Maxwell Equations, Lorentz Force and Coulumb Force

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latot
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Here the 3 set of equations we know, the Maxwell Equations, Lorentz Force, and Coulumb Force, actually I doubt a lot what set of equations represent all the electromagnetic aspects, I try research over the internet and I found a lot of contradictions in the answers, someone says we can get the Lorentz Force from Maxwell and others not...

In the end, I don't know the right answer, so, what set of equations can represent all the proporties of magnetic and electric things (yes, things, electro-magnetic, electros, protons, waves, etc, etc)?

What I can see in Maxwell equations, they are a nice way to set the relations betwen the magnetic and electric field, but isn't too intuitive if there in the middle of it, the Lorent and Coulmb forces are already applied.

Thx.
 

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PeroK
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Maxwell's equations describe the relationship between charges and the electric and magnetic fields:

Charges define the electric and magnetic fields.

The Lorentz force (of which Coulomb's law is a special case) describes how charges move in the presence of electric and magnetic fields:

Electric and magnetic fields tell charges how to move.

Logically, you need both.

It's an interesting question, perhaps, of how little you need to assume about the Lorentz force in order to derive the precise format from Maxwell's equations. For example:

https://arxiv.org/abs/physics/0206022
 
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vanhees71
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You can, however, derive the Lorentz force ##\vec{F}=q(\vec{E}+\vec{v}/c \times \vec{B})## (for a point particle) from the Maxwell equations and energy-momentum conservation. You only need the energy-momentum tensor of the electromagnetic field.
 
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