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The expression $\,\mathbf F_\mathrm G\boldsymbol{=}m\, \mathbf G\,$ and $\,\mathbf F_\mathrm E\boldsymbol{=}q\, \mathbf E\,$ make sense, but is there any derivation for $\,\mathbf F_\mathrm B\boldsymbol{=}q\,\left(\mathbf v\boldsymbol{\times}\mathbf B\right)$?

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All three of the expressions are "derived" from experimental facts. You can in principle try to derive them from Lagrangian dynamics, but that would require you to assume a Lagrangian, which in the first place was written down so that it reproduces the experimentally observed gravitational and Lorentz forces.

Another way in which one could arrive at it, is if one considers quantum mechanics and the principle of gauge invariance as more fundamental. Then one can write down an action for the Schrödinger wave-function and require it to be gauge invariant, which leads to the introduction of a gauge potential, which in turn leads to the Lorentz force. But I have the feeling you're looking for a simpler explanation than that.

At the end of the day, the force on a moving charge in a magnetic field can be traced back to the right hand rule, which is, again, something arrived at from observation.

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Maxwell's equations describe the electromagnetic fields, but they make no statement about force. Although you can find crackpot papers claiming to "derive" the Lorentz force law, it is actually necessary to have a separate equation to relate the fields to the force.

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