Thanks, Higgsy, for your response.
Note that in Maxwell's electrodynamics, considered as a mathematical, logic, system,
not considering the physics, that it uses the symbol "E" for two different things. Yes,
I agree that the so called "static" case is a special case of the "dynamic" case, but
it is using the same symbol for both cases. That leads to a contridiction.
Set that argument aside.
When you see
div E = (1/e0) rho
how do you know that E = -grad phi - @A/@t ("@" here mean partial derivative)
or that E = -grad phi is being refered to? That's what I am pointing
out. By *doing* the math, I find E = -grad phi
is the correct choice, and that E = -grad phi -@A/@t is not the correct choice. So, there
needs to be some indication of this in/with the equation. As I point out on the web page,
there are a couple of different ways to do this. One is to explicitly state with the
equation that @A/@t = 0 is being applied. Nothing wrong with that.
But then this restriction must be applied to all other equations thereafter.
As a system of equation, taken as a whole, you can not impose a restriction just on one equation and not on all other equations.
The second problem with Maxwell's electrodynamics that I point out does not depend on
the first problem (the above problem). Again *doing the math* shows that for Ampere's
circital law with the displacement current term requires (for the least number of restrictions)
the use of E = -grad phi, and not E = -grad phi - @A/@t, together with 2 restrictions. That's math, not physics
(although Ampere's law can be applied to physics, of course). Since it is math, it is provable.
From an experimental point of view, Weber's electrodynamics, Weber's force law, explains
the so-called "exploding wire" experiments of Graneau (see his books), all the experiments
that Ampere performed regarding the development of Ampere's force equation (Ampere's
force equation is a special case of Weber's force equation). Since the Lorentz force
equation is not compatible with Maxwell's electrodynamics, but is compatible with
Weber's electrodynamics, experiments proving Lorentz force equation is supporting
Weber's electrodynamics, not Maxwell's electrodynamics.
As you might guess, I think it is critical to get electrodynamics "correct", in the non-special theory of
relativity regime, since some much depends on it. By "correct" I mean a consistent mathematical system
provable by math and logic.
Also, do you have a reference showing Maxwell's 4 equations for discrete sources? I think I read that there
isn't such a thing because Maxwell's 4 equations is only defined for the continuous source case. (I think
this had something to do with the definition of the magnetic induction field being defined only for
continuous current sources.) Is this true? Note that Weber's electrodynamics is defined first for discrete
sources, including 4 differential Maxwell-like equations. Discrete sources case is the more fundamental since
all charges are discrete. If it is true that Maxwell's electrodynamics (with the 4 differential equations) is not
defined for discrete sources, well, another plus for Weber and a minus for Maxwell.... Let me know if you
have a reference.
I would think people in plasma research would be very interested in this Weber topic....