By Clarence L. Dulaney
2226 Fairgreen Drive
Missouri City, TX 77489
e-mail dulaneyc@flash.net
Abstract: Forces
of Gravity and of Electrodynamics take place over large distances. These forces may be “carried” by an
all-pervading aether, or they may be truly action-at-a-distance (AAD), or they
may be a combination of the two.
It is here proposed that
electrodynamic forces act at a distance at speeds of c or less. The question of whether an aether is
involved is not ruled out, but is not specifically discussed. Certainly, the “super-elastic” aether of
Maxwell is not believed to exist. See
the paper “Why Spin”38for an explanation of an aether that is compatible with
Weber-Ampére Electrodynamics.
The electrodynamic forces of
Ampére and of Weber were derived in the 1820’s and the 1840’s respectively.
With W. Weber using Ampére’s theories as a starting point. Although neither of these theorists knew of
electrons, both developed formulas that are compatible with electronic
conduction of currents. In addition to
Ampére’s theories, Weber based his electrodynamics on Fechner’s Hypothesis,
that current in wires is conducted by positive and negative particles moving in
opposite directions at the same speed.
It should be mentioned that both Ampére’s and Weber’s Forces are
compliant with Newton’s Laws, particularly the Third Law of action and
reaction.
Weber’s Electrodynamics is
shown to be completely compliant with Hertz’s experiments that showed that
electrodynamic waves travel at speed c, and correctly predicts that any
relatively accelerated charge pairs lead to radiation (witness 60 cycle hum
picked up by a cheap AM radio.).
Even Maxwellians admit that
magnetic effects of light in matter are negligible when compared to electric
effects. This was shown by Weber and
Kohlrausch.
Current, (and thus voltage)
waves travel at speed c in low impedance wires. This was predicted by Kirchhoff and by Weber, (both using Weber’s
electrodynamics), and was shown to be true in the 1890’s by Blondot and
separately by others.
The experiments of H. Oersted in 1820 (1), concerning the effect of an electric current on a compass needle led to a considerable amount of research, including that of M. Faraday, and of A. Ampére.
Ampére in 1823 (2) published the results of his work. He theorized the revolutionary idea that magnetism in a compass needle was due to microscopic currents circulating within the permanent magnet.
He also showed that there was a force between two parallel wires each carrying a current, and his Force formula enabled calculation of the (small) force.
Based on the work of Ampére, and the hypothesis of G. Fechner (3) (That a current in a wire is carried equally by positive and negative particles moving at the same speed in opposite directions), W. Weber developed his Electrodynamic Force formula in 1848 (4). This was 8 years before J. Maxwell’s electromagnetic theory was published (5).
There are two different types of electrodynamic formulas. One is based on fields and continuous systems. The second is based on particles and discontinuous systems.
The first is exemplified by the work of J. Maxwell and H. Lorentz. Their formulations were based on reaction of fields with the “luminiferous aether”. The aether is never mentioned in today’s Electrodynamic texts, but the formulations have not changed since early in the 20th century. Not only do the formulations require reaction to aether, there is a lack of compliance with Newton’s Third law (of action and reaction) of many of the basic formulations (cf Assis (6,7) and O’Rahilly (8.120,208)) (Note in (a.xxx), a is the reference # and xxx is the page number).
The second type is demonstrated by the Forces of Ampére and of Weber. These Forces comply completely with Newton’s Third Law, and do not require an aether. (It should be noted they do not completely rule one out.)
Since they are particulate systems, they require that AAD occur. (Generally, when physicists speak of AAD they mean that it takes place “instantaneously”. It will be shown later that such action with Weber’s Force takes place at speeds of c or less.)
In 1846(9) W. Weber developed his Electrodynamic Force Formula. (Note that cw = (2)½c in the notation of Rosenfeld (10)):.
Fw = (qq’/r2 ) (1 – (cw-2)[(dr/dt)2 – 2r(d2r/dt2)])
Fw is the mutual force between charges q and q’ which are separated at a distance r and are moving at a relative speed dr/dt, with a relative acceleration d2r/dt2. As pointed out by Weber and Kohlraush (11), cw is the maximum speed at which the two charges can have relative to each other and still communicate. (This speed was later found to be c)
To show that the Weber Force is conservative, it is necessary to show that it can be derived from a potential. In 1848 (12), Weber derived such a Potential, namely:
U = (qq’/r) (1 – cw-2 (dr/dt)2)
The Weber Force may readily be derived from this potential. One such method was shown by J. Maxwell (13). With modernized notation as given by Assis (14)
Fw = -dU/dr = (qq’/r2)–(qq’/r2cw2) (dr/dt)2 + 2(qq’/rcw2) (d2r/dt2) =
= (qq’/r2)(1- (cw-2[(dr/dt)2 – 2r(d2r/dt2)])
The Fechner Hypothesis was shown to be incorrect for metallic conductors by the experiment of Hall in 1879 (16), which showed that currents in wires were carried by negative particles only.
It would be of interest to derive the Weber potential from basic principles. As shown above, the original Weber Potential leads to the Fechner version of the Weber Force, so it must also be incorrect. In general, the Weber Force has been “corrected” by replacing cw by (2)½c, making the “corrected” Force, Fc = (qq’/r2) (1 – (2c-2[(dr/dt)2 – 2r(d2r/dt2)].
Charge
versus Speed
Elsewhere (14) it is shown that a moving charge decreases with its speed by a factor of (1- (v/c)2)½. (Note that the charge is only property of matter that changes with speed. The decrease of charge goes to create magnetic energy.) When the particle is slowed, the charge reverts to the original value with the magnetic energy decreasing to zero. There is no loss of energy.
Let us derive the potential for a moving charge pair, both negative. For stationary charges, the Coulomb potential is qq’/r. If the charges are moving away from each other at a speed v, then:
Um = (qq’/r)((1 – (v/c)2)½(1 – (v’/c)2)½) = (qq’/r) (1 – (v/c)2)). Note the resemblance to Weber’s Potential.
The Modified Weber Force Wm = (qq’/r2) ( 1 – (c-2[(dr/dt)2 – 2r(d2r/dt2)]) Note that Wm differs from Fw in that cw is replaced by c.
It is here proposed that Wm is the “correct” Weber Force Formula. Note that it has a “built-in” speed correction to take care of Wesley’s concerns (37) about the effect of high speeds on the original Weber formula.
Let us examine with Wm the situation of a steady direct current in a long straight wire. The acceleration would be zero. Let us assume dr/dt is virtually “c”, (otherwise the c-2 would be overwhelming, and the Coulomb term would be very large.). In our hypothetical case, the velocity term would be close to 1, making the Weber force quite small.
To show another way that the velocity must be nearly c, consider the Coulomb force for two electrons separated by 0.001 cm. The force between the two charges would be: Fc = (4.8 x 10-10 esu)2/ (10-3)2 = 2.304 x 10-13 dyne. This would lead to an acceleration of 2.5 x 1014 cm/sec2, an immense value. Remember that the magnetic energy associated with a current is caused by the decrease in charge of the moving electrons.
Most textbooks (18 ) assume there is 1 “free” electron for each metal atom, and that the velocity of the conduction electrons is less than 1 cm/sec. To be “free” electrons would have to be ionized away from the atoms. Ionization energies for most metals is about 7 eV (or roughly 170 Kcal/mole). Note that this is considerably greater than the energy necessary to break Cu-Cu bonds (45.5 Kcal/mole) (19). See my paper on “Longitudinal Forces in Current Carrying Wires” (20). This paper indicates that the current carrying electrons move at speeds nearly c at high enough currents (depending on the resistance of the wire). If the current is applied quite rapidly, the available electrons may not be able to carry the current, and the atomic bonds may be broken, leading to rupturing of the wires (not caused by melting).
A theoretical consideration on
“free’ electrons is given by Hume-Rothery (21). He points out “A metallic crystal may be regarded as
consisting of an array of positive ions held together by attraction to the
common system of negatively charged electrons.” Further, Hume Rothery says “The difference between an
insulator, such as diamond, and a conductor is not that the electrons are bound
in one case and free to move in the other, but rather that in the conductor
the external field can produce a resultant flow in a given direction. This, as we have already explained, means
that the conductor must have unoccupied electron states into which the
electrons may pass when the external field is applied.”
Derivation of Amperè’s Force From Weber’s Force
Historically, The Weber Force was derived from the Ampère Force. It should also be possible to reverse the process. A. Assis (6.84ff ) shows how this can be done His derivation will not be repeated here. See Appendix 1. It will be noted that by use of Fechner’s Hypothesis, the Coulomb Force cancels out, since the positive and negative charges are posited to be equal. Also, the acceleration terms fortuitously cancel out, leaving only the velocity terms.
Firstly, the Coulomb term does not vanish unless the charges are moving at or near c. In this case, beams of negative or of positive ions moving in a straight line at constant speed (near c) are also covered by the discussion.
Secondly, the acceleration term does not vanish unless a steady direct current (in long, straight wires) is considered.
On the basis of these two provisos, the Ampère Force expression comes out to be:
D2Fj,i = (IiIj/c2) (ri,j/ri,j3) [2(dli · dj) –(3/ri,j)(ri,j · dli) (ri,j · dlj)] (See Appendix 1) In the expression, vectors are indicated by bold type, and a unit vector such as one in the a direction would be indicated by a/a. Note that the expression given is indeed Ampère’s Force. See Assis (6.87) for various units concerned with the Force.
The above would also apply to beams of positive or negative ions moving in a straight line at very close to speed c.
Both these cases are beyond the scope of this paper, since effects in such systems are very difficult to predict.
Ionic plasmas containing both positive and negative ions lead to rapid recombination to neutral particles. If the potential is high enough, arcs may occur. See my Longitudinal Current (20) and Cold Lightning (22) papers.
Electrolytic Conduction is complicated by solvation of the ions, and by loss of ions from the solution by plating out on the electrodes as gases or metals. See any book on Electrochemistry.
One of the initial complaints against the Weber Formula was that it contained an acceleration term (8). It was known that accelerated charges radiated energy, which according to the detractors would destroy the conservative nature of the force, which was claimed by Weber.
This is a false claim, since one must consider the entire thermodynamic system (which may be dimensionally quite large). Radiation may well occur, and the energy “lost” to the wire or antenna as “rays” is eventually returned to the system when the “rays” are absorbed, at which point the original energy is recovered, leaving the conservative nature of Wm intact.
The radiation may take place at any frequency to which the charges are accelerated. Witness “60-cycle hum” that is picked up by a cheap AM radio, and the lighting of fluorescent bulbs held up under high voltage transmission lines, for just two examples.
Maxwell’s Equations, for example, do not apply to radiation from alternating current circuits, since they do not have an acceleration term. The only radiation they apply to is at speed c.
The fact that c is an integral part, particularly of Wm, indicates that accelerated charges may lead to light and other “electromagnetic” (better “electrodynamic”) waves. It should be noted that Weber did not initially make the connection between light and c in his formula.
Later Weber and separately Kirchhoff (23) did speculate, using Weber’s Electrodynamics, that current waves with speed c should be propagated on a wire that had small enough impedance. The equation Kirchhoff developed is called the telegraphy equation: ¶2I/¶S2 = (c-2) (¶2I/¶t2) + K(¶I/¶t). The last term vanishes when the impedance is small enough. S is the distance along the wire.
These predictions were confirmed around the turn of the 20th century by Blondot (24) and by others. (25), (26), but by that time, Weber had been thoroughly discredited, so that no mention was made of Weber or of Kirchhoff in any of the papers.
Plane-Wave Equations- Electrodynamic Waves
Almost identical derivations of the Plane-Wave Equations are given by G. Pierce (27) and by F. Crawford, Jr. (28). The derivation by Pierce will be followed here. They are based on Maxwell’s Equations, but it will be shown that they could as well have been based on Weber’s Equation. See Appendix 2.
These equations describe waves occurring in a homogenous (and non-charged) medium involving fluctuations of a variable “M”. the plane-wave in such a case may be represented by the equation:
(V-2)(¶2M/¶T2) = ¶2M/¶S2 Here V is the speed of the wave and S represents distance in the direction of travel of the wave. For “electromagnetic” waves, V may be shown to be c/(me)½ by means of Maxwell’s Equations. The factor m is the magnetic permeability, which for transparent media (and for vacuum) is virtually 1, (out to 5 decimal places for glass) (28.180). This makes V = c/(e)½. Here, e is the dielectric constant of the medium, and (e)½ = n, the refractive index of the medium. Note that the refractive index is frequency dependent, so that the tabulated (static) dielectric constants may not be used to calculate the refractive index (30.594).
The plain-wave equation may be written as: (e/c2) (¶2M/¶t2) = ¶2M/¶S2. M can be either E, H, I or anything that varies with the electrodynamic wave. Note the similarity to the Kirchhoff Telegraphy equation. This again is not mentioned by either Pierce or Crawford, Jr.
The solution of the plane-wave equation (Appendix 2) is that there are two waves, one moving in the +s direction, the other in the –s direction, each with a speed of c/n.
The plane-wave equation is compatible with Wm.
The B Wave
Maxwell claims that
“electromagnetic” waves such as light are alternating E and B
waves of equal strength, but even Maxwellians discount the B wave. For example, F. Crawford, Jr. says (29.191) “Why
do we always consider E and not B? We don’t always, but we often do. Part of the reason we usually express the effect of
electromagnetic waves in terms of E and suppress B from the
formulas is the following: When
electromagnetic waves interact with a charged particle of charge q and velocity
v, the force on the particle is given by the Lorentz force (Vol. II.
Sec. 5.2)
F = qE + (qv/c) XB. In an electromagnetic traveling wave in vacuum, it turns out that E and B have the same instantaneous magnitude. Therefore, the magnitude of the force contributed by B is smaller than that contributed by E by a factor of the order [v/c]. Now, it turns out that when E and B are due to ordinary light, or even due to a powerful laser, the fields E and B are sufficiently weak that the maximum velocity [v] attained in the steady-state motion of driven electrons in a piece of ordinary materials is tiny compared with c. Thus there are a large number of physical situations where we can neglect the force due to B. That is why we emphasize E” (Author’s Note: In the preceding, ordinary face type indicates vectors, ie E, B or v.)
There is an effect of strong magnetic energy on light. This is the “Faraday Effect” where polarized light undergoes a rotation of the plane of polarization when passed between the poles of a strong magnet (several thousand gauss). The primary mechanism for this effect is the different refractive indexes of the two circularly polarized components of plane polarized light. See Glasstone (31) for further discussion. At any rate, Wm can explain these effects certainly as well as the Maxwell equations.
“C” as a Ratio of Units
Quoting Weber and Kohlrausch
(translation by Susan P. Johnson (33)) “The insertion of the values of c
into the foregoing fundamental electrical law makes it possible to grasp, why
the electrodynamic effect of electrical masses, namely
(ee’/rr) (1/cc)[(dr/dt)2 – 2r(ddr/dt2)] compared with the electrostatic ee’/rr always seems infinitesimally small, so that in general the former only remains significant, when, as in galvanic currents, the electrostatic forces completely cancel each other in virtue of the neutralization of the positive and negative electricity.” Remember that in ordinary currents in wires, there are no positive galvanic currents.
Witness a quote from J.
Maxwell(32): “It appears from the
table of dimensions, Art. 628, that the number of electrostatic units of
electricity in one electromagnetic unit varies inversely as the magnitude of
the unit of length and directly as the magnitude of the unit of time which we
adopt.
If, therefore, we determine a
velocity which is represented numerically by this number, then, even if we
adopt new units of length and time, the number representing this velocity will
still be the number of electrostatic units of electricity in one
electromagnetic unit, according to the new system of measurement.
This velocity, therefore,
which indicates the relation between electrostatic and electromagnetic
phenomena, is a natural quantity of definite magnitude, and the measurement of
this quantity is one of the most important researches in electricity.”
This is quite confusing as to what Maxwell is trying to say. He obviously has the dimensions for velocity inverted, so that he seems to say that if the velocity is c, then there would be c electromagnetic units in one electrostatic unit of electricity.
Actually, it makes little difference what the ratio is. The only places the ratio comes into play is in the “electromagnetic” units of electricity, which are not used by physicists (33), and in the philosophical underpinnings of Maxwell’s Equations, where he uses emu/esu = c to intimate equal status for magnetic and electrical effects. This has been shown above to have been rejected by otherwise strong Maxwellians. See “From Weber to Maxwell”(35) for a further discussion of “Units and Dimensions”. For a much more complete discussion, see O’Rahilly, both volumes.
As indicated above, c entered into electrodynamics in the Modified Weber Potential, by virtue of the decrease of charge with speed.
The electrodynamic force formula of Weber was prematurely abandoned. The main argument against it, Instantaneous Action-at-a-Distance was shown to be incorrect, since Weber’s Force takes place at c or less speed. The other argument against Weber was that the acceleration term leads to radiation. This supposedly would obviate the conservative nature of the force. It was also shown to be false, since the entire thermodynamic system must be considered. The fact that radiation is indicated is a definite plus for the formula, since it explains how radiation other than the “electromagnetic” waves can arise. The Weber Force is entirely compatible with the work of H. Hertz (35).
“C” enters into electrodynamics via the decrease of charge with speed, and the inclusion of this phenomenon into the Modified Weber Potential, Um. Wm and the Ampère Force may be derived from Um.
NOTE:All references to http://sites.netscape.net/clarencedulaney/homepage
have been changed to http://mywebpage.netscape.com/clarencedulaney/index.html
APPENDIX I
DERIVATION OF AMPÈRE’S FORCE FROM WEBER’S FORCE
Assis (6) gives a very complete derivation of Ampère’s Force from the Fechner-based Fw. His exposition will not be repeated here. It will simply be said that the derivation succeeds because the Fechner + and – charges lead to cancellation of the Coulombic force terms, and somewhat more obscurely to the cancellation of the acceleration terms.
The Ampère Force is: d2FAj,i = -IiIj (ri,j/c2r3i,l) [2(dli·dlj) – (3/r2i,j)(ri,j·dli) (ri,j·dlj)] In this expression, bold type indicates a vector, r indicates the distance between the charges in the wires, and l is the distance along the wires. A unit vector is indicated by the vector divided by the static value, for example, ri,j/ri,j. See Assis (6.87) for conversion for various unit systems.
It is necessary for our further derivation to convert the Wm to the vector form, following Assis. He shows that vi,j = ri,j·vi,j/ri,j . (1), and dvi,j = (1/r3i,j) ([(vi,j·vi,j) – (ri,j·vi,j)2 + (rij·ai,j)] (2).
Wm = qq’(ri,j/c2r3i,j) [c2 – (vi,j)2 +2(ri,j·ai,j) (3). Substituting from (1) and (2) into (3):
Wm = (qq’ri,j/c2r3i,j)
[c2
– (1/r2i,j)(ri,j·vi,j)2 + (2/r2i,j)[(vi,j·vi,j) – (ri,j··vi,j)2 + (ri,j·ai,j)]]
(4)
Because the charges in Wm all have the same sign, the Coulomb term of (4)
does not vanish, unless the charges are zero, or very nearly so. This can only occur if the charges decrease
with speed. Also, it is necessary that
the acceleration term vanish. This can
only occur if we only consider a steady, direct current in a long, straight
wire, or, as mentioned above by straight beams of positive or negative ions
moving at very nearly speed c.
If both conditions are fulfilled,
Wm » (qq’ri,j/c2r3i,j)[2(vi,j·vi,j) –3(ri,j·vi,j)] (5), and if differential charges dqi and dqj are in wires
I and J, d2WMj,I = -(dqidqjri,j/c2r3i,j)[2(vi,j·vi,j) – 3(ri,j·ai,j)]
(6)
As Assis shows, Ii dli = dqi(vi) and Ij = dqj(vi,j) (7)
Substituting from (7) into
(6), d2FAj,i = -(IiIjri,j/c2r2i,j) [2(dli·dlj) – 3(1/r2i,j)(ri,j·dli)(ri,j·dlj)]
(8)
Equation (8) is the Ampère force. QED
APPENDIX 2
SOLUTION OF THE PLANE-WAVE EQUATION
This explanation is based on Pierce, op. cit., p379ff. Pierce bases his discussion on Maxwell’s
equations, but there is no reason that it could not have as easily been based
on the Weber Force.
A wave passing through an insulating, homogeneous, uncharged medium (ie.
air, water, glass) where a variable, M (E, I, H, etc.) is a function of
distance s and time t only may be described by a “plane-wave” equation: (me/c2) ¶2M/¶t2 = ¶2M/¶t2
(1), where m
is the magnetic permeability of
the medium (very close to 1 for a transparent medium (to 5 decimal points for
glass))and e is the dielectric constant of the
medium.
The solution in general of this second order partial differential
equation (1) is any functions
or arguments of functions such as G which is any function of s and t.
Suppose M = G(s,t). Then ¶M/¶t = aG’,
¶M/¶s
=G’ and ¶2M/¶t2 = a2G’’, ¶2M/¶t2 =
G’’. (2)
Substituting from (2) into (1), (a2me/c2)G’’
= G’’, or a = ±c/(me)½.
Thus M = F(s – ct/(me)½) or, M = G(s – ct + ct/(me)½)
Where F and G are any functions.
Pierce shows that these functions can
describe plane waves moving at v = c/(me)½, the first in the –s direction, and the
second in the +s direction.
As we noted above, for the type of medium we
are considering, m is 1 to a very high
degree, so that v » c/(e)½.
Since the refractive index is n = c/v, so that n » (e)½. It
should be noted that the refractive index is frequency dependent, so that the
tabular values of the dielectric constant cannot be used to calculate the
refractive index.
The fact that m is 1 indicates that the magnetic effects of light are much less than the
electric effects. Remember that M can
be E, I, H, or any other attribute of the wave.
Plane-waves may be derived from WM.
© 7/14/02
Clarence L. Dulaney