Welcome to the Weber Electrodynamics website.

We will here discuss Weber electrodynamics and compare it to Maxwell electrodynamics.

__Terminology__: It is well known that Maxwell never wrote electrodynamics in the form that
today bears his name. He wrote it in component and quaternion form. (See, for example,
the book by Bruce J. Hunt, __The Maxwellians__, Cornell University Press, 1991.)
Heaviside, Gibbs and others rewrote Maxwell's equations, dropping the "speed of the
medium", combining equations, and eventually writting it in vector notation.

Dispite the inaccuracy, we will continue to refer to electrodynamics as described in standard textbooks as "Maxwell's electrodynamics".

__Terminology__: By "Maxwell's electrodynamics" we mean the four Maxwell differential
equations, plus the
field definitions in terms of the scalar and vector potential functions, plus the
definitions of the scalar and vector potential functions written in terms of the
charge density and current density functions, plus the Lorentz force
equation, and the continuity equation.
Warning: As explained below,
this is not the traditional way to present Maxwell's electrodynamics.

Why consider Weber electrodynamics? Because...

- Weber electrodynamics is mathematically consistent. Maxwell's electrodynamics is not.
- The Lorentz force equations is part of the Weber force equation. Maxwell's electrodynamics is incompatible with the Lorentz force equation.
- Weber electrodynamics satisfies (in the regime where time retardation can be ignored)
- Newton's 3rd Law in the strict form,
- Conservation of energy
- Conservation of linear momentum
- Conservation of angular momentum.

- Weber's electrodynamics is completely defined for discrete sources, which is the most fundamental case of electrodynamics, and can smoothly transition to continuous sources. Maxwell's electrodynamics has problems with discrete sources (I haven't seen Maxwell's equations for discrete sources) and has problems transitioning from discrete to continuous sources case.
- Weber's electrodynamics does not have hidden restrictions. Maxwell's electrodynamics has two implicit restrictions: 1) the charge density function must be a constant in time; and 2) the velocity of the test body (the detector) must be zero in the coordinate system chosen.

We take the point of view that Weber's electrodynamics is a correction and extension of Maxwell's electrodynamics. These webpages will demonstrate, through mathematics, that this is an accurate statement.

Unless otherwise stated, we will work under the following restrictions:

- All relational speeds are significantly less than the speed of light.
- Time retardation (time delay) can be ignored.

Some progress toward lifting these restrictions has been made and may be discussed at a future time.

Please feel free to take up this research and extend Weber's electrodynamics beyond these (and any other) restrictions. We look forward to hearing about your progress!

Einstein knew that there was something wrong with Maxwell electrodynamics. In his 1905 paper, "On The Electrodynamics Of Moving Bodies", originally in German as "Zur Elektrodynamik bewegter Korper", in Annalen der Physik, Vol. 17, p. 891, translation available at this link, he writes:

"It is known that Maxwell's electrodynamics -- as usually understood at the present time -- when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise -- assuming equality of relative motion in the two cases discussed -- to electric currents of the same path and intensity as those produced by the electric forces in the former case."

We emphasize: **"The observable phenomenon here depends only on the
relative motion
of the conductor and the magnet, whereas the customary view draws a sharp distinction
between the two cases in which either the one or the other of these bodies is in motion."**

We find that the source of this discrepancy is based on the way that "current" and "current density" have traditionally been defined. Traditionally, "current density" has been defined as the product of the source charge density function and the (average) velocity of the sources $$ \vec{J} \equiv \rho_S \vec{v}_{S} $$

The problem with this definition is that it depends on the reference frame used to measure the source velocity \( \vec{v}_{S} \). Move to a different reference frame and you get a different value for the source current density!

In Weber's electrodynamics, we define the current density function to be
$$ \vec{J}_{TiS} \equiv \rho_S \vec{v}_{TiS} = \rho_S (\vec{v}_{Ti} - \vec{v}_{S}) $$
where \( \vec{v}_{Ti} \) is the velocity of the test charged body (or detector) measured in
the same reference frame that the source current \( \vec{v}_{S} \) is measured in.
Now move to a different reference frame and the current density function does
not change. It **depends only on the relative motion** of the test and source bodies involved.

This definition is then used in the Weber definition of the vector potential function \( \vec{A} \), which is used in the definition of the Weber magnetic induction function \( \vec{B} \). In this way, we have resolved the discrepancy mentioned by Einstein.

And this leads naturally to Weber's electrodynamics. See the following web pages for details.

- Introduction -- We introduce the mathematical notation that we will be using to describe Weber's electrodynamics, as well as a first look at the defining equations.
- Weber electrodynamics: Discrete Case -- We provide a decomposition of the electrodynamics function, as well as differential equations for the discrete sources case.
- Weber electrodynamics: Continuous Case -- We provide a decomposition of the electrodynamics function, as well as differential equations for the continuous sources case.
- From Maxell to Weber electrodynamics -- We show how, by removing the implicit restriction of Maxwell's electrodynamics, and adding two terms to the scalar potential, we obtain Weber's electrodynamics.
- Weber's force equation -- We provide several different expressions for Weber's force equations and show that Ampere's force equation and the Lorentz force equation are part of Weber's force equation.

- Problems with Maxwell electrodynamics -- We provide two examples where the mathematics of Maxwell's electrodynamics, considered as a single mathematical system, is inconsistent.
- Verify Gauss's Law -- We attempt to verifying Gauss's Law for the general case but find that it is not possible.
- Verify Ampere's Law -- We attempt to verifying Ampere's Law for the general case.
- Ampere's Law in Coulomb gauge -- We attempt to verifying Ampere's Law when the Coulomb gauge condition is imposed.

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