Weber Electrodynamics

Continuous Source Case

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

See the Introduction webpage for definitions of the mathematical notation used.

In order to transition Weber electrodynamics from the discrete sources case to the continuous sources case, we define the replacement $$ q_{Sj} \rightarrow \rho_S \; dVol_S $$ where \( \rho_S \) is the volume charge density function and \( dVol_S \) is the differential volume element over the sources. Note that this is the same as in Maxwell's electrodynamics.

Next, we define the replacement $$ q_{Sj} \vec{v}_{TiSj} \rightarrow \vec{J}_{TiS} \; dVol_S $$ where \( \vec{J}_{TiS} \) is the current density function.

The current density function is then defined as $$ \vec{J}_{TiS} \equiv \rho_S \vec{v}_{TiS} = \rho_S (\vec{v}_{Ti} - \vec{v}_{S}) $$ where \( \vec{v}_S \) is the average drift velocity of the source charge bodies that comprise the "continuous" source current through the differential volume \( dVol_S \).

We note that this is not the way that the current density function is defined in Maxwell electrodynamics. In Maxwell electrodynamics, the current density function is defined as $$ \vec{J}_S \equiv \rho_S \vec{v}_{S}. $$ However, from a physics point of view, this either 1) ignores the current due to the velocity \( \vec{v}_{Ti} \) of the detector (the test charged body) in the arbitrary coordinate system choosen, or 2) sets the velocity \( \vec{v}_{Ti} \) to be zero, requiring the coordinate system to be moving with the test charged body, so that the coordinate system is not arbitrary. Recall that this condition, \( \vec{v}_{Ti} = \vec{0} \), was one of the implicit restrictions we discovered and described on the webpage Problems with Maxwell's electrodynamics. (Perhaps this implicit restriction in Maxwell's electrodynmics isn't so "crazy" as it first sounded?)

We note that for the case \( \vec{v}_{Ti} = \vec{0} \) we have \( -\vec{J}_{TiS} = \vec{J}_S \), recovering the Maxwell value of the current density function.

As we showed on the Introduction webpage, Weber's force equation, for the discrete sources case, can be written as $$ \vec{F}_{TiSj} = q_{Ti} \vec{E}_{TiSj}, $$ where \( \vec{E}_{TiSj} \) is the Weber electrodynamics function.

Recall that, for the discrete source case, we can write the Weber electrodynamics function as $$ \vec{E}_{TiSj} \equiv \gamma_W \vec{E}_{TiSj}^{(C)}, $$ where Weber's coupling coefficient is given by $$ \gamma_W \equiv \left( 1 - \dfrac{1}{2 c^2} \dot{r}_{TiSj}^2 + \dfrac{1}{c^2} r_{TiSj} \ddot{r}_{TiSj} \right) $$ and the Coulomb field is $$ \vec{E}_{TiSj}^{(C)} \equiv \dfrac{q_{Sj}}{4 \pi \epsilon_0} \dfrac{\hat{r}_{TiSj}}{r_{TiSj}^2}. $$

Using the replacements described above, we can write Weber's electrodynamics function for the continuous sources case as (we drop the subscript "\( j \)" to indicate continuous source) $$ \vec{E}_{Ti} = \dfrac{1}{4 \pi \epsilon_0} \iiint \rho_S \left( 1 - \dfrac{1}{2 c^2} \dot{r}_{TiS}^2 + \dfrac{1}{c^2} r_{TiS} \ddot{r}_{TiS} \right) \dfrac{\hat{r}_{TiS}}{r_{TiS}^2} \; dVol_S $$

We have discovered, in the continuous sources case, that the Weber electrodynamics function, \( \vec{E}_{Ti} \), can be decomposed into the gradient of a scalar function, minus the time derivative of a vector function, minus a term involving the time derivative of the charge density function. Specifically, $$ 1) \;\;\; \vec{E}_{Ti} = - \nabla_T \Phi_{Ti} - \dfrac{\partial}{\partial t} \vec{A}_{Ti} - \vec{R}_{Ti}^{(E)} $$ where $$ \Phi_{Ti} \equiv \dfrac{1}{4 \pi \epsilon_0} \iiint \rho_S \left( \dfrac{1}{r_{TiS}} + \dfrac{\dot{r}_{TiS}^2}{2 c^2 r_{TiS}} + \dfrac{1}{c^2} \ddot{r}_{TiS} \right)\; dVol_S $$ is the Weber scalar potential function, and $$ \vec{A}_{Ti} \equiv \dfrac{1}{4 \pi \epsilon_0 c^2 } \iiint \dfrac{-\vec{J}_{TiS}}{r_{TiS}} \; dVol_S $$ is the Weber vector potential function, and $$ \vec{R}_{Ti}^{(E)} \equiv \dfrac{1}{4 \pi \epsilon_0 c^2 } \iiint \left( \dfrac{\partial \rho_S}{\partial t} \right) \dfrac{\vec{v}_{TiS}}{r_{TiS}} \; dVol_S . $$

Again, the "extra" minus sign in the definition of the vector potential function is due to the way that we have defined the relative velocity \( \vec{v}_{TiS} \) and the current density function \( \vec{J}_{TiS}\).

It is interesting to note that for the case that the charge density function is a constant in time, \( \partial \rho_S / \partial t = 0\), then Eq. (1) looks like the corresponding Maxwell equation. Of course, the scalar and vector potential function in Weber electrodynamics are defined differently than for Maxwell electrodynamics, so the equations are not the same. Still, recall that this was the other implicit restriction we discovered in Maxwell's electrodynamics.

We now define the Weber magnetic induction function for the continuous sources case to be $$ \vec{B}_{Ti} \equiv \nabla_T \times \vec{A}_{Ti}. $$ Explicitly, this can be written as $$ \vec{B}_{Ti} = \dfrac{1}{4 \pi \epsilon_0 c^2 } \iiint \dfrac{-\vec{J}_{TiS} \times \hat{r}_{TiS}}{r_{TiS}^2}\;dVol_S. $$

Using the above definitions, it can be shown that, for the continuous sources case, (SI units)

\( \nabla_T \bullet \vec{E}_{Ti}^{(C)} = \dfrac{1}{\epsilon_0} \rho_S \left(\vec{r}_{Ti}, t \right) \)
\( \nabla_T \bullet \vec{B}_{Ti} = 0 \)
\( \nabla_T \times \vec{E}_{Ti} = - \dfrac{\partial}{\partial t}\vec{B}_{Ti} - \vec{R}_{Ti}^{\left( \nabla \times E \right)} \)
\( \nabla_T \times \vec{B}_{Ti} = - \mu_0 \vec{J}_{TiS}\left(\vec{r}_{Ti}, t \right) + \mu_0 \epsilon_0 \dfrac{\partial}{\partial t}\vec{E}_{Ti}^{(C)} - \mu_0 \epsilon_0 \vec{R}_{Ti}^{\left( E^{(C)}\right)} \)


\( \vec{R}_{Ti}^{\left( \nabla \times E \right)} \) \( \equiv \nabla_T \times \vec{R}_{Ti}^{(E)} \)
\( \vec{R}_{Ti}^{(E)} \) \( \equiv \dfrac{1}{4 \pi \epsilon_0 c^2 } \iiint \left( \dfrac{\partial \rho_S}{\partial t} \right) \dfrac{\vec{v}_{TiS}}{r_{TiS}} \; dVol_S \)
\( \vec{R}_{Ti}^{\left( E^{(C)}\right)} \) \( \equiv \dfrac{1}{4 \pi \epsilon_0 c^2 } \iiint \left( \dfrac{\partial \rho_S}{\partial t} \right) \dfrac{\hat{r}_{TiS}}{r_{TiS}^2} \; dVol_S \)

We have explicitly listed the arguments to the charge density function and the current density function to explicitly show that these source functions are to be evaluated at the test charge body positions. These are called the free charge density and free current density functions in Maxwell electrodynamics.

For the special case that the charge density function is a constant in time, we have

\( restriction: \partial \rho_S / \partial t = 0 \)
\( \nabla_T \bullet \vec{E}_{Ti}^{(C)} = \dfrac{1}{\epsilon_0} \rho_S \left(\vec{r}_{Ti} \right) \)
\( \nabla_T \bullet \vec{B}_{Ti} = 0 \)
\( \nabla_T \times \vec{E}_{Ti} = - \dfrac{\partial}{\partial t}\vec{B}_{Ti} \)
\( \nabla_T \times \vec{B}_{Ti} = - \mu_0 \vec{J}_{TiS}\left(\vec{r}_{Ti}, t \right) + \mu_0 \epsilon_0 \dfrac{\partial}{\partial t}\vec{E}_{Ti}^{(C)} \)

These four equations look like the corrected Maxwell equations. They are not because the scalar and vector potential functions of Weber electrodynamics are defined to be more general than the Maxwell salar and vector potential functions.

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