Lorentz contrast/EM wave question.

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Discussion Overview

The discussion revolves around the concept of the "Lorentz contrast" in electromagnetic theory, specifically focusing on the Lorenz condition related to the scalar and vector potentials. Participants explore the implications of making certain terms in the wave equation equal to zero and the significance of gauge choices in the context of electromagnetic wave propagation.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant mentions deriving a non-homogenous wave equation and questions the meaning and utility of making a specific term equal to zero.
  • Another participant clarifies that the term in question is related to the Lorenz condition, suggesting that its elimination leads to a simpler interpretation where the potentials satisfy the ordinary wave equation.
  • A subsequent reply posits that making the term disappear is necessary for the potentials to propagate at the speed of light, along with the electric and magnetic fields.
  • In contrast, another participant argues that while making the term disappear is convenient, it is not strictly necessary, emphasizing the non-uniqueness of the scalar and vector potentials and the role of gauge choices.
  • This participant explains that the Lorenz condition decouples the potentials into wave equations, while the Coulomb gauge restricts the divergence of the vector potential but leads to an instantaneous scalar potential.
  • There is a correction regarding the terminology, clarifying that it is "Lorenz" and not "Lorentz," with references to common confusions in literature.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of eliminating the term in the wave equation, with some arguing for its importance while others suggest it is a matter of convenience. The discussion remains unresolved regarding the implications of gauge choices and the historical attribution of the Lorenz condition.

Contextual Notes

Participants note the non-uniqueness of potentials and the implications of different gauge conditions, highlighting that the discussion may depend on specific definitions and interpretations of gauge invariance.

Lavabug
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In my EM course I've encountered what is called the "Lorentz contrast". If I derive the wave equation using the scalar and vector potential, I end up with a non-homogenous wave equation with the term:

d0354dcaff89f84df81a4002c8dde6f9.png

(left hand side), or more precisely, the divergence of said term.

What does it mean and why is making it equal to zero important/useful/possible?
 
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You mean Lorentz condition, right? It's both important and useful to make this term disappear because what you're left with is easy to interpret: A and φ both satisfy the ordinary wave equation, which describes something that can propagate at the speed of light.
 
So making it disappear is necessary because a change in potential also needs to "propagate" at the speed of light, along with E and B?
 
It's not necessary, it's a convenient solution to an overriding problem. The real problem is that the scalar and vector potentials are non-unique. We can apply an arbitrary transformation to them and still result in the same electric and magnetic fields. The reason why the Lorenz condition is used is that it decouples the scalar and vector potentials. The new decoupled potential equations are wave equations. However, we can still achieve uniqueness by using a different gauge. The Couloumb gauge only restricts that the divergence of the vector potential be zero. The difference with the Coulomb gauge is that the scalar potential is now the instantaneous Coulombic potential. On the other hand, the Lorenz gauge requires that the potentials be retarded potentials since they satisfy a wave equation.

And it's Lorenz, not Lorentz. People mix that one up A LOT. In fact, looking at Jackson's text shows that the section title uses "Lorenz" but the section title at the page header is "Lorentz." *SIGH*
I also see that Wikipedia mixes the two even on the same page.

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5672647&tag=1
 
Why this nice gauge condition should be attributed to the Danish physicst Ludvig Lorenz and not the Dutch physicicsts Hendrik Antoon Lorentz you can read in

Jackson, J.D., and Okun, L.B.: Historical roots of gauge invariance, Rev. Mod. Phys. 73, 663 (2001)
 

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