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I've seen similar questions asking about expressing Maxwell's equations with relativistic formulation, but my question is about the physical interpretation it may give for their conventional form.

I'm only familiar with the intuition of special relativity, but explanations I've seen for the origins of the magnetic field describe it as an electric field in a non-stationary reference frame caused by the length contraction of electrons increasing negative charge density. This results in a radial "electric" field in the reference frame of a moving charge that we view as caused by a magnetic field in a stationary frame. How do Maxwell's equations relating the curl of the electric and magnetic fields to the time derivatives of one another mathematically express this physical relationship? Credit to https://www.youtube.com/watch?v=1FE0Z4lov7Y Credit to https://www.youtube.com/watch?v=1FE0Z4lov7Y for image and beautiful basic explanation.

Lambda
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    I'd really recommend not to do that. There are plenty of invariant quantities, and they show that if you started with a purely magnetic field, there is no way to transform it to a pure electric field. – naturallyInconsistent May 22 '23 at 02:21
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    @naturallyInconsistent well $ E^2 -B^2$ is Lorentz invariant ($c=1$), so ofc you can't. – JEB May 22 '23 at 04:04
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    at 11:27, the video is wrong. The moving electrons in the wireframe are not Lorentz contracted because of their motion. That they have separation $d_0$ is stipulated b/c the wire is neutral in its rest frame. This is basically a version of Bell's Spaceship Paradox, that almost every you tuber gets wrong. If the spaceships remain the same distance apart in the 'rest' frame, the string breaks. period. – JEB May 22 '23 at 04:19
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    The fact that electric and magnetic fields have Lorentz transformations that “mix” them does not mean that in any particular frame they are not conceptually distinct fields. Energy and momentum mix under Lorentz transformations, but they aren’t the same thing. The notion that a magnetic field “is an electric field” is simply wrong. – Ghoster May 22 '23 at 04:27
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    You need only look at the Lorentz force law to see that the two fields have completely different effects on charged particles and thus can be operationally distinguished. – Ghoster May 22 '23 at 04:30
  • And it is also very complicated; the field in a current carrying wire is not the same as the superposition of the field of stationary ions and of flowing electrons. – naturallyInconsistent May 22 '23 at 04:31
  • Related : Why don't stationary charge feel force from a current carrying wire? – Frobenius May 22 '23 at 11:58
  • Ah, I should have thought of $E^2-B^2$: this definitely disproves that one field can be completely transformed into the other, but then couldn't the magnetic field still be viewed as a relativistic correction to the electric field, where both still exist but the magnetic field increases at higher velocities? – Lambda May 26 '23 at 23:24
  • As far as I know, it is possible to derive Maxwell's equations and the existence of a magnetic field from electrostatics and special relativity, see for example Rosser classical elelctromagnetism via relativity. But while that is on my reading list, I didn't yet find the time to read it so I'm not sure how well this book is written etc. – Tarik May 31 '23 at 15:07

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