The Electrotonic Field

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    Imagine that someone has established an electric charge of magnitude q1 on a particle that they have constrained to remain motionless in the inertial frame that we occupy. Visualize further in the laboratory of your mind a particle upon which we have established an electric charge of magnitude q2 and notice that we have positioned it far enough from q1 that the force exerted between the two particles stands close enough to zero that we can ignore it for our calculations as negligible: in such a case we say that, relative to q1, q2 lies at infinity (bearing in mind, of course, that the word infinity does not denote an actual number or place).

    Moving q2 and bringing it to rest at some point close to q1 involves doing work on the particle (if the force between the particles repels them from each other) or gaining work from the particle (if the force between the particles attracts them to each other). The particle thus gains or loses energy in the amount

(Eq'n 1)

in which φ1 represents the potential of the electrostatic field emanating from q1. At the same time q1 gains or loses energy vis-a-vis q2 in the amount

(Eq'n 2)

Using Coulomb's law to represent the potential,

(Eq'n 3)

lets us see readily that E1=E2. Of course, that statement conforms to what we would have expected based on the law of conservation of energy, which Hermann Ludwig Ferdinand von Helmholtz (1821 Aug 31 - 1894 Sep 08) first presented in "▄ber die Erhaltung der Kraft" (On the Conservation of Force, 1847). We see how that law works when we bring q2 close to q1 and thereby gain or lose E2 and then we move q1 away from q2 and thereby lose or gain E1, which lose or gain must bring our total store of energy to its original value.

    Our understanding of Relativity tells us that a gain or loss of energy corresponds to a gain or loss of inertial mass. In this imaginary experiment our two particles thus gain or lose the same amount of mass,

(Eq'n 4)

If our source particle, carrying q1, moves with some velocity v, then that change in the particle's mass also corresponds to a change in the particle's linear momentum,

(Eq'n 5)

That change in momentum did not come about as a consequence of an externally-applied force actually pushing on the particle and accelerating it, but came from another particle and its attendant electric field coming close to the source particle. That fact means that we have no external force provider to receive the equal and oppositely directed reaction mandated by Newton's third law of motion. We must thus assume that the test particle, bearing the charge q2, manifests the necessary reaction as a virtual momentum,

(Eq'n 6)

    We want to describe the force that a change in that virtual momentum would exert upon q2, but we want to express it in terms of things that we can control, the properties of q1. Thus we define

(Eq'n 7)

and then express the force on q2 as

(Eq'n 8)

We can reconceive that force as if q2 were immersed in an electric field conforming to the description

(Eq'n 9)

In that equation v2 denotes the motion of the particle bearing q2.

    In that derivation the vectorfield A looks very much like what Michael Faraday (1791 Sep 22 -- 1867 Aug 25) described when he wrote:

    "While the wire is subject to either volta-electric or magneto-electric induction it appears to be in a peculiar state, for it resists the formation of an electric current in it; whereas, if in its common condition, such a current would be produced; and when left uninfluenced it has the power of originating a current, a power which the wire does not possess under ordinary circumstances. This electrical condition of matter has not hitherto been recognized, but it probably exerts a very important influence in many if not most of the phenomena produced by currents of electricity. ... I have, after advising with several learned friends, ventured to designate it as the electro-tonic state."

    In the vectorfield A we have a mathematical description of Faraday's electrotonic state. Now we refer to the phenomenon itself as the electrotonic field and its mathematical description as the magnetic vector potential.

    If we include the negative gradient of the electrostatic potential, Equation 9 gives us the mathematical description of the Lorentz force. The first term in Equation 9 represents electromagnetic induction, the phenomenon that Faraday discovered on 1831 Aug 29: this term describes the part of the electrodynamic field generated by changes in the source of the electrotonic field itself, either in its motion or its arrangement of electric charge. The second term represents the interaction of the test charge, as an electric current, with the electrotonic field's magnetic induction B, which we define as

(Eq'n 10)

This term gives us what we normally think of as the magnetic force. And the third term gives us the negative gradient of what we might regard as a kind of magnetic scalar potential acting on a particle in an inertial frame separated from ours by the velocity v2. For the total force acting on our test particle, then, we have

(Eq'n 11)

The fields represented in that equation conform to the six laws that James Clerk Maxwell listed in his paper "On Faraday's Lines of Force" (Dec 1855 and Feb 1856):

    "Law I: The entire electro-tonic intensity round the boundary of an element of surface measures the quantity of magnetic induction which passes through that surface, or, in other words, the number of lines of magnetic force which pass through that surface."

    That law expresses the integral form of Equation 10. If we have a surface of area S with a boundary MS, then we calculate the flux of the magnetic induction field through that surface by integrating the dot product of the B-field through the surface with the oriented differential element of area over the entire surface; that is,

(Eq'n 12)

in which we have used Stokes' theorem to replace the integral of the curl of A over the surface with the equivalent integral of A around the boundary of the surface.

    "Law II: The magnetic intensity at any point is connected with the quantity of magnetic induction by a set of linear equations, called equations of conduction."

    In this law Maxwell expressed the magnetic analogue of Ohm's law of electric conduction. Just as Ohm's law applied to electric currents flowing in closed circuits, so the magnetic analogue applies to magnetic field intensities "flowing" in closed magnetic circuits. The magnetic analogue differs from Ohm's law primarily in the fact that, while the electric conductivity of vacuum equals zero, thereby keeping the electric currents confined within the metal in an electric circuit, the magnetic permeability of vacuum, the magnetic-field analogue of electric conductivity, does not equal zero; thus, we can only create magnetic circuits properly analogous to electric circuits by using materials with extremely high magnetic permeabilities (ferromagnetic materials, like iron, generally have permeabilities greater than one hundred times that of vacuum, so an iron ring gives us a good approximation of a perfect simple magnetic circuit).

    We start our analysis of this law by stating that the magnetomotive force imposed in a magnetic circuit equals the magnetic flux that appears in the circuit multiplied by the reluctance of the circuit. The magnetomotive force equals the total electric current flowing in a closed curve that encloses part of the magnetic circuit and also equals the magnetizing force (H) integrated around the length of the circuit, so we have

(Eq'n 13)

in which N represents the number of times the wire carrying the electric current loops around the magnetic element and I represents the amount of electric current that we put into the wire. In accordance with that description, we generally measure magnetomotive force in ampere-turns. We then have the magnetic analogue of Ohm's law as

(Eq'n 14)

in which Φm represents the flux of the magnetic field in the circuit and R represents the circuit's reluctance, which we calculate as

(Eq'n 15)

in which μ represents the magnetic permeability of the circuit element and S represents the cross-sectional area of the circuit element whose length equals dl.

    The magnetic induction (B), also known as the flux density, conforms to the statement that

(Eq'n 16)

If we restrict our analysis to one element of the circuit of length dl, we have Equation 14 as

(Eq'n 17)

which gives us

(Eq'n 18)

If we have the necessity of representing the permeability with a tensor (essentially a three-by-three matrix), then Equation 18 becomes a trio of linear equations, the equations of conduction, corresponding to Equation 14 and analogous to Equations 22.

    "Law III: The entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface."

    We call this Ampere's law in its form as a precursor of the fourth of Maxwell's Equations and we can represent it mathematically by slightly modifying Equation 13. That equation expresses the fact that if we have an electric current flowing in a coil of wire, in essence a single current I=Ni flowing around a loop, then that current produces a magnetizing force, which, when integrated around a closed curve that passes through the coil, equals the total current. Now we can reverse that statement based on the fact that the closed curve along which we integrate the magnetizing force is the boundary of a surface through which the electric current must pass. We thus get Law III in the form

(Eq'n 19)

in which j represents the electric current density at any point. If we use Stokes' theorem to replace the first integral in that equation with the equivalent integration over the bounded surface and then differentiate the resulting equation with respect to area on the surface, we get

(Eq'n 20)

which differs from the fourth Maxwell equation only in lacking the electric displacement term.

    "Law IV: The quantity and intensity of electric currents are connected by a system of equations of conduction."

    In Maxwell's writing the phrase "intensity of electric current" corresponds to what we call the voltage in the circuit. Law IV, then, corresponds to Ohm's law. But Maxwell didn't use the simple scalar law

(Eq'n 21)

that we got from the German physicist Georg Simon Ohm (1789 Mar 16 -- 1854 Jul 06) in 1827; Maxwell contemplated electric current flowing in bulk conductors and thus had to use something with broader application. He represented electric currents as vectors -- so much flowing in the x-direction, so much flowing in the y-direction, and so much flowing in the z-direction at each point inside the conducting medium -- and did the same with the driving voltage. In that circumstance he had to replace the scalar resistance R with components of a three-by-three matrix so that he obtained a trio of linear equations that describe the conduction of electricity in a bulk medium, the equations of conduction,

(Eq'n 22)

    "Law V: The total electro-magnetic potential of a closed circuit is measured by the product of the quantity of the current multiplied by the entire electro-tonic intensity estimated in the same direction around the circuit."

    Electric current denotes the rate at which electric charge passes any given point, I=dQ/dt. In a closed one-wire circuit I has the same value everywhere along the wire. If we represent Maxwell=s potential with Π, the Law V states

(Eq'n 23)

in which dQ/dl indicates the amount of conductible electric charge per unit length in the wire and dl/dt indicates the speed at which that charge flows through the wire. Because dQ/dt represents a constant of the circuit, we can rewrite it inside the integration, thereby getting

(Eq'n 24)

In that equation shifting the differential time does not change the value of Π, but it transforms the integral from an integration of the product of current and the electrotonic field with respect to distance around the circuit into an integration of the product of the electrotonic field and the velocity of the conductible charge with respect to the total conductible charge in the circuit. That gives us the potential that we see reflected in the fourth term in the right side of Equation 11.

    "Law VI: The electro-motive force on any element of a conductor is measured by the instantaneous rate of change of the electro-tonic intensity on that element, whether in magnitude or direction."

    In mathematical formalism we have that statement as

(Eq'n 25)

which we recognize as one form of the third of Maxwell's Equations, which we also call Faraday's law.

    That law sums up a more elaborate statement that Maxwell made in describing the electrotonic field. In his paper "On Faraday's Lines of Force", read in two parts (Dec 1855 and Feb 1856) for the Cambridge Philosophical Society, Maxwell wrote in Part II;

    "When a conductor moves in the neighborhood of a current of electricity, or of a magnet, or when a current or magnet near the conductor is moved, or altered in intensity, then a force acts on the conductor and produces electric tension, or a continuous current, according as the circuit is open or closed. This current is produced by changes of the electric or magnetic phenomena surrounding the conductor, and as long as these are constant there is no observed effect on the conductor. Still the conductor is in different states when near a current or magnet, and when away from its influence, since the removal or destruction of the current or magnet occasions a current, which would not have existed if the magnet or current had not been previously in action.

    "Considerations of this kind led Professor Faraday to connect with his discovery of the induction of electric currents the conception of a state into which all bodies are thrown by the presence of magnets and currents. This state does not manifest itself by any known phenomena as long as it is undisturbed, but any change in this state is indicated by a current or tendency towards a current. To this state he gave the name of the "Electro-tonic State," and although he afterwards succeeded in explaining the phenomena which suggested it by means of less hypothetical conceptions, he has on several occasions hinted at the probability that some phenomena might be discovered which would render the electro-tonic state an object of legitimate induction. ..."

    The astonishing part of that statement consists of its first sentence. That sentence describes almost exactly the imaginary experiment that Albert Einstein described at the beginning of his original paper on Relativity, "On the Electrodynamics of Moving Bodies" (1905).

    The electromagnetic potentials, φ and A, come to us as Štherial constructs and thus seem to us as unreal as any figment of the imagination. We may conceive them as nothing more than a convenient way of encoding the more real-seeming electric and magnetic forces. But it turns out that the potential fields -- the electrostatic field and the electrotonic field -- play a more fundamental role in Reality than do the electric and magnetic fields. We can see this proposition demonstrated more clearly in the following three examples.

1. Feynman's Carousel

    Imagine that we have a lazy-susan, which consists of a turntable made of polystyrene mounted on a stand through a bearing that allows the turntable to rotate effectively without friction. On the turntable's rim we have mounted a series of metal pegs or small blocks and near those we lay a circular coil of wire that draws current from a battery or fuel cell at the turntable's center. We put a certain amount of electric charge on each and all of the pegs and then, with the turntable initially motionless, we open the circuit of which the wire is a part. What happens as a consequence of the electric current suddenly going to zero?

    In the early 1940's, while he was working on the Manhattan Project, Richard Feynman got this little problem from Ted Welton, a friend he had met at M.I.T. when he was an undergraduate. When we look at this problem naively we get a Feynmanian paradox, a problem that gives us one result when we analyze it one way and a different result when we analyze it a different way. Here are two analyses that Feynman started with:

    1. When current flows in the wire it generates a magnetic field that surrounds the wire and envelopes the pegs. When the current diminishes, that field collapses and generates an electric field at every point it occupies. At each peg the electric field points in the direction parallel to the turntable's rim at that point, so the electromotive forces exerted upon the complete set of pegs exerts a torque upon the turntable and makes it rotate.

    2. When the experiment begins the turntable sits motionless, so the lazy-susan possesses zero angular momentum (note that we don't count the electric current as conferring angular momentum on the system, because we could, in concept, create the current from equal quantities of electrons and positrons moving in opposite directions). We know that angular momentum must obey a conservation law, so we infer that after the current ceases to flow in the wire the lazy-susan will still have zero angular momentum and, thus, the turntable won't rotate.

    So does the turntable rotate or not? If we analyze the problem through the electromagnetic theory, we get one result and if we analyze it through Newtonian mechanics, we get a different result that excludes the first. How can we possibly dissolve this paradox?

    We dissolve it in essentially the same way that Feynman did. We do that by noting that our assumption that the motionless lazy-susan at the beginning of our imaginary experiment has zero angular momentum turns out to stand false to fact. In our deduction of the existence of the electrotonic field I noted that an electrotonic field confers a virtual linear momentum upon any electrically charged particle immersed in it. Thus, we can see that the electrotonic field generated by the electric current in our device puts the charged pegs, as a system, into a state of non-zero virtual angular momentum. When the current stops flowing, the consequent collapse of the electrotonic field converts that virtual angular momentum into actual angular momentum (or converts electromagnetic angular momentum into mechanical angular momentum, if you prefer a less abstract approach) and the turntable rotates. So either analysis, properly conceived and executed, yields the same result.

    That view of the electrotonic field sounds very much like the description Maxwell gave of the phenomenon when he described it as "a state into which all bodies are thrown by the presence of magnets and currents. This state does not manifest itself by any known phenomena as long as it is undisturbed, but any change in this state is indicated by a current or tendency towards a current." Things have changed a bit since 1856: we now know of two phenomena that we can interpret as manifestations of an unchanging electrotonic field -- the Marinov motor and the Aharonov-Bohm effect.

2. The Marinov Motor

    Invented by the Bulgarian physicist Stefan Marinov (1930 Feb 01 -- 1997 Jul 15), the Marinov motor gives us a physical manifestation of the interaction between an electric current and a pair of vortical electrotonic fields. The core of the device consists of a permanent magnet in the form of a rectangular loop (which we may approximate with a pair of bar magnets joined to each other at both ends by iron keepers), in which the magnetic field remains confined within the metal. We imagine so mounting this core on a base that the two bar magnets rise vertically. The second component of the device consists of a copper annulus whose inner diameter exceeds the width of the core. We suspend that annulus from a bearing that holds it in the core's horizontal plane of symmetry and leaves it free to rotate about its center, which coincides with the center of the core. And the third component of the device consists of two straight wires that touch opposite sides of the annulus, either on the outer rim or on its inner rim. For the device to work properly, the wires must lie in the vertical plane of symmetry that includes both bar magnets' centerlines.

    Imagine that we have set this device up in such a way that from our point of view the bar magnet on the left side of the core has its north pole at its top and its south pole at its bottom. If we look on that configuration from above, the electrotonic field of the left magnet will circulate in the counterclockwise sense and the electrotonic field of the right magnet will circulate in the clockwise sense. Because no magnetic field exists outside the bar magnets (ideally), each magnet's external electrotonic field must be irrotational and thus has the form of a vortex,

(Eq'n 26)

in which A0 represents the strength of the field at the magnet's surface, a distance r0 from the magnet's centerline, and r represents the distance of the field point from the magnet's centerline. Of course, in the assembled device the electrotonic field at any point is the vector superposition of the electrotonic fields of the two magnets.

    Now let electric current flow into the device from the left. If the brushes conveying current into and out of the annulus touch the outer rim of the annulus, the annulus will turn clockwise. If the brushes touch the inner rim of the annulus, the annulus will rotate counterclockwise.

    In the first case the current must move partly inward as it spreads across the annulus and moves longitudinally, in both directions, toward the brush taking current out of the annulus. That inward motion subjects each element of current to an increase in the electrotonic field in the counterclockwise direction, so that element comes under a force in the clockwise direction. On the opposite side of the device the current has a component of motion away from the core as it goes to the brush that will take it off the annulus. That outward motion takes each element of current through a decrease in the clockwise circulating electrotonic field, thereby subjecting that element to a clockwise-oriented force. Each element of current will also encounter forces due to its longitudinal motion and longitudinal changes in the electrotonic field, but for each element those forces add up to a net zero due to their symmetry. Thus the annulus comes under the exertion of forces that produce a net clockwise torque, which makes the annulus turn clockwise. In the second case, putting the brushes against the inner rim of the annulus reverses the radial motions of the current and thus reverses the forces, which results in the annulus turning counterclockwise.

    That analysis suggests a simple improvement in the Marinov motor. We need only cut radial slots in the annulus from the rim that the brushes touch to points just short of the other rim and then fill them with some electrically insulating material. By thus restricting the electric current to move only radially until it reaches the rim opposite the one the brushes touch we can maximize the electrotonic force exerted upon it. While it doesn't look like a very powerful motor, the Marinov motor may find application in situations in which we want a low-torque, vibration-free DC motor.

3. The Aharonov-Bohm Effect

    Werner Ehrenberg (no dates found) and Raymond Eldred Siday (1912 -- 1956) discovered this effect in 1949, but it gained so little attention that neither Yakir Aharonov (1932 -- ?) nor David Joseph Bohm (1917 Dec 20 -- 1992 Oct 27) knew of it when they described it anew in 1959. While I don't disparage the practice of giving it the names of those who described it, I believe that we might better call the effect quantum electrotonic refraction, because that name gives us an indication of the effect that an electrotonic field has on the wave nature of fundamental particles.

    Imagine a long, straight pipe made of a superconducting material that has an electric current circulating around its circumference. Both inside and outside the pipe the current generates an electrotonic field whose lines of electrotonicity have the form of circles concentric with the pipe's centerline. Inside the pipe the intensity of the electrotonic field stands in direct proportion to distance from the centerline of the pipe: that part of the field has a non-zero curl, so inside the pipe we have a magnetic field whose lines of force run parallel to the centerline. Outside the pipe the intensity of the electrotonic field stands in inverse proportion to distance from the centerline of the pipe (as in Equation 26): that part of the field has a curl equal to zero, so outside the pipe we have no magnetic field whatsoever.

    Imagine putting a wide, flat, opaque barrier near the pipe and parallel to it. The barrier has two long, straight, narrow slits cut into it in a direction parallel to the pipe and so located in the barrier that particles passing through them in directions at or near that perpendicular to the barrier will not hit the pipe. At some distance from the pipe, opposite the barrier, we set up a wide, flat screen parallel to the barrier; that screen will fluoresce where and when an electron hits it. As seen from the screen the pipe lies between the two slits. Now, from a source some distance from the barrier on the side opposite the pipe, shoot imaginary electrons at the barrier.

    In the quantum theory particles move in conformity with the propagation of a wave function that describes the particle's position and linear momentum by a probability distribution in accordance with Heisenberg's indeterminacy principle. Each of our imaginary electrons has a propagation function cos(kĽx-ωt), which describes how the particle moves in the x-direction with the elapse of time t. That monochromatic wave function gives us a gross oversimplification of the situation that I have described, but a proper representation of the situation involves a linear sum of such wave functions, so our analysis of one of them gives us a good, if only approximate, insight into the phenomenon of electrotonic refraction of particle trajectories. In particular, it tells us that the wave propagates through the two slits in the barrier and then interferes with itself. At any given point on the screen the phase difference between the parts of the wave that followed the two possible paths to that point determines whether the electrons paint a bright spot on the screen or leave it dark. Werner Ehrenberg and Raymond Eldred Siday and then Yakir Aharonov and David Bohm discovered that a vortical electrotonic field will shift that phase difference.

    The vector k, the wave number, corresponds to the particle's linear momentum in accordance with k=p/S , so when an electron enters an electrotonic field it gains or loses virtual linear momentum and its wave vector changes in accordance with

(Eq'n 27)

in which the lower-case ee represents the fundamental unit of electric charge, the charge on the electron. We calculate the amplitude of that propagation vector by dividing two pi radians by the wavelength of the wave.

    For any given path, from the electron's source in our experiment, for example, to the electron detector (a given point on the screen in this case), we can calculate the total angle contained in the wave train extending along that path at any instant. We simply integrate the propagation vector over the length of the path,

(Eq'n 28)

in which N represents the number of whole waves on the path and θ represents a partial wave, a phase difference between the source and the detector. Between the source and any given point on the screen each electron has two paths that its associated wave can follow: the phase difference between those two paths,

(Eq'n 29)

determines whether the two waves interfere constructively or destructively and by how much. In that calculation, when we make the appropriate substitution from Equation 27, the terms involving p cancel each other, so we retain only the electrotonic terms. Because both paths have the same endpoints, they for a closed loop: subtracting the integral of k along one path from the integral of k along the other path equals the integral of k around the loop, so we have for the phase difference

(Eq'n 30)

    If we extinguish the electrotonic field, such as by raising the temperature of the pipe above the superconducting transition temperature of its material (blowing warm helium through the pipe would do the job nicely), thereby causing normal electrical resistance to reduce the electric current to zero, then that phase difference vanishes. Applied point by point across the screen, that vanishing of the electrotonic phase difference corresponds to a shift of the interference pattern across the screen. Experiments conducted in the 1980's confirmed that such a pattern shift does, in observed fact, occur when the electrotonic field vanishes.

    Equation 30 gives us a very difficult calculation. Fortunately we have a sneaky trick that we can use to make it simpler and easier. Stokes' theorem tells us that integrating a vectorfield dot product around a closed loop gives us the same result that we get from integrating the dot product of that vectorfield's curl over the surface bounded by the loop, so we can rewrite Equation 30 as

(Eq'n 31)

in which Φ represents the total magnetic flux passing through the surface S. In our present example the magnetic field exists only inside the pipe (which passes through the surface bounded by our electrons' two paths) and its intensity has the same value throughout the pipe, so we calculate the flux by multiplying the field strength at any point inside the pipe by the pipe's cross-sectional area. Thus we get

(Eq'n 32)

in which R represents the radius of the pipe.

    In the latter two of these three examples we see how an electrotonic field can influence the motions of electrically charged particles even where we have zero electric and magnetic fields. That fact implies that, along with the electrostatic potential field, the electrotonic field is more fundamental to Reality than is the electromagnetic field. That implication gets a further boost when we express Maxwell's Equations in terms of the potentials instead of the forcefields;

(Eq'n 33)

Those wave equations (with sources) show more clearly than do the standard Maxwell's Equations the exquisite spatio-temporal symmetry underlying the phenomena of electricity and magnetism. In that picture we can conceive the electrostatic-potential and electrotonic fields as the temporal component and the spatial components, respectively, of a four-vector, as distinct from the 16-element tensor that we must use to give proper representation to the electromagnetic field. Because we want the laws of physics to evolve out of the simplest possible description of Reality, in accordance with Ockham's Razor, elegance implies fundamentality.

Appendix: A Review of the Electrostatic Field

    In several other essays I have shown how to deduce certain features of the electrostatic field, whose relativistic extension gives us the electrotonic field. Some deductions, such as those exploiting Newton's third law of motion, were discovered so long ago that we cannot say for certain who devised them; others, such as the use of conservation of angular momentum to infer the inverse radial distance property of the potential, we can trace to specific people (James Clerk Maxwell in this case); and in yet others, such conservation of electric charge, I have had to fill the gap myself. Here I present a checklist of those deductions, putting each feature under the principle from which we deduce it.

    1. The finite-value theorem:

        a) No conserved quantity can manifest an infinite change or an infinite rate of change (even for an infinitesimal elapse of time). Thus, the force exerted by one particle upon another must be exerted over a finite distance. The force must conform to some function of that particle's coordinates and the coordinates of the particle receiving the force;

(Eq'n A-1)

        b) No conserved quantity can manifest itself in an infinitesimal amount or change by an infinitesimal amount. That fact necessitates that electric charge have a smallest unit of finite, definite value that it can manifest and by which it can change.

    2. Conservation of Linear Momentum:

        Given a property Q by virtue of which it exerts a force upon a particle carrying q of the same property, the particle exerts a force proportional to some f(Q). The force-receiving particle must exert a force proportional to f(q). The forces must have equal magnitudes, though act in opposite directions, so we must have a force conforming to

(Eq'n A-2)

    3. Newton's Zeroth Law:

        The field must spread smoothly and continuously out from its source, so over a minuscule distance we must have a ratio of F(r+dr)-F(r) to dr equal to a finite, definite value. Thus the derivatives of F must exist.

    4. Conservation of Angular Momentum:

        a) Lest they constitute a couple that creates angular momentum ex nihilo, the forces that the two particles exert upon each other must act along the straight line connecting the two particles. For a particle lying at the origin of a coordinate frame we then have as true to Reality

(Eq'n A-3)

        b) Maxwell's deduction. Because gradf=0 (because f(Q) is not a function of the coordinates), we have

(Eq'n A-4)

for an irrotational field (one that does not make a uniform distribution of charge spin). That fact necessitates that the dot product FĽdr be an exact differential, which necessitates in turn the existence of a potential field φ such that

(Eq'n A-5)

    5. Helmholtz's Theorem:

        A nonconstant irrotational vectorfield must emanate from sources of some density distribution ρ such that

(Eq'n A-6)

the Laplace-Poisson equation (also known as the first of Maxwell's Equations), stands true to Reality. The integral version of that equation gives us Gauss's law, which tells us that

(Eq'n A-7)

For the smallest unit of electric charge we have the density distribution function as the product of the charge itself and the Dirac delta.

    6. Conservation of Electric Charge:

        a) We must have

(Eq'n A-8)

If that statement did not stand true to Reality, then bringing two or more unit charges together would create additional charge ex nihilo. By the conservation law the creation of an equal amount of the opposite charge, at the same place at the same time, would have to accompany the creation of new charge. Consequently, the bringing of charges together yields the same result that we get from the simple linear addition of charges. Thus, if f(Q) follows a rule different from Equation A-8, its actual manifestation is identical to and indistinguishable from Equation A-8.

        b) At any point in space the electric fields emanating from two or more particles located elsewhere in space combine in a linear vector sum, a simple superposition. If that statement did not stand true to Reality, the overlapping of two or more electric fields would generate an additional electrostatic field whose necessarily existent discontinuities (sources by Helmholtz's theorem) correspond to equal and opposite electric charges created ex nihilo, but not at the same time and place as required by Relativity.

    Finally, I want to review our deduction of the law pertaining to the conservation of electric charge. We know that because the lines of electric force must spread out in all available directions from the force-exerting property, the electric charge, any discontinuity in the field represents the presence of an electric charge. The electric field of a single charge cannot decline to zero, except at a discontinuity, and it cannot reach the boundary of space (the boundary cannot have an electric charge or any other property). Thus we infer that every electric field has two ends (discontinuities) of opposite polarity. The Gauss's law version of Helmholtz's theorem tells us that a given charge produces a field with an invariant flux, so the electric charges at the beginning and the end of a field must have equal magnitudes. Therefore, we infer that the net electric charge of the Universe equals zero, always has, and always will.

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