When was the ratio between electric and magnetic forces in an electromagnetic field established?

We know that the magnetic force on a particle moving in a magnetic field was found by J.J. Thomson in 1881, with a slight mistake, and then corrected by Heaviside in 1885 to $$F_M = q\,v\times B$$.

Can you tell me when was discovered that its ratio to the electric force was $$v^2/c^2$$, and how was it determined? I suppose it could have been done only in a cyclotron.

Then, lastly, do you know when and who decided that $$F_M$$ was just $$F_E$$ seen in a different system exploiting the resemblance of the ratio with a Lorentz transformation, unifying the two forces ?

• Hi, we prefer one question per post. Your second question probably has a rather long answer, so it would be easier to deal with in a separate post. Oct 18 '19 at 13:21
• I am not sure what ratio you are referring to. Lorentz derived the formula for the total force exerted by the electromagnetic field on a moving charged particle, now called the Lorentz force, as $F=q\,E+q\,v\times B$ in 1895. He did not need a cyclotron, only Maxwell's equations. But the ratio of the components does not need to be $v^2/c^2$. Oct 19 '19 at 4:51
• @Conifold, lorentz has nothing to do with my question. He simply joined in one formula the electric and magnetic force acting on a charge. My question is about the ratio between the two forces (FM)/E. Two electrons circling a ciclotron at C/2 feel two forces E, repulsive, and M, attractive, which is 1/4 of E Oct 19 '19 at 12:04
• @CarlWitthoft, it is one question because who joined the forces did it because of the resemblance of v^2/C^2 with relativity transformation Oct 19 '19 at 12:06