In a particle accelerator, if the force applied to the particles is constant, what approximate values do $F$, $m$ and $a$ have, in $F = ma$?
Or if not, is $F = \dfrac{dp}{dt}$ tested?
Either way, to what accuracy is Newton's second law (of acceleration) tested by the accelerations involved?
This of course means the relativistic version of the law.
Thank you.
The equation $F=ma$ is a non-relativistic approximation of Newton's second law. The most general statement of the law is that the force is equal to the instantaneous rate of change of momentum: $$F=\frac{dp}{dt}.$$ This equation is exact both in non-relativistic and relativistic regimes and has been extremely well tested. In non-relativistic mechanics inertia of a body is independent of its velocity so we can write: $$F=\frac{d(mv)}{dt}=m\frac{dv}{dt}=ma.$$ However, in relativistic mechanics inertia increases with velocity so we cannot simply move the mass in front of the derivative. Instead, we have to differentiate the relativistic momentum $$p=\gamma{mv}$$ with respect to time, where $\gamma$ is the Lorentz factor. What we get is the relativistic version of Newton's second law: $$F=\gamma^3{ma}.$$
J.J. Thomson balanced the cathode rays between the electric and magnetic forces. This assumes Maxwell's equations are correct. Note that this does not depend on the electron mass.
With modern tech it would be possible to electrostatically accelerate electrons through a given voltage and measure their time-of-flight over a distance. This does depend on knowledge of the electron mass to calculate the theoretical velocity. I don't know whether such an experiment has been done.
