- #1

0xDEADBEEF

- 816

- 1

the B-Field of a moving point charge is:

[tex]\mathbf{B}=\frac{\mu_0}{4\pi}q \frac{\mathbf{v}\times\mathbf{r}}{\left|r\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.