Why are Maxwell's Equations a complete description of EM?

[LucasGB]

It is a well-known fact that Maxwell's Equations, along with Lorentz's Force Law, form a complete description of classical electromagnetism. But why is that? I mean, I can understand that Lorentz's Law is necessary for describing the interaction between matter and electromagnetic fields, and I also understand the meaning of the other four equations but could you please explain to me why it is that they say everything which needs to be said about electromagnetism? Is it because the four equations, using the concepts of divergence and curl, completely describe the spatial configurations of the electric and magnetic vector fields, and how they relate to charges and to each other? How do physicist know they're complete?

 

[Nabeshin]

How do physicist know they're complete?

In short, because they describe all electromagnetic phenomenon we observe. The Maxwell's equations were constructed by adding on terms and equations to explain observed effects. This was Maxwell's chief contribution, actually. One could suppose that the second derivative of the electric field with respect to time equals the double curl of the B-field, or some other such nonsense, but such an equation would not be in accord with our experiences.

 

[Dale]

They are not complete. They do not describe the photoelectric effect.

 

[Andy Resnick]

It is a well-known fact that Maxwell's Equations, along with Lorentz's Force Law, form a complete description of classical electromagnetism. But why is that? I mean, I can understand that Lorentz's Law is necessary for describing the interaction between matter and electromagnetic fields, and I also understand the meaning of the other four equations but could you please explain to me why it is that they say everything which needs to be said about electromagnetism? Is it because the four equations, using the concepts of divergence and curl, completely describe the spatial configurations of the electric and magnetic vector fields, and how they relate to charges and to each other? How do physicist know they're complete?

If I instead re-state your first sentence as "Maxwell's equations completely specify the electromagnetic field in a vacuum", then the explanation is simple- because each two equations completely specify a vector field:

http://en.wikipedia.org/wiki/Helmholtz_decomposition

However, you are also correct that additional equations are needed to explain the interaction of the electromagnetic field with matter. One of these is the Lorentz force equation, but others exist as well- constitutive equations relating (for example) the E and D fields.

 

[D H]

They are not complete. They do not describe the photoelectric effect.

The photoelectric effect is not classical. Re-read the original post (emphasis mine):

It is a well-known fact that Maxwell's Equations, along with Lorentz's Force Law, form a complete description of classical electromagnetism.

Maxwell's equations collectively describe how electromagnetic waves are generated, propagate through space, and interact with matter. They are complete in the sense that Maxwell's equations describes all of these phenomena, but only from a macroscopic point of view. They are not complete when one delves into the microscopic world (e.g., the photoelectric effect).

 

[mgb_phys]

It is a well-known fact that Maxwell's Equations, along with Lorentz's Force Law, form a complete description of classical electromagnetism.

Isn't that a rather circular argument?

Maxwell's Equations perfectly describe classical electromagnetism.
Classical electromagnetism being defined as that which follows Maxwell's Equations!

 

[LucasGB]

In short, because they describe all electromagnetic phenomenon we observe.

That's interesting, I can see that.

If I instead re-state your first sentence as "Maxwell's equations completely specify the electromagnetic field in a vacuum", then the explanation is simple- because each two equations completely specify a vector field:

http://en.wikipedia.org/wiki/Helmholtz_decomposition

However, you are also correct that additional equations are needed to explain the interaction of the electromagnetic field with matter. One of these is the Lorentz force equation, but others exist as well- constitutive equations relating (for example) the E and D fields.

Very interesting, but I have a few questions here:

1. Why did you have to specify that they were in a vacuum?
2. I apologize for my poor understanding of this, but from what I understand, Helmholtz Decomposition states that any smooth vector field can be resolved into the sum of a curl-free and a divergence-free vector field. But Maxwell's Equations predict the electric field is not divergence-free everywhere, and the electric and magnetic field are not curl-free under all circumstances.
3. Could you please list the other equations needed to build a complete picture of the interaction between the electromagnetic field and matter?

Thank you all for your patience!

 

[GRDixon]

It is a well-known fact that Maxwell's Equations, along with Lorentz's Force Law, form a complete description of classical electromagnetism.

They are not complete. As Richard Feynman points out in "The Feynman Lectures on Physics", V2, Sect. 28-1, "But we want to stop for a moment to show you that this tremendous edifice, which is such a beautiful success in explaining so many phenomena, ultimately falls on its face." Feynman is referring to the dual facts that (1) U, the total energy in the electrostatic field of a resting spherical shell of charge is a constant times 1/8, whereas electromagnetic mass times c^2 (i.e., mc^2) is the same constant times 1/6. In brief, U=(3/4)mc^2. As Feynman points out, "This formula was discovered before relativity, and when Einstein and others began to realize that it must always be that U=mc^2, there was great confusion."
Poincare suggested that the solution to this conundrum lies in the fact that there are stresses in the spherical shell distribution, and such stresses have an energy density that is not mentioned in Maxwell's theory. You can read all about it in Chapter 28 of Feynman's Lectures, V2.

 

[Andy Resnick]

<snip>
Very interesting, but I have a few questions here:

1. Why did you have to specify that they were in a vacuum?
2. I apologize for my poor understanding of this, but from what I understand, Helmholtz Decomposition states that any smooth vector field can be resolved into the sum of a curl-free and a divergence-free vector field. But Maxwell's Equations predict the electric field is not divergence-free everywhere, and the electric and magnetic field are not curl-free under all circumstances.
3. Could you please list the other equations needed to build a complete picture of the interaction between the electromagnetic field and matter?

Thank you all for your patience!

Here's my answers:

1) Because E and H are the fields in matter-free (source-free) space. The fields in matter, D and B, can only be related back to E and H by constitutive relations.

2) Both of those sentences are true. See, for example, the section "fields with prescribed divergence and curl" on that wiki page.

3) It's still an open research question. I like E. J. Post's "Formal structure of electromagnetics" (Dover) as a reference for questions like this. Landau and Lifgarbagez vol. 8, Penfield and Haus, "Electrodynamics of moving Media" and Truesdell's "Classical Field Theories" (Handbuch of Physics, vol III/I) also have good information.