4.2 Deduction of the Weber force from the Weber-Maxwell force
The Weber-Maxwell force (
2.3.1) is very general. This makes some calculations unnecessarily complicated. In the case of direct current, a significant simplification is possible, because in this case a force-generating point charge $q_s$ usually moves so slowly that the trajectory in the time interval between emitting the force at time $\tau$ and reaching the receiver charge $q_d$ at time $t$ is almost a straight line. This means that the distance vector $\vec{s} := \vec{r}_d(t) - \vec{r}_s(t)$ between $q_d$ and $q_s$ at time $t$ can be expressed by the equation
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$$\vec{s} = \vec{r} + \vec{v}\,(t-\tau).$$
| (4.2.1) |
Furthermore, the acceleration $\vec{a}$ disappears and the velocity $\vec{v}$ is a constant. The Weber-Maxwell force can therefore be simplified to
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$$\vec{F} = \frac{q_d\,q_s\,\gamma(v)\,\left(\vec{r}\,c + r\,\vec{v}\right)\left(c^2 - v^2\right)}{4\,\pi\,\varepsilon_0\,\left(r\,c + \vec{r}\cdot\vec{v}\right)^3}.$$
| (4.2.2) |
Because of $c^2 - v^2 = c^2/\gamma(v)^2$ this can be rearranged to
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$$\vec{F} = \frac{q_d\,q_s\,\left(\frac{\vec{r}}{r} + \frac{\vec{v}}{c}\right)}{4\,\pi\,\varepsilon_0\,\gamma(v)\,r^2\,\left(1 + \frac{\vec{r}}{r}\cdot\frac{\vec{v}}{c}\right)^3}.$$
| (4.2.3) |
Because of equation (
2.3.5), $t - \tau = r/c$. Substituting this into the equation (
4.2.1) gives
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$$\frac{\vec{s}}{r} = \frac{\vec{r}}{r} + \frac{ \vec{v}}{c}.$$
| (4.2.4) |
Hence, the equation (
4.2.3) becomes
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$$\vec{F} = \frac{q_d\,q_s\,\vec{s}}{4\,\pi\,\varepsilon_0\,\gamma(v)\,r^3\,\left(1 + \frac{\vec{r}}{r}\cdot\frac{\vec{v}}{c}\right)^3}.$$
| (4.2.5) |
In addition, equation (
4.2.4) yields the relation
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$$ \frac{\vec{r}}{r}\cdot\frac{\vec{s}}{r} = 1 + \frac{\vec{r}}{r}\cdot\frac{\vec{v}}{c}.$$
| (4.2.6) |
This turns equation (
4.2.5) into
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$$\vec{F} = \frac{q_d\,q_s\,\vec{s}}{4\,\pi\,\varepsilon_0\,\gamma(v)\,\left(\frac{\vec{r}}{r}\cdot\vec{s}\right)^3}.$$
| (4.2.7) |
Now the only remaining problem is the term $\vec{r}/r\cdot\vec{s}$. Because of equation (
4.2.1) and because of $t-\tau = r/c$ we get
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$$\frac{\vec{r}}{r}\cdot\vec{s} = \frac{\vec{s} - \vec{v}\,(t-\tau)}{c\,(t-\tau)}\cdot\vec{s} = \frac{s^2}{c\,(t-\tau)} - \frac{\vec{s}\cdot\vec{v}}{c}.$$
| (4.2.8) |
From equation (
2.3.5) follows
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$$t - \tau = \frac{r}{c} = \frac{\Vert\vec{s} - \vec{v}\,(t-\tau)\Vert}{c},$$
| (4.2.9) |
i.e.
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$$t - \tau = \frac{s^2}{\vec{s}\cdot\vec{v} + \sqrt{(\vec{s}\cdot\vec{v})^2 + s^2\,(c^2 - v^2)}}.$$
| (4.2.10) |
This can be substituted into the equation (
4.2.8) and we get
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$$\frac{\vec{r}}{r}\cdot\vec{s} = \frac{1}{c}\,\sqrt{(\vec{s}\cdot\vec{v})^2 + s^2\,(c^2 - v^2)} = \sqrt{s^2 - \frac{1}{c^2}\Vert\vec{s}\times\vec{v}\Vert^2}.$$
| (4.2.11) |
This allows equation (
4.2.7) to be transformed to
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$$\vec{F} = \frac{q_d\,q_s\,\vec{s}}{4\,\pi\,\varepsilon_0\,\gamma(v)\,\left(s^2 - \frac{1}{c^2}\Vert\vec{s}\times\vec{v}\Vert^2\right)^{3/2}}.$$
| (4.2.12) |
This equation now only depends on the current distance vector $\vec{s}$ between $q_d$ and $q_s$ at time $t$. Variables that depend on the past time $\tau$ are no longer present, which considerably improves the practical value of the equation.
Equation (
4.2.12) is not only a special solution of Maxwell's equations but also an alternative representation of the classical Weber force (
2.2.1). To demonstrate this, we perform the substitution $\vec{v} \to u\,\vec{v}$ and develop the term into a Taylor series. We get
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$$\vec{F} = \frac{q_d\,q_s\,\vec{s}}{4\,\pi\,\varepsilon_0\,s^3}\,\left(1 - \frac{ s^2\,v^2 - 3\,\Vert\vec{s}\times\vec{v}\Vert^2}{2\,s^2\,c^2}\,u^2 + \mathcal{O}(u)^3\right).$$
| (4.2.13) |
Now we can set $u = 1$ and thus cancel the substitution $\vec{v} \to u\,\vec{v}$. By means of the relation
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$$\Vert\vec{s}\times\vec{v}\Vert^2 = s^2\,v^2 - \left(\vec{s}\cdot\vec{v}\right)^2$$
| (4.2.14) |
we finally obtain the approximation
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$$\vec{F} = \frac{q_d\,q_s\,\vec{s}}{4\,\pi\,\varepsilon_0\,s^3}\,\left(1 + \frac{v^2}{c^2} - \frac{3}{2}\,\left(\frac{\vec{s}}{s}\cdot\frac{\vec{v}}{c}\right)^2\right),$$
| (4.2.15) |
which corresponds exactly to the Weber force (
2.2.1). This clearly shows that Weber electrodynamics is a subset of Weber-Maxwell electrodynamics. It is also evident that Weber electrodynamics only applies under certain conditions, namely
- when the transit time of the force from $q_s$ to $q_d$ is so small that both point charges move almost uniformly within this time span and
- the differential speed between the two point charges is very small compared to the speed of light.
In electrical engineering, this is the case with direct currents and low-frequency alternating currents. Low frequency means that the distance between the current and the measuring point is much smaller than the wavelength. If this condition is not satisfied, Weber electrodynamics is no longer applicable and Weber-Maxwell electrodynamics is required.
