# explanation for lorentz force (current in magnetic field)

by serverxeon
Tags: current, explanation, field, force, lorentz, magnetic
 P: 98 i am looking for a why. not a description of how to calculate or how to find its direction. i came across this question in a school question. "why does the wire move up" and then i thought. is there really a 'why'? or it just happens?
 P: 882 It just happens. You may explain it in terms of more fundamental laws. E.g. by relativistic transformation of electrostatic force. Is the electrostatic force more fundamental than magnetic one? Asking 'why' you come to hen and egg. Those more fundamental laws were formulated such to match empirical results of multiple different experiments, among them the Oersted's one.
 Mentor P: 14,667 What sort of answer would satisfy you? (An intrinsic issue with all "why" questions)
P: 102

## explanation for lorentz force (current in magnetic field)

Richard Feynman on magnets and the question of "Why?":

P: 146
 Quote by Vanadium 50 What sort of answer would satisfy you? (An intrinsic issue with all "why" questions)
I think I understand OP's question, because I've wondered the same thing -- but I'm not sure I can phrase it much better, either.

Maybe the way to put it would be "what is the smallest set of first-principles from which the Lorentz force can be derived?" One reason the Lorentz force doesn't sit well for me comes from the usual pedagogical development of electromagnetism, something like...

1) Introduce Coulomb's law
2) Define electric fields in terms of Coulomb force on 'test charges'
3) Introduce Maxwell Equations for dynamics of these fields
4) Introduce Lorentz force as 'real' Coulomb force

So, it's sort of a circular definition (with Coulomb's law acting as a bootstrap)...though a covariant formulation definitely shows that they're more deeply related.

Also, historically, the Lorentz force took a long time to develop correctly -- if I'm not mistaken, several incorrect forms of it were introduced even AFTER Maxwell's equations...so it would maybe be interesting to see how it was originally derived. I know E. T. Whittaker wrote about this, but it seems to be out-of-print (WARNING: the book IS in the public domain, but only the FIRST PART has been scanned...so all of the "new" copies on Amazon are in all likelihood going to be just that first part -- since these parasites invariably just use scans from archive.org, rather than actually doing the work themselves...I'm convinced some of them don't even know what they're selling)....so -- anyone have any insight on that?
 Mentor P: 10,699 In classical electromagnetism, the Lorentz force law and Maxwell's equations for the E and B fields are generally considered to be first principles, whose consequences are tested and verified by experiment.
P: 617
 Quote by jtbell In classical electromagnetism, the Lorentz force law and Maxwell's equations for the E and B fields are generally considered to be first principles, whose consequences are tested and verified by experiment.
I agree. In the end, first principles exist because everything derived from them matches experiment.
Mentor
P: 14,667
 Quote by jtbell In classical electromagnetism, the Lorentz force law and Maxwell's equations for the E and B fields are generally considered to be first principles,
Indeed, I would say that the Lorentz force law defines E and B.
P: 146
 Quote by Vanadium 50 Indeed, I would say that the Lorentz force law defines E and B.
Exactly -- and that's how it's traditionally taught, but it's also plainly inadequate...we all agree that fields have existence independent of a charged particle to act on via the Lorentz force, right? So while you could provisionally define fields in terms of how they act on charged particles, that says nothing, for example, about how they act in the absence of charged particles.

Of course, you'll say that's what the Maxwell equations are for -- but then we're back to the circular definition.

One example of a 'first principle' derivation might be something contrived like deducing the form that form of interaction equations of a spin 1 field must take, but I'm convinced there are more elegant ones: the covariant formulation is so simple that it's amazing that we don't see these equations arising all over the place in many and various ways...

So, I'm just looking for one of these many possible derivations / explanations that doesn't get at the Lorentz force by just taking it as an axiom or as "empirical" -- particularly since it when all we had was empirical evidence, minds as great as JJ Thompson got it wrong by a factor of two. Empirical first principles aren't things that you get wrong by a factor of two...the correct form was eventually derived theoretically by Heaviside -- so it would be interesting to see, for example, how he did so.
 Mentor P: 10,699 If you're prepared to move beyond classical electrodynamics to quantum electrodynamics, postulating that particle (e.g. electron) fields have local U(1) gauge symmetry requires the existence of a "new" field that has exactly the properties of the electromagnetic (photon) field. See for example http://en.wikipedia.org/wiki/Gauge_t...lectrodynamics The Quantum Physics forum here is probably a better place to pursue this train of thought.
P: 146
 Quote by jtbell If you're prepared to move beyond classical electrodynamics to quantum electrodynamics, postulating that particle (e.g. electron) fields have local U(1) gauge symmetry requires the existence of a "new" field that has exactly the properties of the electromagnetic (photon) field. See for example http://en.wikipedia.org/wiki/Gauge_t...lectrodynamics The Quantum Physics forum here is probably a better place to pursue this train of thought.
That's certainly one way of getting there -- but it seems like it shouldn't be necessary to go beyond classical physics...Heaviside, for instance, clearly didn't base his derivation of the Lorentz force on gauging the Dirac equation.

I do think some of the geometrical insights that gauge theory brings in could provide some direction though -- perhaps if you could somehow strip them of all of the quantum-theoretic trappings, which does seem possible given how purely geometric it is.

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