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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.

Hritik Narayan
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user141183
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2 Answers2

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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: $$F=\gamma^3{ma}.$$

Marko Leljak
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  • How long does it take to accelerate a particle from zero to near c in an accelerator? Am I right in assuming it's just a few seconds, or are the particles made to make a large number of circuits before they reach relativistic speeds? – user141183 Jul 03 '17 at 10:49
  • These are technical details which depend on the type of the accelerator. According to Wikipedia, the LHC operates like this: "Prior to being injected into the main accelerator, the particles are prepared by a series of systems that successively increase their energy. The first system is the linear particle accelerator LINAC 2 generating 50-MeV protons, which feeds the Proton Synchrotron Booster (PSB). There the protons are accelerated to 1.4 GeV and injected into the Proton Synchrotron (PS), where they are accelerated to 26 GeV. Finally the Super Proton Synchrotron (SPS) is used to... – Marko Leljak Jul 04 '17 at 06:53
  • ...further increase their energy to 450 GeV before they are at last injected (over a period of several minutes) into the main ring. Here the proton bunches are accumulated, accelerated (over a period of 20 minutes) to their peak energy, and finally circulated for 5 to 24 hours while collisions occur at the four intersection points." – Marko Leljak Jul 04 '17 at 06:54
  • Thank you. There are four rings of increasing size. It seems you're saying about half an hour from near zero to near c. Is the speed known accurately at every stage on the way? If so, how is it known? – user141183 Jul 04 '17 at 11:12
  • I'm really not qualified to give a more detailed response, maybe someone else is. – Marko Leljak Jul 04 '17 at 11:16
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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.

Keith McClary
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