On this webpage, we introduce some of the notation we will be using to describe Weber's electrodynamics. Our point of view is that conciseness at the expense of clarity is not a virtue. Therefore, the notion we use is loaded with subscripts. This may be tedious to use, but we feel it will help clarify the equations.
Consider two charged bodies. These will be labeled \( q_{Sj} \) and \( q_{Ti} \). The charged body \( q_{Sj} \) will be called the "source" body and the charged body \( q_{Ti} \) will be called the "test" body. The subscripts \( j \) and \( i \) are used when there is more than one source or test body.
We use the terminology "charged body" or "body" to emphasize that we recognize that these are not point particles. We take the point of view that charged bodies are not zero dimensional point particles but are instead open, dynamic systems existing in 3D-space. As such, they are not "solid things" giving rise to fields. Instead, like air and a tornado, the seeming appearance of a local "charged body" is the field dynamically defining a locally stable dynamic system. We will not pursue this concept further, but will simply say that "charged body", although better than "point particle", is still inacurate.
We assume that the source charged body's location can be mathematically represented and approximated to be at the position \( \vec{r}_{Sj} \) and that the test charged body's location can be represented and approximated to be \( \vec{r}_{Ti} \). Then their relative position, in an arbitrary coordinate system, is given by $$ \vec{r}_{TiSj} \equiv \vec{r}_{Ti} - \vec{r}_{Sj}. $$
The magnitude of this relative position vector is given by (we will use positive roots unless otherwise stated) $$ r_{TiSj} \equiv \sqrt{ (\vec{r}_{Ti} - \vec{r}_{Sj}) \bullet (\vec{r}_{Ti} - \vec{r}_{Sj}) } . $$
The unit vector from the source body \( q_{Sj} \) to the test body \( q_{Ti} \) is given by $$ \hat{r}_{TiSj} \equiv \dfrac{\vec{r}_{TiSj}}{r_{TiSj}}. $$
The relative velocity is given as $$ \vec{v}_{TiSj} \equiv \dfrac{d}{dt}\vec{r}_{TiSj} = \vec{v}_{Ti} - \vec{v}_{Sj} . $$
The relative acceleration vector is given by $$ \vec{a}_{TiSj} \equiv \dfrac{d}{dt}\vec{v}_{TiSj} = \vec{a}_{Ti} - \vec{a}_{Sj} . $$
The "relational" speed is defined to be $$ \dot{r}_{TiSj} \equiv \dfrac{d}{dt}r_{TiSj}. $$
The relational speed can be written as $$ \dot{r}_{TiSj} = \hat{r}_{TiSj} \bullet \vec{v}_{TiSj}. $$
And the "relational" acceleration is defined to be $$ \ddot{r}_{TiSj} \equiv \dfrac{d^2}{dt^2}r_{TiSj}. $$
The relational acceleration can be written as $$ \ddot{r}_{TiSj} = \dfrac{1}{r_{TiSj}} ( \vec{v}_{TiSj} \bullet \vec{v}_{TiSj} - \dot{r}_{TiSj}^2 + \vec{r}_{TiSj} \bullet \vec{a}_{TiSj} ) . $$
These relational quantities are very important. They have the same value regardless of the reference system choosen. Note that the reference system need not be an "inertial" reference system. See the books by Prof. Assis: Weber's Electrodynamics, Kluwer, 1994; Relational Mechanics and Implementation of Mach's Principle with Weber's Gravitational Force, Apeiron, 2014.
We will write the nabla vector operator as $$ \nabla_T \equiv \dfrac{\partial}{\partial x_T} \hat{x} + \dfrac{\partial}{\partial y_T} \hat{y} + \dfrac{\partial}{\partial z_T} \hat{z}. $$
Additional notation will be defined as needed.
With the above definitions, Weber's potential energy function, for the potential energy involved with two charged bodies, \( q_{Sj} \) and \( q_{Ti} \), is given as $$ U_{TiSj} = \dfrac{q_{Ti} q_{Sj} }{4 \pi \epsilon_0} \dfrac{1}{r_{TiSj}} \left( 1 - \dfrac{\dot{r}_{TiSj}^2}{2 c^2} \right). $$
All of Weber's electrodynamics comes from this single equation!
Weber's force equation for the force on the test charged body due to the source charged body is calculated from the equation $$ \vec{F}_{TiSj} = - \hat{r}_{TiSj} \dfrac{d}{d r_{TiSj}}U_{TiSj}. $$
Performing the time derivative gives $$ \vec{F}_{TiSj} = q_{Ti} \left( 1 - \dfrac{1}{2 c^2} \dot{r}_{TiSj}^2 + \dfrac{1}{c^2} r_{TiSj} \ddot{r}_{TiSj} \right) \dfrac{q_{Sj}}{4 \pi \epsilon_0} \dfrac{\hat{r}_{TiSj}}{r_{TiSj}^2}. $$
The way we have writting Weber's force equation suggests the following form $$ \vec{F}_{TiSj} = q_{Ti} \gamma_W \vec{E}_{TiSj}^{(C)}, $$
where we define $$ \gamma_W \equiv \left( 1 - \dfrac{1}{2 c^2} \dot{r}_{TiSj}^2 + \dfrac{1}{c^2} r_{TiSj} \ddot{r}_{TiSj} \right) $$ to be the "Weber coupling coefficient", and $$ \vec{E}_{TiSj}^{(C)} \equiv \dfrac{q_{Sj}}{4 \pi \epsilon_0} \dfrac{\hat{r}_{TiSj}}{r_{TiSj}^2} $$ is the Coulomb field.
From the way that we have written the Weber force equation, in terms of the coupling coefficient \( \gamma_W \), and the Coulomb field, \( \vec{E}_{TiSj}^{(C)} \), we suggest that there is only one field, the Coulomb field. From this point of view (which is not critical to Weber electrodynamics) we consider all other "fields" to be convenient functions and not physical fields.
Note that both \( U_{TiSj} \) and \( \vec{F}_{TiSj} \) are relational functions. This means that they have the same value in any reference frame, not just in an inertial reference frame.
Another way that we can write Weber's force equation is $$ \vec{F}_{TiSj} = q_{Ti} \vec{E}_{TiSj} $$ where we now define the "Weber electrodynamics function" (not a physical field) to be $$ \vec{E}_{TiSj} \equiv \gamma_W \vec{E}_{TiSj}^{(C)}. $$ Again, this interpretation of what is a "field" and what is a "function" is not critical to Weber's electrodynamics. It is just one way to think of the equations.
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