# Lorentz Force Question

1. May 8, 2012

### thegreenlaser

Essentially I'm wondering about coupling with Maxwell's equations. It seems that, for application of the Lorentz force equation to make sense, the E and B fields used should not include the E and B fields generated by the charge in question, since a charge won't exert force on itself. However, if I try to couple with Maxwell's equations by modifying my charge density and current density functions to account for a point charge moving in space, then the E and B fields that appear in Maxwell's equations do include the ones generated by the charge, so it would seem to me that using those same E and B fields in the Lorentz equation to find the force on that charge wouldn't work (and if I'm not mistaken, the fields would approach nonsense values at the exact location of the charge). Am I thinking correctly? If so, how exactly would one couple Maxwell's equations with the Lorentz force equation? (Maybe subtract the field generated by the charge in question in the Lorentz equation?) If not, where am I going wrong?

2. May 8, 2012

### Hassan2

If own field does not exert force on the particle, it implies that we can accelerate a particle without spending energy.

Also a moving charge radiates energy. Assume a moving charge in ZERO applied field. If the charge's own field doesn't stop it, it keeps radiating the same flow of energy energy forever which is against law of conservation of energy .

3. May 8, 2012

4. May 8, 2012

### TrickyDicky

How about using the Maxwell stress tensor, it looks like the obvious tool to relate Maxwell equations and the Lorentz force. It is a rank 2 3X3 symmetric tensor, that has the peculiarity of being diagonal (divergence-free) with xx,yy,zz components similar to the pressure components of the regular stress tensor and vanishing shear components like a perfect fluid precisely due to the properties derived from the Maxwell eq. (one can always rotate coordinates so that any component is normal).
Being a symmetric tensor I think the issues raised in the OP either don't arise or are automatically solved.
This tensor is actually the 3X3 ij components part of the EM stress-energy tensor that appears in the electrovacuum solutions of the EFE. This trace-free tensor in its null version refers to radiation, with pressure components 1/3 of the energy density component. A non null solution would correspond to charged bodies I think (wouldn't be trace-free in this case).

Last edited: May 8, 2012
5. May 8, 2012

### Hassan2

Thanks for correcting me. Actually I'm not knowledgeable in this field and should not have made a comment. So my the second argument is not valid. But I remember I read something about the particle's own field retarding the particle's acceleration. Do you confirm that?

6. May 9, 2012

### PhilDSP

In at least one version/variation of the original Maxwell equations (the ones actually written by J. C. Maxwell) the Lorentz Force terms are embedded in the "Faraday's Law" equation triplets. I believe it's in one section of the first volume of the 3rd edition of his treatise (but probably in every edition).

That term is necessary to fully account for the EMF in one aspect of Faraday's experiments. You can find good detail in Wikipedia on that. The reason Maxwell generally dropped that term in other versions of the equations might be related to the ambiguity of the coupling you mention and difficulty for students to unravel the situation. And also because he may have realized that a complete theory for moving charges was a bit beyond what could be summoned at the time apparently. Unfortunately, he left no clear instructions on the matter that I could find.

PS. This thread indicates some of the details and issues with that: