General Classical Electrodynamics, part two

Abstract

General Classical Electrodynamics part two. The energy density of electrodynamic sources are evaluated in the context of Maxwell's theory and in the context of General Classical Electrodynamics.

Full text

General Classical Electrodynamics, part two
Koen J. van Vlaenderen1,2,∗
1Ethergy B.V, Reseach, Hobbemalaan 10, 1816GD Alkmaar, North Holland, The Netherlands
2Associate professor at the Institute for Basic Research, P.O. Box 1577, Palm Harbor, FL 34684 U.S.A.
∗Correspondence email address: koen@ethergy.com
Abstract General Classical Electrodynamics part two. The energy density of electrodynamic sources are eval-
uated in the context of Maxwell’s theory and in the context of General Classical Electrodynamics.
Keywords General Classical Electrodynamics, Scalar Fields, Energy density
1 Introduction
Generalized energy interaction terms are derived by means of General Classical Electrodynamics (GCED). Firstly,
the following is defined.
δ(x) = −1
4π∆1
|x|(1.1)
F(x) = Z
V0
F(x0)δ(x−x0) d3x0(1.2)
The fundamental theorem of vector algebra is as follows: a vector function F(x) can be decomposed into two
unique vector functions Fl(x) and Ft(x), such that
F(x) = Fl(x) + Ft(x) (1.3)
Fl(x) = −1
4π∇Z
V0
∇0·F(x0)
|x−x0|d3x0(1.4)
Ft(x) = 1
4π∇×Z
V0
∇0×F(x0)
|x−x0|d3x0(1.5)
The lowercase subindexes ’l’ and ’t’ will have the meaning of longitudinal and transverse in this paper. The
longitudinal vector function Flis curl free (∇×Fl=0), and the transverse vector function Ftis divergence free
(∇·Ft= 0). We assume that Fis well behaved (Fis zero if |x|is infinite). Let us further introduce the following
notations and definitions.
1

ρNet electric charge density, in C/m3
J=Jl+JtNet electric current density, in A/m2
Φ Net electric charge (scalar) potential, in V
A=Al+AtNet electric current (vector) potential,
in V·s/m
EΦ=−∇Φ Electric field, in V/m
EL=−∂tAlField induced divergent electric field
ET=−∂tAtField induced rotational electric field
E=EΦ+EL+ET
=−∇Φ−∂tASuperimposed electric field
BΦ=−∂tΦ Field induced scalar field, in V/s
BL=−∇·AlScalar magnetic field, in T = V·s/m2
BT=∇×AtVector magnetic field, in T = V·s/m2
B=−1
c2∂tΦ− ∇·ASuperimposed scalar magnetic field
φ02
0µ0Polarizability of vacuum, in F·s2/m3
µ0Permeability of vacuum: 4π10−7H/m
0Permittivity of vacuum: 8.854−12 F/m
(x, t) = (x, y, z, t) Place and time coordinates
∂t=∂
∂t Partial time differential
∇=∂
∂x ,∂
∂y ,∂
∂z Del operator
∆ = ∇ · ∇ Laplace operator
∆Φ = ∇·∇Φ,∆A=∇∇·A− ∇ ×∇×A
2 The energy density of electrodynamic sources
The electric power density of sources in the power theorem of GCED in the Whittaker premise are expressed as
follows:
P=−J·E−c2ρB
=J· ∇Φ + ρ∂tΦ + J·∂tA+c2ρ∇·A(2.1)
Integration by parts of the first and the fourth term of this expression, over the entire space, results into:
P=−Φ∇·J+ρ∂tΦ + J·∂tA−c2A· ∇ρ(2.2)
The first term can be combined with the second term, by means of the equation of continuity of charge:
P=∂t(ρΦ) + J·∂tA−c2A· ∇ρ(2.3)
In case the specific (non general) condition 1
c2∂tJ+∇ρ=0is satisfied as well, then we can combine all terms into
a single term:
P=∂t(ρΦ + J·A) (2.4)
The energy density of the electrodynamic sources is simply ρΦ + J·Afor this particular case. Compare this
result with the ’interaction terms’ in the Lagrange theory (the Lagrangian) of the incorrect Maxwell-Lorentz
electromagnetism.
2

3 The special case of net electric current, as the product of net charge
and its velocity
In general one cannot attribute a particular velocity, vto the net charge density, ρ, that is present in macroscopic
material (plasma, gas, fluid, solid), such that J=vρ. For instance, the net charge, ρ, on a metallic wire that
carries a current with density J, mainly depends on the electric potential from the electric power source that is
attached to the wire, so the net charge on the wire does not relate directly to the current by means of J=vρ.
Just the conservation of charge is generally true, which also relates the net charge density and net current density,
and in most cases J6=vρ.
In case of a dynamic object that has a measurable velocity, v, and that is electrically charged, one can say the
object carries a current such that J=vρ. If we consider the electrodynamic force on that object in the context of
GCED, we can calculate the ’work density’, W, for instance of time ’dt’:
∂tW=v·[ρE+ρv×BT+ρvBL] (3.1)
=J·E+ρv2BL
This expression must be equal to the power density expression of GCED in the Whittaker premise:
J·E+ρv2BL=J·E+ρc2B(3.2)
such that:
B=v2
c2BL(3.3)
v2BL=BΦ+c2BL
BΦ= (v2−c2)BL
3