Electromagnetic Momentum of a Current
(22) We may begin by considering the state of the field in the neighbourhood of an electric current. We know that magnetic forces are excited in the field, their direction and magnitude depending according to known laws upon the form of the conductor carrying the current. When the strength of the current is increased, all the magnetic effects are increased in the same proportion. Now if the magnetic state of the field depends on motions of the meditum, a certain force must be exerted in order to increase or diminish these motions, and when the motions are excted they continue, so that the effect of the connexion between the current and the electromagnetic field surrounding it, is to endow the current with a kind of momentum, just as the connexion between the driving-point of a machine and a fly-wheel endows the driving-point with an additional momentum, which may be called the momentum of the fly-wheel reduced to the driving-point. The unbalanced force acting on the driving-point increases this momentum, and is measured by the rate of its increase.
In the case of electric currents, the resistance to sudden increase or diminution of strength produces effects exactly like those of momentum, but the amount of this momentum depends on the shape of the conductor and the relative position of its different parts.
The momentum described here is provided by the primary angular momentum of the electron and photon. It is seen in the APM that what we consider at the macro level of existence (that angular momentum is a property of something), at the quantum level angular momentum is a thing itself. It is because electrons and photons are primary angular momentum that they can appear as either waves or particles. The structures of the electron and photon are explained in Secrets of the Aether.
Primary angular momentum is a more primary level of existence than are atoms. The property of solidity of solid particles only begins at the level of bound particles (such as atoms). More primary existences such as electrons and photons appear less solid, yet they still possess (and are) angular momentum. The Aether is even more primary than electrons and protons, and there are yet more primary levels of existence than Aether, such as the Gforce and dark matter.
Maxwell's point is that current has momentum and there is a delay in its effect upon the magnetic field when the current is changed in magnitude.
Mutual Action of two Currents
(23) If there are two electric currents in the field, the magnetic force at any point is that compounded of the forces due to each currently separately, and since the two currents are in connextion with every point of the field, they will be in connexion with each other, so that any increase or diminution of the one will produce a force acting with or contrary to the other.
Dynamical Illustration of Reduced Momentum.
(24) As a dynamical illustration, let us suppose a body C so connected with two independent driving-points A and B that its velocity is \(p\) times that of A together with \(q\) times that of B. Let \(u\) be the velocity of A, \(v\) that of B, and \(w\) that of C, and let \(\delta x\), \(\delta y\), \(\delta z\) be their simultaneous displacements, then by the general equation of dynamics[1],
\begin{equation}C\frac{dw}{dt}\delta z=X\delta x+ Y\delta y \end{equation}
where X and Y are the forces acting at A and B.
But
\begin{equation}\frac{dw}{dt}=p\frac{du}{dt}+q\frac{dv}{dt} \end{equation}
and
\begin{equation}\delta z=p\delta x + q\delta y\end{equation}
Substituting, and remembering that \(\delta x\) and \(\delta y\) are independent,
\begin{equation}\begin{array}\\X =\frac{d}{dt}(Cp^{2}u+Cpqv)\\ Y=\frac{d}{dt}(Cpqu+Cq^{2}v) \end{array} \end{equation}
We may call Cp2u+Cpqv the momentum of C referred to A, and Cpqu+Cq2v its momentum referred to B; then we may say that the effect of the force X is to increase the momentum of C referred to A, and that of Y to increase its momentum referred to B.
If there are many bodies connected with A and B in a similar way but with different values of p and q, we may treat the question in the same way by assuming
\begin{equation}L=\sum(Cp^{2}), M=\sum(Cpq), and N=\sum(Cq^{2}), \end{equation}
where the summation is extended to all the bodies with their proper values of C, p, and q. Then the momentum of the system referred to A is
and referred to B,
and we shall have
where X and Y are the external forces on A and B.
The above is a mechanical analogy that Maxwell provides for quantifying reduced momentum. By Maxwell's definition, A and B are already velocities because p times A and q times B are velocities and p and q are dimensionless. His initial logic is flawed in that p and q are magnitudes of "driving- points" A and B. There is no scientific basis for separating a magnitude from its unit. The variables p and q cannot be used for defining L, M, N based upon the logic presented. Regardless, Maxwell clearly intended L, M, and N to be units of mass.
The "driving points" Maxwell are talking about would be the points of connection between matter and Aether. Matter is inherently encapsulated by Aether. The two meet at a boundary, which quantifies as the unit of conductance. Conductance is a property of all surfaces, whether the surfaces are of matter or space. Actually, conductance is strictly a property of the surface of space. The material counterpart to conductance is magnetic flux. Magnetic flux is reciprocal to conductance in the Aether Physics Model.
If the velocity of A be increased at the rate du/dt, then in order to prevent B from moving a force, ?=d/dt(Mu) must be applied to it.
This effect on B, due to an increase of the velocity of A, corresponds to the electromotive force on one circuit arising from an increase in the strength of a neighbouring circuit.
This dynamical illustration is to be considered merely as assisting the reader to understand what is meant in mechanics by Reduced Momentum. The facts of the induction of currents as depending on the variations of the quantity called Electromagnetic Momentum, or Electrotonic State, rest on the experiments of Faraday[2], Felici16, &c.
One cannot simply discuss electrical processes, which are based upon dimensions of charge, in terms of mechanical processes, which are based upon units of angular momentum. In order to equate mechanical processes with electrical processes, one must directly correlate magnetic charge (not electrostatic charge) with mass, as there is an absolute mass to magnetic charge ratio in the Universe. Magnetic charge was not recognized during Maxwell's time, as it was quantified as a moving electrostatic charge. The ratio of mass to electrostatic charge is not constant, as it changes due to varying quantities of electrons, protons, and neutron in matter.
Coefficients of Induction for Two Circuits
(26) In the electromagnetic field the values of L, M, N depend on the distribution of the magnetic effects due to the two circuits, and this distribution depends only on the form and relative position of the circuits. Hence, L, M, N are quantities depending on the form and relative positions of the circuits, and are subject to variation with the motion of the conductors. It will be presently seen that L, M, N are geometrical quantities of the nature of lines, that is, of one dimension in space; L depends on the form of the first conductor, which we shall call A, N on that of the second, which we shall call B, and M on the relative position of A and B.
(27) Let ? be the electromotive force acting on A, x the strength of the current, and R the resistance, the \(Rx\) will be the resisting force. In steady currents the electromotive force just balances the resisting force, but in variable currents the resultant force \(\xi=Rx\) is expended in increasing the "electromagnetic momentum," using the word momentum merely to express that which is generated by a force acting during a time, that is, a velocity existing in a body.
Maxwell has just defined what is today known as Ohm's law. Since Maxwell was on the cutting edge of technology at his time, we can forgive the misuse of the phrase "electromotive force" in describing the potential. Force and potential are different units of physical behavior. In today's detailed and extended body of scientific knowledge, we must be explicit with our terminology in physics.
In the case of electric currents, the force in action is not ordinary mechanical force, at least we are not as yet able to measure it as common force, but we call it electromotive force, and the body moved is not merely the electricity in the conductor, but something outside the conductor, and capable of being affected by other conductuctors in the neighbourhood carrying currents. In this it resembles rather the reduced momentum of a driving-point of a machine as influenced by its mechanical connexions, than that of a simple moving body like a cannon ball, or water in a tube.
The "something" outside the conductor is the Aether, and it structurally quantified in the Aether Physics Model.
Electromagnetic Relations of two Conducting Circuits.
(28) In the case of two conducting circuits, A and B, we shall assume that the electromagnetic momentum belonging to A is
\begin{equation}Lx + My\end{equation}
and that belonging to B,
\begin{equation}Mx + Ny\end{equation}
where L, M, and N correspond to the same quantities in the dynamical illustration, except that they are supposed to be capable of variation when the conductors A or B are moved.
If L, M, and N are the same quantities as the dynamical illustration, then they are in units of mass. However, the equations following only work if L,M, and N are in units of inductance. There is no physical basis for this arbitrary change from mass to inductance.
Then the equation of the current x in A will be
\begin{equation}\xi=Rx+\frac{d}{dt}(Lx+My)\end{equation}
and that of \(y\) in B
\begin{equation}\eta=Sy+\frac{d}{dt}(Mx+Ny)\end{equation}
where \(\xi\) and \(\eta\) are the electromotive forces, x and y are the currents, and R and S the resistances in A and B respectively.
The units do not match in the above equations. Maxwell apparently forgot to account for \(dt\).
Induction of one Current by another.
(29) Case 1st. Let there be no electromotive force on B, except that which arises from the action of A, and let the current of A increase from 0 to the value \(x\), then
\begin{equation}Sy+\frac{d}{dt}(Mx+Ny)=0\end{equation}
whence
\begin{equation}Y=\int_{0}^{t}ydt=-\frac{M}{S}x\end{equation}
that is, a quantity of electricity Y, being the total induced current, will flow through B when \(x\) rises from 0 to \(x\). This is induction by variation of the current in the primary conductor. When M is positive, the induced current due to increase of the primary current is negative.
The above equation is false since the units don't match. The simplified integral must also be false. Also, since L, M, and N are simply the masses of something (as defined in section (24), it makes no sense to describe M in terms of being positive or negative.
Induction by Motion of Conductor
(30) Case 2nd. Let \(x\) remain constant, and let M change from M to M', then
\begin{equation}Y=-\frac{M'-M}{S}x\end{equation}
so that if M is increased, which it will be by the primary and secondary circuits approaching each other, there will be a negative induced current, the total quantity of electricity passed through B being Y.
This is induction by the relative motion of the primary and secondary conductors.
Whether by using Standard Model units or APM quantum measurements units, the "quantity of electricity" passing through B and labeled Y has no meaning. There is no empircally measured unit having three dimensions of charge, let alone over a two dimensional area.
The bad habit of Maxwell to constantly reuse the same variables for different units tend to hide his errors from most readers. However, by systematically going through the paper in a math program, such as MathCAD, it is easier to keep track of the units and catch errors.
There can be no doubt that Y has been redefined in this case to be a "quantity of electricity," but there is no basis for a unit with cubed charge over a surface.
Based upon Maxwell's subsequent theory following below, the best that can be guessed is that Maxwell changed the units of L, M, and N from mass to inductance without giving any reason for it. Even still, the equation in section (30) produces a unit of charge squared (coul^2 in MKS units, coul^4 in QMU units), which has not been empirically defined.
Equation of Work and Energy
(31) To form the equation between work done and energy produced, multiply (1) by \(x\) and (2) by \(y\), and add
\begin{equation}\xi x+\eta y=Rx^{2}+Sy^{2}+x\frac{d}{dt}(Lx+My)+y\frac{d}{dt}(Mx+Ny)\end{equation}
Here \(\xi x\) is the work done in unit of time by the electromotive force \(\xi\) acting on the current \(x\) and maintaining it, and \(\eta y\) is the work done by the electromotive force \(\eta\). Hence the left-hand side of the equation represents the work done by the electromotive forces in unit of time.
We begin to see Maxwell's errors incorporated into his theory. He is squaring the current, but not the resistance. No physical basis is given for squaring the current. Also, the units do not match in the above equation (in any system of units).
From the perspective of the APM, it is apparent Maxwell misinterpreted the unit of Charge Resonance (current squared in MKS units) due to the incorrect notation of charge dimensions in the units. The APM quantum measurement unit of Charge Resonance is:
\begin{equation}chrs={e_{emax}}^{2}\cdot {F_{q}}^{2}\end{equation}
Charge resonance is equal to current times frequency.
\begin{equation}chrs=curr \cdot freq\end{equation}
Heat produced by the Current
(32) On the other side of the equation we have, first,
\begin{equation}Rx^{2}+Sy^{2}=H \end{equation}
Another error in judgment by Maxwell is in assuming that power is temperature. In the APM, temperature is specifically the unit of velocity squared.
\begin{equation}temp=velc^{2} \end{equation}
Which represents the work done in overcoming the resistance of the circuits in unit of time. This is converted into heat. The remaining terms represent work not converted into heat. They may be written
\begin{equation} \frac{1}{2}\frac{d}{dt}(Lx^{2}+2Mxy+Ny^{2})+\frac{1}{2}\frac{dL}{dt}x^{2}+\frac{dM}{dt}xy+\frac{1}{2}\frac{dN}{dt}y^{2}\end{equation}
The above terms do not produce the unit of energy, or any other known physical unit. This, again, is a clear example of a mistake in Maxwell's work.
Intrinsic Energy of the Currents
(33) If L, M, N are constant, the whole work of the electromotive forces which is not spent against resistance will be devoted to the development of the currents. The whole instrinsic energy of the currents is therefore
\begin{equation}\frac{1}{2}Lx^{2}+Mxy+\frac{1}{2}Ny^{2}=E \end{equation}
In order for the above equation to produce the unit of energy, L, M, and N would have to be in units of inductance. Maxwell gives no clue that L,M, and N have been changed to inductance. Even after switching to inductance, the equation could only work when the charge dimension in the unit of current is single dimension. Since a strong case can be made that all charge should always be distributed, this equation of Maxwell's (even after being corrected for inductance) has no physical basis.
This energy exists in a form imperceptible to our senses, probably as actual motion, the seat of this motion being not merely the conducting circuits, but the space surrounding them.
The energy as he quantified it is simply non-existent since the equation has no foundation.
(34) The remaining terms,
\begin{equation}\frac{1}{2}\frac{dL}{dt}x^{2}+\frac{dM}{dt}xy+\frac{1}{2}\frac{dN}{dt}y^{2}=W \end{equation}
Represent the work done in unit of time arising from the variations of L, M, and N, or, what is the same thing, alterations in the form and position of the conducting circuits A and B.
Now if work is done when a body is moved, it must arise from ordinary mechanical force acting on the body while it is moved. Hence this part of the expression shows that there is a mechanical force urging every part of the conductors themselves in that direction in which L, M, and N will be most increased.
The existence of the electromagnetic force between conductors carrying currents is therefore a direct consequence of the joint and independent action of each current on the electromagnetic field. If A and B are allowed to approach a distance \(ds\), so as to increase M from M to M' while the currents are \(x\) and \(y\), then the work done will be
\begin{equation}(M'-M)xy \end{equation}
Apparently, Maxwell intends to change L, M, and N to inductance
We will also change the units to single dimension charge for the moment
and the force in the direction of ds will be
and this will be an attraction if x and y are of the same sign, and if M is increased as A and B approach.
It appears, therefore, that if we admit that the unresisted part of electromotive force goes on as long as it acts, generating a self-persistent state of the current, which we may call (from mechanical analogy) its electromagnetic momentum, and that this momentum depends on circumstances external to the conductor, then both induction of currents and electromagnetic attractions may be proved by mechanical reasoning.
Clearly, Maxwell is wrong. He has not provided a mechanical analogy linked to an electromagnetic analogy. He switched the definitions of the L,M, and N variables without giving the proper logic, which if intentional, would be unethical in science. Regardless of the switch, the charge dimensions of the units are wrong due to an error that preceded Maxwell in converting cgs units to MKS units. And even when using Maxwell's own units his equations do not always balance.
What I have called electromagnetic momentum is the same quantity which is called by Faraday the electrotonic state of the circuit, every change of which involves the action of an electromotive force, just as change of momentum involves the action of mechanical force.
Maxwell is clearly wrong. Potential is not force. In the Aether Physics Model, potential is equal to pressure times volume per strong charge.
Potential in the APM may also view as force across a length per strong charge:
Since potential is a volumetric pressure, potential may only exist as a force as long is the force is longitudinal. That is, the force inherent to potential is applied across a radius and in all directions.
If, therefore, the phenomena described by Faraday in the Ninth Series of his experimental Researches were the only known facts about electric currents, the laws of Ampere relating to the attraction of conductors carrying currents, as well as those of Faraday about the mutual induction of currents, might be deduced by mechanical reasoning.
In order to bring these results within the range of experimental verification, I shall next investigate the case of a single current, of two currents, and of the six currents of the electric balance, so as to enable the experimenter to determine the values of L, M, N.
Case of a single Circuit
(35) The equation of the current x in a circuit whose resistance is R, and whose coefficient of self-induction is L, acted on by an external electromotive force ?, is
Whenever the distributed charge based units (inductance for example) are equated with single dimension charge based units, the equations are wrong according to the APM. For the time being, we will use Maxwell's units.
When ? is constant, the solution is of the form
where a is the value of the current at the commencement, and b is the final value. the total quantity of electricity which passes in time t, where t is great, is
The value of the integral of x2 with respect to the time is
The actual current changes gradually from the initial value of a to the final value b, the values of the integrals of x and x2 are the same as if a steady current of intensity ½(a+b) were to flow for a time 2 L/R, and were then succeeded by the steady current b. The time 2 L/R is generally so minute a fraction of a second, that the effects on the galvanometer and dynamometer may be calculated as if the impulse were instantaneous.
Just because humans cannot easily discern a small effect does not mean one can arbitrarily assume instantenous or null results. This assumption only introduces errors (or covers them up, as the case may be.)
If the circuit consists of a battery and a coil, then, when the circuit is first completed, the effects are the same as if the current had only half its final strength during the time 2 L/R. This diminution of the current, due to induction, is sometimes called the counter-current.
(36) If an additional resistance r is suddenly thrown into the circuit, as by the breaking contact, so as to force the current to pass through a thin wire of resotance r, then the original current is a=?/R, and the final current is b=?/R+r.
The current of induction is then
and continues for a time 2L/R+r. This current is greater than that which the battery can maintain in the two wires R and r, and may be sufficient to ignite the thin wire r.
When contact is broken by separating the wires in air, this additional resistance is given by the interposed air, and since the electromotive force across the new resistance is very great, a spark will be forced across.
If the electromotive force is of the form E sin pt, as in the case of a coil revolving in a magnetic field, then
where ?2 = R2 + L2p2, and tan ? = Lp/R.
Case of Two Circuits
(37) Let R be the primary circuit and S the secondary circuit, then we have a case similar to that of the induction coil.
The equations of currents are those marked A and B, and we may here assume L, M, N as constant because there is no motion of the conductors. The equations then become
To find the total quantity of electricity which passes, we have only to integrate these equations with respect to t; then if x0, y0 be the strengths of the currents at time 0, and x1, y1, at time t, and if X, Y be the quantities of electricity passed through each circuit during time t,
1. Lagrange, Méc. Anal. ii. 2. § 5.
