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4 Proofs

4.1 Derivation of the Weber-Maxwell force from Maxwell's equations

Starting point

The Weber-Maxwell force (2.3.1) can be derived from Maxwell's equations. We start with Maxwell's equations from the perspective of a resting test charge $q_d$. The Lorentz force (2.1.5) simplifies to
$$\vec{F}_M = q_d\,\vec{E}$$ (4.1.1.1)
in the rest frame of the test charge. Furthermore, we want to assume that the fields $\vec{E}$ and $\vec{B}$ are produced by a moving point charge $q_s$. The charge density of this point charge is
$$\rho = q_s\,\delta(\vec{r} - \vec{r}_s(t)),$$ (4.1.1.2)
with $\vec{r}_s(t)$ being the location of the point charge $q_s$ at time $t$. The current density is
$$\vec{j} = \dot{\vec{r}}_s(t)\,\rho.$$ (4.1.1.3)


Derivation of the wave equation

The derivative of the fourth of Maxwell's equations (2.1.4) with respect to time is
$$\nabla\times\frac{\partial\vec{B}}{\partial t} = \mu_0\,\frac{\partial\vec{j}}{\partial t} + \frac{1}{c^2}\,\frac{\partial^2\vec{E}}{\partial t^2}.$$ (4.1.2.1)
Substituting the third Maxwell equation (2.1.3) gives
$$\frac{1}{c^2}\,\frac{\partial^2\vec{E}}{\partial t^2} = -\mu_0\,\frac{\partial\vec{j}}{\partial t} - \nabla\times\left(\nabla\times\vec{E}\right).$$ (4.1.2.2)
Because of $\nabla\times\left(\nabla\times\vec{E}\right) = \nabla\left(\nabla\cdot\vec{E}\right) - \nabla^2\vec{E}$, we get equation
$$\left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2\right)\,\vec{E} = - \frac{1}{\varepsilon_0}\,\left(\frac{1}{c^2}\,\frac{\partial\vec{j}}{\partial t} + \nabla\,\rho\right)$$ (4.1.2.3)
using the first Maxwell equation (2.1.1) and $\mu_0 = 1/(\varepsilon_0\,c^2)$. Applying the equations (4.1.1.1), (4.1.1.2) and (4.1.1.3), we obtain the wave equation
$$\left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2\right)\,\vec{F}_M = - \frac{q_d\,q_s}{\varepsilon_0}\,\left(\frac{1}{c^2}\,\frac{\partial}{\partial t}\,\dot{\vec{r}}_s(t)\,\delta(\vec{r} - \vec{r}_s(t)) + \nabla\,\delta(\vec{r} - \vec{r}_s(t))\right).$$ (4.1.2.4)


Solution of the wave equation

The wave equation (4.1.2.4) can be solved. The solution method is described in [20] section IV, but can also be found in some textbooks, such as Jackson [24]. The solution is
$$\vec{F}_M = -q_d\,\left(\frac{\partial}{\partial t} \vec{A} + \nabla\,\Phi\right)$$ (4.1.3.1)
with
$$\Phi = \frac{q_s\,c}{4\,\pi\,\varepsilon_0\,\left(c^2\,(t - \tau) - \dot{\vec{r}}_s(\tau)\cdot\left(\vec{r} - \vec{r}_s(\tau)\right)\right)},$$ (4.1.3.2)
$$\vec{A} = \frac{1}{c^2}\,\dot{\vec{r}}_s(\tau)\,\Phi$$ (4.1.3.3)
and
$$\tau = t - \frac{1}{c}\left\Vert\vec{r}-\vec{r}_s(\tau)\right\Vert.$$ (4.1.3.4)
The potentials $\Phi$ and $\vec{A}$ are referred to as Liénard-Wiechert potentials.

Simplification of the Liénard-Wiechert potentials

When using the Liénard-Wiechert potentials, two derivatives with respect to $\vec{r}$ and $t$ have to be determined in order to calculate the actual force $\vec{F}_M$ in equation (4.1.3.1). However, the potentials $\Phi$ and $\vec{A}$ contain dependencies on $\tau$, with $\tau$ being a mostly complicated and unknown function of $\vec{r}$ and $t$. For this reason, the Liénard-Wiechert potentials are considered to be the final result in the scientific literature. It was not known that they can be further simplified by applying a mathematical trick [22].

First, we calculate the derivatives of the potentials $\Phi$ and $\vec{A}$ and obtain by using the definitions
$$h_1(\tau) := \left(\vec{r} - \vec{r}_s(\tau)\right)\cdot\dot{\vec{r}}_s(\tau) - c^2\,(t - \tau)$$ (4.1.4.1)
and
$$h_2(\tau) := c^2-\dot{\vec{r}}_s(\tau)\cdot\dot{\vec{r}}_s(\tau) + \left(\vec{r} - \vec{r}_s(\tau)\right)\cdot\ddot{\vec{r}}_s(\tau)$$ (4.1.4.2)
the equations
$$\nabla\,\Phi = \frac{q_s\,c\,h_2(\tau)\,\nabla\,\tau + q_s\,c\,\dot{\vec{r}}_s(\tau)}{4\,\pi\,\varepsilon_0\,h_1(\tau)^2}$$ (4.1.4.3)
and
$$\frac{\partial}{\partial t}\,\vec{A} = \frac{q_s\,\left[h_2(\tau)\,\dot{\vec{r}}_s(\tau)-h_1(\tau)\,\ddot{\vec{r}}_s(\tau)\right]\,\frac{\partial\tau}{\partial t} - q_s\,c^2\,\dot{\vec{r}}_s(\tau)}{4\,\pi\,\varepsilon_0\,c\,h_1(\tau)^2}.$$ (4.1.4.4)
We now insert the equations (4.1.4.3) and (4.1.4.4) into the equation (4.1.3.1) and obtain
$$\vec{F}_M = -\frac{q_d\,q_s}{4\,\pi\,\varepsilon_0}\,\frac{c^2\,h_2(\tau)\,\nabla\,\tau + \left[h_2(\tau)\,\dot{\vec{r}}_s(\tau)-h_1(\tau)\,\ddot{\vec{r}}_s(\tau)\right]\,\frac{\partial\tau}{\partial t}}{c\,h_1(\tau)^2}.$$ (4.1.4.5)
The remaining derivatives in equation (4.1.4.5) can be determined by applying the differential operators to both sides of the equation (4.1.3.4). This gives us
$$\nabla\,\tau = \frac{\vec{r}-\vec{r}_s(\tau) + \left(\vec{r}-\vec{r}_s(\tau)\right)\cdot\dot{\vec{r}}_s(\tau)\,\nabla\,\tau}{c\,\left\Vert\vec{r}-\vec{r}_s(\tau)\right\Vert}$$ (4.1.4.6)
and
$$\frac{\partial\tau}{\partial t} = 1 + \frac{\left(\vec{r}-\vec{r}_s(\tau)\right)\cdot\dot{\vec{r}}_s(\tau)}{c\,\left\Vert\vec{r}-\vec{r}_s(\tau)\right\Vert}\,\frac{\partial\tau}{\partial t}.$$ (4.1.4.7)
By solving the equations using the equation $\left\Vert\vec{r}-\vec{r}_s(\tau)\right\Vert = c\,(t-\tau)$ and the definition (4.1.4.1), we obtain the unexpectedly simple equations
$$\nabla\,\tau = \frac{\vec{r}-\vec{r}_s(\tau)}{h_1(\tau)}$$ (4.1.4.8)
and
$$\frac{\partial\tau}{\partial t} = -\frac{c^2\,(t-\tau)}{h_1(\tau)}.$$ (4.1.4.9)
These can now be inserted into the equation (4.1.4.5). This results in
$$\vec{F}_M = -\frac{q_d\,q_s}{4\,\pi\,\varepsilon_0}\,\frac{c^2\,h_2(\tau)\,\left(\vec{r}-\vec{r}_s(\tau)\right) - \left[h_2(\tau)\,\dot{\vec{r}}_s(\tau)-h_1(\tau)\,\ddot{\vec{r}}_s(\tau)\right]\,c^2\,(t-\tau)}{c\,h_1(\tau)^3}.$$ (4.1.4.10)


Generalization to moving receivers

The last step is to assume that the receiver charge $q_d$ is moving as well and that the trajectory is given by $\vec{r}_d(t)$. In the center of momentum frame of this receiver charge, it then appears as if the source charge $q_s$ is moving with trajectory $\vec{r}_s(t) - \vec{r}_d(t)$, provided that the differential velocity between $q_s$ and $q_d$ is so small that the Lorentz transformation can be approximated by a Galilean transformation. Furthermore, $\vec{r}$ is zero in the center of momentum frame. We therefore first perform the substitution
$$\vec{r} \to \vec{0}$$ (4.1.5.1)
and then the substitutions
$$\vec{r}_s(\tau) \to \vec{r}_s(\tau) - \vec{r}_d(\tau) =: -\vec{r},$$ (4.1.5.2)
$$\dot{\vec{r}}_s(\tau) \to \dot{\vec{r}}_s(\tau) - \dot{\vec{r}}_d(\tau) =: -\vec{v},$$ (4.1.5.3)
and
$$\ddot{\vec{r}}_s(\tau) \to \ddot{\vec{r}}_s(\tau) - \ddot{\vec{r}}_d(\tau) =: -\vec{a}.$$ (4.1.5.4)
This gives us the equation
$$\vec{F}_M = -\frac{q_d\,q_s}{4\,\pi\,\varepsilon_0}\,\frac{c^2\,\left(c^2-v^2 - \vec{r}\cdot\vec{a}\right)\,\vec{r} - \left[-\left(c^2-v^2 - \vec{r}\cdot\vec{a}\right)\,\vec{v}+\left(-\vec{r}\cdot\vec{v} - c^2\,(t - \tau)\right)\,\vec{a}\right]\,c^2\,(t-\tau)}{c\,\left(-\vec{r}\cdot\vec{v} - c^2\,(t - \tau)\right)^3}.$$ (4.1.5.5)
Now we use the relation $c\,(t - \tau) = r$ and summarize further. This brings us to
$$\vec{F}_M = \frac{q_d\,q_s}{4\,\pi\,\varepsilon_0}\,\frac{\left(\vec{r}\,c + r\,\vec{v}\right)\,\left(c^2-v^2 - \vec{r}\cdot\vec{a}\right) + \vec{a}\,r\,\left(r\,c + \vec{r}\cdot\vec{v}\right)}{\left(r\,c + \vec{r}\cdot\vec{v}\right)^3}.$$ (4.1.5.6)
By comparing the force $\vec{F}_M$ with the Weber-Maxwell force (2.3.1), we find that
$$\vec{F} = \gamma(v)\,\vec{F}_M$$ (4.1.5.7)
applies. The force is therefore identical to the Weber-Maxwell force except for a scalar Lorentz factor $\gamma(v)$ that depends only on the differential velocity $v = \Vert\dot{\vec{r}}_d(\tau) - \dot{\vec{r}}_s(\tau)\Vert$.

The reason for using a Lorentz factor instead of the magnetic part of the Liénard-Wiechert potentials is a consequence of Weber electrodynamics. Carl Friedrich Gauss recognized already in 1835 that a magnetic field is not actually required, but that all magnetostatic effects can be explained by a specific central force. The formula (4.1.5.6) has exactly the same properties as Gauss's central force. The only thing missing is the correct normalization. By adding the scalar Lorentz factor, Gauss's idea is subsequently integrated into Maxwell's electrodynamics. This avoids having to handle the B-field components of the Liénard-Wiechert potentials, which on the one hand simplifies the mathematics considerably and on the other hand removes contradictions and inconsistencies.