Lorentz contrast/EM wave question.

[Lavabug]

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:

(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?

 

[Bill_K]

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.

 

[Lavabug]

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?

 

[Born2bwire]

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

 

[vanhees71]

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)

 

[jtbell]

The proofreader's nightmare: the Lorentz-Lorenz equation!

(or is that Lorenz-Lorentz?)