Redundant cross product removed from Maxwell equation?

In summary, Maxwell's equation specifies the value of the electric field intensity when the particle is moving in a magnetic field. If the particle is not moving in a magnetic field, the equation is the same as the modern equation for the E field. Maxwell's equation is a mystery and further study is warranted.
  • #1
PhilDSP
643
15
I'm trying to track down the rationale for removing the cross product of velocity and magnetic field intensity from Maxwell's equation which specifies the value of the electric field intensity. In the third edition of "A Treatise on Electricity and Magnetism" Maxwell specifies (in modern terminology) that E = cross product of velocity and B minus the derivative with respect to time of the vector potential minus the gradiant of the scalar potential.

Was the assumption that the potential already contains the changing value of the magnetic field so that the cross product is redundant? It seems that Maxwell was second-guessed when Gibbs and Heaviside developed the modern variant of the equations.
 
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  • #2
The equation intended by Maxwell is: [itex] E= v [/itex] x [itex] B - dA/dt - \nabla \phi [/itex]
 
  • #3
I think you're referring to the Lorentz force equation, where q=1C:
http://en.wikipedia.org/wiki/Lorentz_force

If Wikipedia is to be believed, this paragraph addresses your concerns:
[PLAIN]http://en.wikipedia.org/wiki/Lorentz_force said:
Although[/PLAIN] [Broken] this equation is obviously a direct precursor of the modern Lorentz force equation, it actually differs in two respects:

It does not contain a factor of q, the charge. Maxwell didn't use the concept of charge. The definition of E used here by Maxwell is unclear. He uses the term electromotive force. He operated from Faraday's electro-tonic state A,[6] which he considered to be a momentum in his vortex sea. The closest term that we can trace to electric charge in Maxwell's papers is the density of free electricity, which appears to refer to the density of the aethereal medium of his molecular vortices and that gives rise to the momentum A. Maxwell believed that A was a fundamental quantity from which electromotive force can be derived.[7]
 
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  • #4
Yes, that's exactly what I was referring to. Thanks for the reference.

Maxwell says that the [itex] v [/itex] x [itex] B [/itex] term is only needed when the particle is moving in a magnetic field. Otherwise the equation is the same as the modern equation for the E field defined with respect to the potentials I believe. Seems to be a mystery how to interpret what Maxwell was thinking and further study seems warranted.
 
  • #5
This is a good reason why I don't bother reading original papers. Apart from the fact that their notation is outdated, the authors are often wedded to the the prevailing paradigms of their time, which may end up confusing the reader.
 

1. What is the redundant cross product in Maxwell's equations?

The redundant cross product refers to a mathematical term that appears in Maxwell's equations, specifically in the Ampere's law equation. It involves the cross product of the electric field and magnetic field, which can be simplified using vector identities to eliminate this term.

2. Why was the redundant cross product removed from Maxwell's equations?

The redundant cross product was removed from Maxwell's equations in order to simplify and streamline the equations. It was found that it did not add any new information or physical insight, and could be eliminated without affecting the accuracy or validity of the equations.

3. How does removing the redundant cross product affect the interpretation of Maxwell's equations?

Removing the redundant cross product does not affect the interpretation of Maxwell's equations. The equations still describe the fundamental relationship between electric and magnetic fields, and their interactions with each other and with charges and currents.

4. Are there any cases where the redundant cross product is still used?

In most cases, the redundant cross product is not used in modern formulations of Maxwell's equations. However, there are some specialized cases where it may still be used, such as in certain theoretical or mathematical analyses.

5. How does removing the redundant cross product impact the practical applications of Maxwell's equations?

The removal of the redundant cross product has no impact on the practical applications of Maxwell's equations. The equations are still used in a wide range of fields, including electromagnetism, electrical engineering, and communications, with the same level of accuracy and usefulness as before.

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