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If two stationary charges experience 1N of force, two moving charges should experience 1N of force as well right? (Due to the fact the the moving charges see each other as stationary.)

So can we use only coulomb law to explain every phenomenon in electromagnetism especially magnetism?

Qmechanic
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Physics33
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    Related/possible duplicates: https://physics.stackexchange.com/q/3618/50583, https://physics.stackexchange.com/q/126518/50583 – ACuriousMind Jan 10 '22 at 10:34
  • I think you can’t since $\mathbf E$ is not invariant. A nice and similar discussion about what is magnetism was done here. Please read the comments there. – J. Manuel Jan 10 '22 at 10:34

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Indeed, what you say is true and this is exactly the approach that is taken in the undergrad textbook Electricity and Magnetism by Purcell and Morin. The entire chapter 5 is dedicated to showing that the magnetic field has to be consequence of electrostatic force and special relativity.

Cyrus Tirband
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  • So is it possible to eliminate the concept of magnetism from our textbook? Just be like replacing mass with energy. (In particle physics) – Physics33 Jan 10 '22 at 09:56
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    Not exactly. Although special relativity and coulomb force shows that the magnetic force must exist, this does not mean that it is always possible to find a reference frame where the magnetic force is zero. For a full description of the electromagnetic field in any reference frame, the magnetic field is necessary. For example, two lorentz invariant quantities are $\mathbf{E}^2-\mathbf{B}^2$ and $\mathbf{E}\cdot\mathbf{B}$ which should be telling – Cyrus Tirband Jan 10 '22 at 10:01
  • E^2-B^2, it means that electric force increases, magnetic force increases as well? Can I interpret like this? Situation 1: For two stationary charges, electric force =5N and magnetic force=0N Situation 2: For two moving charges, electric force=6N and magnetic force 1N. (It become 6N because of length contraction or squeezing effect of coulomb field) – Physics33 Jan 10 '22 at 10:07
  • The invariant quantity relates to the field strengths. $\mathbf{E}^2-\mathbf{B}^2$ being invariant means that if the 2-norm of the E-field is larger than the 2-norm of the B-field in one frame, then it is larger in any frame. If they are equal in one frame, they are equal in any frame. This does not relate one-to-one to the force on a particle, as that depends on the speed of the particle. – Cyrus Tirband Jan 10 '22 at 10:20
  • On a sidenote, in SI units the invariant is $\frac{\mathbf{E}^2}{c^2}-\mathbf{B}^2$. – Cyrus Tirband Jan 10 '22 at 10:23
  • @Physics33: since the Lorentz transform of special relativity is a global transform, but the Coulomb-Lorentz force (and with it the electric and magnetic field) possesses local values for all space, all you can generally hope for is to eliminate the magnetic field for exactly one point in space, but nowhere else. Only under special circumstances can you "transform away" the magnetic field everywhere, and that is the specific scenario you have mentioned in your question. – oliver Jan 10 '22 at 10:23
  • Oh I finally understood. My professor claims that two stationary charges experience different force compared to two moving charges. (Given that charge and distance are not changing) This is the wrong statement right? Magnitude of lorentz force/coulomb force should be invariant right? – Physics33 Jan 10 '22 at 10:28
  • @Physics33 Depends on the context. Omitting relativistic effects (the speed of the charges is much smaller than c), the two forces are equal. If your professor is teaching a class on special relativity, then it is necessary to make the distinction that it is the magnitude of the four-force that is equal, not just the usual space components. – Cyrus Tirband Jan 10 '22 at 10:37
  • @Physics33 Special relativity implies the exact opposite, all inertial frames are equivalent and thus, the magnetic field is just as real as the electric field and you can't do away with using only electricity. Moreover, nobody replaces the concept of mass with energy in any part of physics, mass is a Lorentz invariant whereas energy is very much not, so you obviously can't replace one with the other. Yes, for massive bodies, mass is equal to rest-frame energy. However, not everything is a massive body and by relativity, the rest-frame of an object is no special than any other inertial frame. –  Jan 10 '22 at 11:37
  • Interaction of static charges is a quite an exotic, a particular case. We cannot proceed from it since generally charges are moving relative each other, so magnetic field is as fundamental as electric one. – Vladimir Kalitvianski Jan 10 '22 at 13:42
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Indeed you do not need the Lorentz force to describe the force between stationary charges. For such cases a scalar Coulomb potential suffices in the rest frame of the charges. For all other cases you need tbe Lorentz force. Notably, induction cannot be explained in any frame with only the Coulomb force.

my2cts
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If two stationary charges experience 1N of force, two moving charges should experience 1N of force as well right? (Due to the fact the the moving charges see each other as stationary.)

So can we use only coulomb law to explain every phenomenon in electromagnetism especially magnetism?

If two stationary charges experience 1N of force, two similar co-moving charges will exhibit less than 1N of force.

Due to the fact the the moving charges see each other as stationary, the charges themselves see the force as 1 N.

So we can use only coulomb law to explain this phenomenon in electromagnetism, if we say that same Coulomb force is different in different frames.

(This phenomenon has the old name "Lorentz force")

stuffu
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  • The magnitude of force is not the same? – Physics33 Jan 10 '22 at 12:25
  • @Physics33 Yes. See post #25 here: https://www.physicsforums.com/threads/how-do-relativistic-effects-change-oscillation-of-the-balance-wheel-in-mechanical-watch-causing-it-to-tick-slower.1011025/ – stuffu Jan 10 '22 at 13:10