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Suppose we have a magnetic field directed into the computer screen. Assume that there is no gravitational field. Now suppose we have a stationary positively charged particle in the magnetic field. The magnetic force is given by $$\mathbf{F}=q\mathbf{v}\times\mathbf{B}$$

Since $\mathbf{v}=0, \mathbf{F}=0$.

Now pick a reference frame moving to the left with constant velocity. In this frame, the particle is moving to the right with constant velocity. So, the magnetic force is directed upwards.

From what I've learnt in physics, forces are independent of choice of reference frames as long as the frames are inertial. But in the 2 inertial reference frames I have chosen, I get 2 different magnetic forces. How do we reconcile this?

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The resolution to this paradox is that the electric and magnetic fields need to be transformed when you change from one inertial frame to another, through rules fixed by special relativity. This is explained in this Wikipedia page or in your favourite EM textbook's relativity section, but the short of it is that in the frame transformation you describe, the magnetic field is tranformed into an electric field given by $$ \mathbf E'= \gamma \, \mathbf u \times\mathbf B, $$ where $\mathbf u$ is the velocity of the new frame with respect to the old one and $\gamma = 1/\sqrt{1-u^2/c^2}$. This new electric field acts on the particle to fully cancel out the effect of the magnetic force it feels on the new frame - electric and magnetic fields change from one inertial frame to the next, but if the total force is zero in one frame it must be zero in all frames.

It can feel pretty funny that one needs to invoke relativity to deal with what was originally a galilean reference-frame transformation, but that's just the way the cookie crumbles: electromagnetism was a fully relativistic theory from the time Maxwell formulated it, and in fact it was precisely these sorts of symmetries in electromagnetic analyses from different reference frames that motivated Einstein to develop special relativity.

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Observations and conclusions sometimes are different things. First time electromagnetic induction was observed for a wire wound into a coil. A changing magnetic field induces a current in the wire. And the opposite case is an induction process too: A current carrying wire induces a magnetic field. Since the electrons from this current are moving in circles (on a spiral path to be precise) it could be concluded that accelerated charges induce a magnetic field.

Unfortunately the observation of the attraction or repealing of two current carrying straight wires led to the conclusion that any current is responsible for the induction of a magnetic field. This doesn't take in account some phenomena:

  1. The electron possesses a magnetic dipole moment. This is an intrinsic - means a permanently existing- property. Normally the separation of charges doesn't show at the same time the macroscopic property of a magnetic field from the electrons magnetic dipole moments. The magnetic dipole moments are randomly distributed in space. And this - as well as the existence of this magnetic dipole moment - was unknown to Faraday and Ampere.
  2. The induction of a magnetic field around a straight wire has a prehistory. The current has to be switched on. This *accelerates** the electrons. So again it can be concluded that accelerated electrons induce a magnetic field.
  3. Could we conclude that the induced magnetic field is the sum of the aligned magnetic dipole moments from the involved electrons? An indication for this conclusion is the fact that for permanent magnetic materials their magnetic fields is reasoned by the alignment of magnetic dipole moments from the involved electrons (see ferromagnetism).
  4. What is a self inductance? Holding a permanent magnet to a ferromagnetic material could magnetize this material and the alignment of the magnetic dipole moments of the involved electrons is selfholding (latching?). After a current is switched on even a DC current could self induce (latch) a magnetic field as the sum of the aligned electrons magnetic dipole moments.
  5. About how this alignment of the magnetic dipole moments happens see EM induction in a rotating disc with external magnetic field.

So your statement

From what I've learnt in physics, forces are independent of choice of reference frames as long as the frames are inertial.

is right even without any Lorentz contraction or frame transformation.

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    $\begingroup$ This answer is an off-topic rant that has nothing to do with the question as asked. $\endgroup$ – Emilio Pisanty May 4 '17 at 19:49
  • $\begingroup$ @EmilioPisanty From a scientist I'm - a non-trained participant of PSE - awaiting a more foundated criticism. Without my last sentence, what is wrong from the point of your physics education? $\endgroup$ – HolgerFiedler May 4 '17 at 19:55
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    $\begingroup$ None of the claims in this answer is at all relevant to the question as posed. While all known fundamental charges have a magnetic dipole moment, it is perfectly possible to have a system with nonzero charge and zero magnetic dipole moment - just take two electrons with oppositely-aligned spins. The question concerns such a system (nonzero charge, zero magnetic dipole moment), and as such all considerations of magnetic moments are irrelevant. Instead, this answer is a rant by OP trying to push their personal theories, as in multiple other answers on this site. $\endgroup$ – Emilio Pisanty May 4 '17 at 20:00
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    $\begingroup$ @EmilioPisanty Indeed the theory of supraconducting magnets involves Cooper pairs with oppositely-aligned magnetic dipole moments. The question is about magnetic field in different frames and in the question is nothing about paired electrons which you mention. BTW in your answer $\gamma = 1/\sqrt{1-u^2/c^2}$ doesn't compensate the differences in the two frames (or I'm not aware how this coefficient do this). $\endgroup$ – HolgerFiedler May 4 '17 at 20:23

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