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August 27

Lorentz force and moving magnetic flux

Just like Faraday where you can move a magnet in a coil, or the coil towards this magnet and obtain an electric current in both cases. With the Lorentz force, according to the same principle of relativity, can we move the magnetic field rather than the charges and also obtain the Lorentz force? Has anyone had this experience? — Preceding unsigned comment added by Malypaet (talk • contribs) 21:17, 27 August 2022 (UTC)[reply]

When a charged particle moves through a magnetic field, it indeed experiences a Lorentz force, causing an acceleration. If we change to a frame of reference in which the charged particle is initially stationary, it must still experience some acceleration, which cannot be a Lorentz force, as the Lorentz force is proportional to the velocity, which is now zero. If you do the proper maths with a Lorentz transformation, you'll see that we now have an electric field in addition to the magnetic field, providing an electric force. The maths are pretty complex, but yes, you can move the magnet instead of the charged particle and you'll see the same effect. PiusImpavidus (talk) 09:27, 28 August 2022 (UTC)[reply]

Why apply Lorentz transformations ? For the Lorentz force to apply, must the velocity of the charges be relativistic in a magnetic flux ? Malypaet (talk) 19:38, 28 August 2022 (UTC)[reply]

Your question involves a transformation from a stationary coordinate frame in spacetime to a moving frame, so a Lorentz transformation should be applied to obtain the answer (see Lorentz transformation § Transformation of other quantities and Four-current § Motion of charges in spacetime). The classical formulas for the Lorentz force apply at velocities for which relativistic effects can be ignored. Otherwise, one should use the relativistic form of the Lorentz force. --Lambiam 07:48, 29 August 2022 (UTC)[reply]

You don't need a relativistic speed for the Lorentz force to act on a charged particle (it works quite well even at modest speeds of a few hundred km/s), but to make it work, you need Lorentz transformations anyway. I suppose there's a reason why the force and the transformations were named after the same guy. Being able to make the force go to zero means that you cannot neglect the corrections coming from relativistic effects (compared to zero, nothing can be neglected), which is indeed how classical electromagnetism paved the way towards relativity. Einstein too was standing on the shoulders of giants.

When looking at the Lorentz transformation of the electromagnetic four-potential of a pure and constant magnetic field, you can see that at low velocities the resulting time-like component is tiny. However, for the resulting classical electric potential, you have to multiply that with the speed of light, so that even at non-relativistic velocities, the electric field is significant. PiusImpavidus (talk) 08:48, 29 August 2022 (UTC)[reply]