Interaction between two point charges

[universal_101]

Let's suppose, we have two ideal point charges $q_{1}, m_{1}$ and $q_{2}, m_{2}$.

One of them comes from almost infinite distance with relative velocity $v_{0}$, w.r.t another charge. I'm curious how can we analyze this situation mathematically, i.e the equations of the motions of these particles.

Please post some references too.

Thanks

 

[Studiot]

What are your thoughts on the dynamics of the situation?

 

[universal_101]

What are your thoughts on the dynamics of the situation?

I think it is fairly hard situation to analyse mathematically,

We know how to calculate the Electric Field produced by a moving charge, but this Electric field is w.r.t a fixed point.

For example we have Heaviside's equation for uniform motion as

$E = \frac{q}{4\pi\epsilon_{0} r^3} \frac{\left(1- v^2/c^2\right)}{\left(1- \left(v^2/c^2\right)sin^2\theta\right)^{3/2}}$ $r$

and Feynman's equation for all motion as

$E = \frac{q}{4\pi\epsilon_{0}}\left(\frac{e_{r^\acute{}}}{{r^\acute{}}^2} + \frac{r^\acute{}}{c} \frac{d}{dt}\left(\frac{e_{r^\acute{}}}{{r^\acute{}}^2}\right) - \frac{1}{c^2} \frac{d^2}{{dt}^2}\left({e_{r^\acute{}}}\right) \right)$

where all the symbols have their usual meaning, but since these Electric Field values are only for observing from an inertial system or stationary system, I don't know how to proceed for non-inertial system, i.e a simple two point charge problem.

 

[Studiot]

I don't know how to proceed for non-inertial system, i.e a simple two point charge problem.

Your original question had mass so why is it non inertial?

 

[universal_101]

Your original question had mass so why is it non inertial?

Non-inertial as in SR frame of reference Non-inertial

Obviously, both charges would accelerate/ decelerate as they start interacting with each other, I posted equations for calculating Electric Field in order to get the force on each other, and then dividing by mass we can always get the acceleration.

That is multiplying E by q/m should give the acceleration, but i don't know how to get E at the first place.

Since, we are looking for the dynamics of the situation, we need to have the accelerations of each to calculate their speed or position after some time.

 

[mfb]

In the non-relativistic limit, you can reduce the problem to a 1-body-motion around the common center of mass, and you get the Kepler problem. If that is not precise enough, you can try to add correction terms for relativistic mechanics.

With relativistic velocities, it is still possible to reduce the problem, but the formulas are not as simple as in the Kepler problem. You get radiation and so on.

 

[universal_101]

In the non-relativistic limit, you can reduce the problem to a 1-body-motion around the common center of mass, and you get the Kepler problem. If that is not precise enough, you can try to add correction terms for relativistic mechanics.

With relativistic velocities, it is still possible to reduce the problem, but the formulas are not as simple as in the Kepler problem. You get radiation and so on.

Thanks mfb,

I would really appreciate some easy to get references, for the two body problem with relativistic speeds for point charges.