How to Calculate Force Between Moving Charged Particles?

[Nusc]

Homework Statement

In a frame S, two identical particles with electric charge q move abreast along lines parallel to the x-axis, a distance r apart and with velocity v. Determine the force in S that each exerts on the other, by use of the force law for a uniformly moving charge.



I would need a lorentz forc e law in 4 vector notation,

what is it? Then where do I proceed?

Homework Equations

The Attempt at a Solution

 

[diazona]

My first thought would be to make a Lorentz transformation into the frame where both particles are at rest.

 

[Nusc]

We know its just the Coulomb force,

we have the equation for field of a moving charge:

$$e= \frac{Q \bf{r_o}}{\gamma^2 r_o^2 (1-u^2/c^2 sin^2(\theta))^{3/2}}$$

and we can determine the magnetic field in this case,
$$cb = \frac{\bf{v} \times \bf{e}}{c}$$

I'm not sure at this point

 

[Nusc]

The above equation was obtained by this:

$$\bf{e} = \frac{Q}{(cr-\bf{r}\bf{u})^3}\frac{c^2}{\gamma^2}(c\bf{r}-r\bf{u})$$

I'm not sure of I"m on the right track.