How Does the Lorentz Force Fit into Classical Electromagnetism?

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The Lorentz force describes the force exerted on a charged particle moving through electric and magnetic fields, fitting within the framework of classical electromagnetism. It is not a separate force but rather a manifestation of the electromagnetic force, which encompasses both electric and magnetic interactions. The Lorentz force equation, F = q(E + v × B), illustrates how electric fields (E) and magnetic fields (B) influence charged particles (q) in motion (v). Historical naming conventions have led to some confusion, but the Lorentz force is fundamentally part of electromagnetic theory. Understanding this relationship clarifies its role in classical physics.
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I thought in classical physics, only four forces account for all mechanics:

1 gravitational attraction
2 electromagnetic force
3 nuclear strong force
4 nuclear weak force

There is also a Lorentz force on a charged particle moving through a magnetic field. How does classical EM account for this force? It does not fall in one of the above categories. I must be missing something important. What?
 
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If you mean what I think you mean, the Lorentz force falls under the electromagnetic force. They just call it the Lorentz force equation for historical reasons. For reference, this is the equation I'm thinking of:

\vec{\mathbf{F}} = q \left( \vec{\mathbf{E}} + \vec{\mathbf{v}} \times \vec{\mathbf{B}} \right)
 
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