Weber Electrodynamics - Discrete Source Case

On this webpage, we write out the equations for Weber's electrodynamics for the discrete sources case. As far as we know, Weber's electodynamics has not previously been written in this form.

As we showed on the Introduction webpage, Weber's force equation can be written as $$ \vec{F}_{TiSj} = q_{Ti} \gamma_W \vec{E}_{TiSj}^{(C)}, $$ 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.

Another way that we can write Weber's force equation is $$ \vec{F}_{TiSj} = q_{Ti} \vec{E}_{TiSj} $$ where we define the "Weber electrodynamics function" to be $$ \vec{E}_{TiSj} \equiv \gamma_W \vec{E}_{TiSj}^{(C)}. $$

We have discovered that the Weber electrodynamics function can be decomposed into the gradient of a scalar function and the time derivative of a vector function. Specifically, $$ 1) \;\;\; \vec{E}_{TiSj} = - \nabla_T \Phi_{TiSj} - \dfrac{d}{d t} \vec{A}_{TiSj} $$ where $$ \Phi_{TiSj} \equiv \dfrac{q_{Sj} }{4 \pi \epsilon_0} \dfrac{1}{r_{TiSj}} \left( 1 + \dfrac{1}{2 c^2} \dot{r}_{TiSj}^2 + \dfrac{1}{c^2} \ddot{r}_{TiSj} \right) $$ will be called the "Weber scalar potential function", and $$ \vec{A}_{TiSj} \equiv \dfrac{-q_{Sj}}{4 \pi \epsilon_0 c^2 }\dfrac{\vec{v}_{TiSj}}{r_{TiSj}} $$ will be called the "Weber vector potential function".

The "extra" minus sign in the definition of the vector potential function is due to the way that we have defined the relative velocity $$ \vec{v}_{TiSj} \equiv \vec{v}_{Ti} - \vec{v}_{Sj}. $$

WARNING: The decomposition, Eq. (1), is only valid for the discrete sources case. In the continuous sources case an additional term appears. This will be specified on the page dealing with the continuous source case.

Next, we can define the "Weber magnetic induction function" to be $$ \vec{B}_{TiSj} \equiv \nabla_T \times \vec{A}_{TiSj}. $$ Explicitly, this can be written as $$ \vec{B}_{TiSj} = \dfrac{-q_{Sj}}{4 \pi \epsilon_0 c^2 } \dfrac{\vec{v}_{TiSj} \times \hat{r}_{TiSj}}{r_{TiSj}^2}. $$

Note that in Weber's electrodynamics, it is natural to define a magnetic induction function for a discrete charged body. In Jackson's book, and in Griffths' book, defining a magnetic induction function for discrete charged bodies is problematic.

$ \nabla_T \bullet \vec{E}_{TiSj}^{(C)} = \dfrac{q_{Sj}}{\epsilon_0} \delta(\vec{r}_{TiSj}) $

$ \nabla_T \bullet \vec{B}_{TiSj} = 0 $

$ \nabla_T \times \vec{E}_{TiSj} = - \dfrac{d}{dt}\vec{B}_{TiSj} $

$ \nabla_T \times \vec{B}_{TiSj} = - \mu_0 q_{Sj} \vec{v}_{TiSj}\delta(\vec{r}_{TiSj}) + \mu_0 \epsilon_0 \dfrac{d}{dt}\vec{E}_{TiSj}^{(C)} $

This table of equations looks a lot like the corrected Maxwell equations. (See the Problems with Maxwell's electrodynamics web page.) However, the Maxwell equations are for continuous sources, and they have two restrictions. Here, these Weber equations are for the discrete sources case.