Newtons third law states that there is a counter force to every force. Unfortunately this doesn't seem to work for moving point charges. The Coulomb force cancels out but
the BField of a moving point charge is:
[tex]\mathbf{B}=\frac{\mu_0}{4\pi}q \frac{\mathbf{v}\times\mathbf{r}}{\leftr\right^3}[/tex]
And the Lorenz force is
[tex]\mathbf{F}=q\, \mathbf{v}\times \mathbf{B}[/tex]
Lets assume that the two charges have velocities [itex]\mathbf{v}_1,\mathbf{v}_1[/itex]
Therefore the two Lorenz forces are
[tex]\mathbf{F}_1=k(r)\, \mathbf{v}_1 \times (\mathbf{v}_2 \times\mathbf{r}) [/tex]
and
[tex]\mathbf{F}_2=k(r)\, \mathbf{v}_2 \times (\mathbf{v}_1 \times ( \mathbf{r})) [/tex]
Due to the Jacobi identity the sum of the two forces is not zero
[tex]\mathbf{F}_1+\mathbf{F}_2= k(r)\, \mathbf{r}\times(\mathbf{v}_1\times \mathbf{v}_2)[/tex]
What is the solution here? The Pointing vector? Relativity? I think that the basic formulas must be correct for slowly moving charges. So it shouldn't be due to non linear trajectories, neglected acceleration or some such thing.
