potential in the charge's rest frame to zero, the result is:
Integrated over the path taken by the charge yields same voltages that were predicted by Cyril Smith,[4] but here we use a correct expression and without modifying Einstein's theory of Special Relativity.
This new result has inspired a development beyond the S.H.O. Drive. A much more powerful and compact permanent magnet system is now envisioned, the viability of which is rests on the condition that [math]\langle \nabla \varphi' \rangle = 0[/math] each charge [math]q[/math]. The preliminary name is Makerarc (MAY-KERR-ARK), which stands for:
