Electrical Force Between Two Moving Charges

[samieee]

We all know the coulomb's law of electrical force between two charges q_1,q_2 equals to \normalsize F= Kq_1q_2/d^2. But this law is not precisely true when the two charges are moving. We know for a single charge q moving with a velocity v the force is \ F=q(E+v×B)

What is the straightforward formula of electrical force between two moving charges?

 

[PhysTech]

I don't think you have a straightforward formula to compute the force between two charges. If some elaborate formula exists I would imagine that it would be highly nontrivial and far from straightforward. There is, however, a more crude and obvious approach. One can determine the force between the two charges $F(t)$ in an iterative way. Given the initial ($t=0$) velocities of the two charges, their initial separation, and the charge they carry, one can determine $F(0)$. Then using basic kinematics one can determine their position and velocity after a small time $\Delta t$. One can repeat this for subsequent $\Delta t$. As you might have suspected, $\Delta t$ must be chosen very carefully. Your $\Delta t$ must account for the worst possible scenarios. Otherwise you would produce garbage results. This may not be a foolproof technique. Although I don't recall any examples at this moment, it is reasonable to expect that there may be some exotic and chaotic systems where implementation of this naïve technique would not work. And one may have to employ some clever techniques to account for the complexity of the system.