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General Classical Electrodynamics Universal Journal of Physics and Application 10(4): 128-140, 2016 DOI: 10.13189/ujpa.2016.100404 http://www.hrpub.org General Classical Electrodynamics Koen J. van Vlaenderen Ethergy B.V, Research, Hobbemalaan 10, 1816GD Alkmaar, North Holland, The Netherlands Institute for Basic Research, P.O. Box 1577, Palm Harbor, FL 34684 U.S.A. Copyright c 2016 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Maxwell’s Classical Electrodynamics (MCED) The fundamental theorem of vector algebra is as follows: a suffers several inconsistencies: (1) the Lorentz force law of vector function F(x) can be decomposed into two unique MCED violates Newton’s Third Law of Motion (N3LM) in vector functions Fl(x) and Ft(x), such that case of stationary and divergent or convergent current distri- butions; (2) the general Jefimenko electric field solution of MCED shows two longitudinal far fields that are not waves; (3) the ratio of the electrodynamic energy-momentum of a charged sphere in uniform motion has an incorrect factor F(x) = Fl(x) + Ft(x) (1.3) 4 Z 0 0 of . A consistent General Classical Electrodynamics 1 r · F(x ) 3 0 3 Fl(x) = − r d x (1.4) (GCED) is presented that is based on Whittaker’s reciprocal 4π jx − x0j 0 force law that satisfies N3LM. The Whittaker force is V 1 Z r0 × F(x0) expressed as a scalar magnetic field force, added to the F (x) = r× d3x0 (1.5) t 4π jx − x0j Lorentz force. GCED is consistent only if it is assumed V 0 that the electric potential velocity in vacuum, ’a’, is much greater than ’c’(a c); GCED reduces to MCED, in case we assume a = c. Longitudinal electromagnetic waves and superluminal longitudinal electric potential waves are The lowercase subindexes ’l’ and ’t’ will have the meaning predicted. This theory has been verified by seemingly of longitudinal and transverse in this paper. The longitudinal unrelated experiments, such as the detection of superluminal vector function Fl is curl free (r×Fl = 0), and the trans- Coulomb fields and longitudinal Ampere` forces, and has a verse vector function Ft is divergence free (r·Ft = 0). We wide range of electrical engineering applications. assume that F is well behaved (F is zero if jxj is infinite). Let us further introduce the following notations and definitions. Keywords Classical Electrodynamics, Longitudinal Ampere` Force, Scalar Fields, Longitudinal Electric Waves, Superluminal Velocity, Energy Conversion ρ Net electric charge density, in C/m3 2 J = Jl + Jt Net electric current density, in A/m 1 Introduction Φ Net electric charge (scalar) potential, in V A = Al + At Net electric current (vector) potential, An alternative to Maxwell’s [1, 2] Classical Electrodynam- in V·s/m ics (MCED) theory is presented, called General Classical EΦ = −∇Φ Electric field, in V/m Electrodynamics (GCED), that is free of inconsistencies. For EL = −@tAl Field induced divergent electric field the development of this theory we make use of the fundamen- ET = −@tAt Field induced rotational electric field tal theorem of vector algebra. The proof of this fundamental B = −@ Φ theorem is based on the three dimensional delta function δ(x) Φ t Field induced scalar field, in V/s 2 and the sifting property of this function, see (1.1 – 1.2): BL = −∇·Al Scalar magnetic field, in T = V·s/m 2 BT = r×At Vector magnetic field, in T = V·s/m −1 1 δ(x) = ∆ (1.1) 2 2 3 4π jxj φ0 0µ0 Polarizability of vacuum, in F·s /m Z −7 F(x) = F(x0) δ(x − x0) d3x0 (1.2) µ0 Permeability of vacuum: 4π10 H/m −12 V 0 0 Permittivity of vacuum: 8.854 F/m Universal Journal of Physics and Application 10(4): 128-140, 2016 129 µ Z J(x0) A(x) = 0 d3x0 (2.1) (x; t) = (x; y; z; t) Place and time coordinates 4π r @ V 0 @ = Partial time differential 0 t @t r = x − x @ @ @ r = jx − x0j r= ; ; Del operator @x @y @z Since @ A = 0 for stationary currents, the electric field ∆ = r · r Laplace operator t equals E = EΦ = −∇Φ, such that the Gauss law for General ∆Φ = r·rΦ; ∆A = rr·A − r×r×A Magnetostatics is given by 1 The permittivity, permeability and polarizability of vacuum r·EΦ(x; t) = ρ(x; t) (2.2) are constants. The charge- and current density distributions, 0 the potentials and the fields, are functions of place and not The magnetostatic vector field BT (x) is defined by Biot- always functions of time. Time independent functions are Savart’s law as follows: called stationary or static functions. Basically there are three types of charge-current density distributions: BT (x) = r×At(x) = r×A(x) µ Z 1 A. Current free charge J = 0 = − 0 J(x0) × r d3x0 4π r B. Stationary currents @tJ = 0 V 0 1. closed circuit @ J = 0 ^ r·J = 0 t µ Z 1 2. open circuit @ J = 0 ^ r·J 6= 0 = 0 J(x0) × r d3x0 (2.3) t 4π r3 C. Time dependent currents @tJ 6= 0 V 0 The magnetic field is indeed static, since the current density The charge conservation law (also called ’charge continuity’) is stationary. From the continuity of charge (1.6), and (2.2), is true for all types of charge-current density distributions: it follows that Jl = −0@t(EΦ), hence the Ampere` law for General Magnetostatics is as follows: @ρ + r·J = 0 (1.6) @t @EΦ r×BT − 0µ0 = µ0J (2.4) The physics of current free charge density distributions is @t called Electrostatics (ES): @ ρ = −∇·0 = 0. The physics t 2.1 The Lorentz force of stationary current density distributions (@tJ = 0) is called General Magnetostatics (GMS). A special case of GMS are The magnetic force density, fT (x), that acts transversely divergence free current distributions (r · J = 0), and this on current density J(x) at place x, is given by: is widely called Magnetostatics (MS) in the scientific educa- tional literature. In case of Magnetostatics, the charge density fT (x) = J(x) × BT (x) distribution has to be static as well: @tρ = −∇·J = 0, such µ Z 1 that the electric field and the magnetic field are both static. = 0 J(x) × J(x0) × r d3x0 4π r3 The Maxwell-Lorentz force law satisfies Newton’s third V 0 law of motion (N3LM) in case of Electrostatics and Magne- µ Z 1 h i = 0 J(x) · rJ(x0) − J(x0) · J(x)r d3x0 tostatics, however, this force law violates N3LM in case of 4π r3 General Magnetostatics. A violation of N3LM means that V 0 Z momentum is not conserved by GMS systems, for which µ0 0 3 0 = fT (x; x ) d x (2.5) there is no experimental evidence! This remarkable inconsis- 4π V 0 tency in classical physics is rarely mentioned in the scientific educational literature. This is the Lorentz force density law for Magnetostatics; it This is not the only problematic aspects of MCED. Jefi- is assumed that the electric force densities are negligible. 0 menko’s electric field expression that is derived from MCED Notice that the integrand is non-reciprocal: fT (x; x ) 6= 0 theory, shows two longitudinal electric field terms that do not −fT (x ; x), and that r changes into −r by swapping x and 0 interact by induction with other fields, therefore these electric x . This means that the Lorentz force law agrees with N3LM, fields cannot be field waves and nevertheless these electric but only if one calculates the total force on closed on-itself fields fall off in magnitude by distance, as far fields, which is current circuits (magnetostatics is usually defined for diver- 4 gence free currents only), which is proven as follows. inconsistent. A third inconsistency is the problematic 3 fac- tor in the ratio of the electric energy and the electromagnetic Consider two non-intersecting and closed current circuits C 0 0 momentum of a charged sphere. In the next sections we de- and C , that carry the stationary electric currents I and I , 0 scribe these related inconsistencies of MCED theory in more see figure1. The currents I and I are equal to the surface detail, and how to resolve them. integral of the current density over a circuit line cross sec- tion of respectively circuits C and C0. The total force acting on circuit C is a double volume integral of the Lorentz force density. We assume that the currents in C and C0 are con- 2 General Magnetostatics stant for each circuit line cross section, therefore we replace the double volume integral by a double line integral over the 0 Let J(x) be a stationary current distribution. The vector circuits C and C , in order to determine the force, FC , acting potential A(x) at place vector x is given by: on C: 130 General Classical Electrodynamics Etc ... It is not at all straightforward to find practical examples of a measured force exerted on a stationary and perfectly closed- on-itself current circuit. The Meisner effect might be such an example: a free falling permanent magnet approaches a su- 0 dl perconductor, which induces a closed-circuit current in the I0 superconductor. The magnet falls until the induced currents r and magnetic field of the superconductor perfectly opposes I the field of the magnet, which causes the magnet to levitate, and from that moment on the superconductor current is sta- x x0 tionary and divergence free.
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