Consider the problem of two charges, one positive and one negative, both located on the $y$ axis at time $t=0$, at finite separation. Their initial velocities are exactly equal and along the $x$ axis (positive or negative direction) and have speed $v$. I'm interested in the values of $v$ such that the electric attraction is perfectly cancelled by the magnetic interaction, which would result in the charges continuing to move with the same velocity.
I find that the solution is $v=\pm\sqrt{1/\epsilon_0\mu_0}$ when the electric force between the charges and the magnetic field created by each charge are $$\mathbf{F}_E=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2};\quad\mathbf{B}=\frac{\mu_0}{4\pi}\frac{q\mathbf{v}\times\mathbf{r}}{r^3}.$$
Why is this value of $v$ equal the speed of an EM wave? Is there an intuitive reason for why this is the case? Is the reasoning robust when we consider a non-vacuum medium?
Best Answer
Boost yourself into the rest frame of the two charges. Since the charges are not moving, the force between them is purely electric, which causes them to accelerate towards each other at some rate. Now return to the lab frame. Here, the particles have relativistic speed, which means time dilation comes into effect. This will result in the charges accelerating towards each other at a slower rate. This slower acceleration can also be described as a repulsive magnetic force. The time dilation explanation and the magnetic force explanation are equivalent, since electric and magnetic fields transform into each other in different reference frames.
In your calculation, the charges stop accelerating at the speed of light (ignoring for a moment that all known charged particles have mass and so cannot reach the speed of light). This is because objects traveling at the speed of light do not experience time, and so cannot accelerate. At any sub-light speed, the charges have a rest frame, so there must be attraction between them in all reference frames.
In non-vacuum mediums, the situation is complicated by interactions with the medium like Cherenkov radiation that causes charged particles to emit radiation when they exceed the speed of light in that medium.