Lorentz Force: Coupling w/Maxwell's Eqs

In summary, the conversation discusses the issue of coupling Maxwell's equations with the Lorentz force equation. It is mentioned that the E and B fields used in the Lorentz force equation should not include the fields generated by the charge in question, but when trying to couple with Maxwell's equations, these fields are included. It is questioned whether using these fields in the Lorentz equation to find the force on the charge would work, and suggestions are made such as using the Maxwell stress tensor. There is also a mention of a term in the original Maxwell equations that accounts for the Lorentz force, but it is not included in later versions. Overall, the conversation highlights the complexities and uncertainties surrounding the coupling of these equations.
  • #1
thegreenlaser
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16
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?
 
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  • #2
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
Hassan2 said:
Also a moving charge radiates energy.

An accelerating charge radiates energy.
 
  • #4
thegreenlaser said:
... 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?

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).
 
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  • #5
jtbell said:
An accelerating charge radiates energy.

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
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:
https://www.physicsforums.com/showthread.php?t=252215
 
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FAQ: Lorentz Force: Coupling w/Maxwell's Eqs

1. What is the Lorentz force?

The Lorentz force is the force that is exerted on a charged particle when it moves through an electromagnetic field. It is a combination of the electric force, which is caused by the electric field, and the magnetic force, which is caused by the magnetic field.

2. How is the Lorentz force related to Maxwell's equations?

The Lorentz force is directly related to Maxwell's equations, which are a set of four equations that describe the behavior of electric and magnetic fields. The third equation, also known as the Ampere-Maxwell law, describes the relationship between the magnetic field and the current density. When combined with the first and second equations, which describe the electric field, the Lorentz force can be derived.

3. Can you give an example of the Lorentz force in action?

One example of the Lorentz force in action is a charged particle moving through a magnetic field. The particle will experience a force perpendicular to its velocity, causing it to move in a circular path. This is the principle behind particle accelerators, such as the Large Hadron Collider, which use magnetic fields to accelerate charged particles to high speeds.

4. How is the Lorentz force used in practical applications?

The Lorentz force has many practical applications, such as in electric motors and generators. In these devices, the Lorentz force is used to convert electrical energy into mechanical energy. It is also used in particle accelerators, as mentioned before, and in medical equipment such as MRI machines, which use magnetic fields to produce images of the body.

5. Is the Lorentz force always present in electromagnetic interactions?

The Lorentz force is always present whenever a charged particle moves through an electromagnetic field. However, its effects may be negligible in some cases, such as when the particle's velocity is very low or when the magnetic field is very weak. In these cases, the electric force may dominate and the Lorentz force may not be noticeable.

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