Electrodynamic vector potential wave equations in free space.

In summary: Gv1sRgEIwfDQIn summary, David Bohm argues that in empty space the choice of divison of a leads to the Rayleigh-Jeans law. However, this is simply Laplace's equation, and there is only one regular solution, which is phi = 0.
  • #1
Peeter
305
3
In David Bohm's "Quantum Theory" (an intro topic building up to the Rayleigh-Jeans law), he states:

"We now show that in empty space the choice div a = 0 also leads to \phi = 0 ...

But since div a = 0, we obtain

[tex]
\nabla^2\phi = 0
[/tex]

This is, however, simply Laplace's equation. It is well known that the only solution of this equation that is regular over all space is \phi = 0. (All other solutions imply the existence of charge at some points in space and, therefore, a failure of Laplace's equation at these points.)"

Now, everything leading up to the Laplacian I understand fine, but I'm not clear on the argument that requires \phi = 0. In particular I'm not sure what regular means in this context.

Bohm is trying to arrive at the wave equation for the vector potential. I'd seen this done differently before by picking the gauge

[tex]
\partial_\mu A^\mu = 0
[/tex]

to arrive at the four wave equations

[tex]
\partial_\mu\partial^\mu A^\nu = J^\nu
[/tex]

and doing so one has no requirement for [itex]A^0 = 0[/itex] for the free space case. Are these two approaches equivalent in some not obvious to me?
 
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  • #2
Peeter said:
Now, everything leading up to the Laplacian I understand fine, but I'm not clear on the argument that requires \phi = 0. In particular I'm not sure what regular means in this context.

It's taken me a few days to understand what your sticking point was, so you may already know the answer.

The only meaning of "regular", as applied to harmonic functions, that I have found, means infinitely differentiable. This doen't seem to fit--exactly.

I think Bohm is saying that no harmonic function exists over all R^3 space (including infinity) that is singularity free. Translated into physics, this means that phi is either constant over all space, or the space contains charge--even if that charge is located at infinity. (Bohm must have been mistaken, and should been speaking of grad(phi)=0, as this is the most general condition that allows space to be charge free.)

You might try Liouville's Boundedness Theorem if you can follow it.
http://mathworld.wolfram.com/LiouvillesBoundednessTheorem.html
 
  • #3
thanks Phrak. I think you are right. It must be grad(phi) = 0 here that he means. That makes two things make more sense. One is the wave equation result he's trying to get. If I do that calculation myself, using div(a) = 0, I get:

[tex]
\begin{align*}
0 &= \nabla^2 \phi \\
0 &= \nabla^2 \mathbf{a} - \partial_{00}{\mathbf{a}} - \partial_0 \nabla \phi \\
\end{align*}
[/tex]

So, with grad(phi) = 0 one gets the vector potential wave equation, and there isn't any requirement to make phi itself equal to zero.

With the correction of phi =0 -> grad(phi) =0, the use of the term regular makes some sense too. In the context of complex numbers , by writing i = e_1 e_2 as a Clifford product and factoring out one of the vectors, grad(phi) = 0 can be used as a compact way of expressing the Cauchy Reimann equations, the condition for a complex number to be differentiable at a point, ie: regular.
 
  • #4
You lost me with the clifford algebra, but good luck.
 
  • #5

1. What is the electrodynamic vector potential in free space?

The electrodynamic vector potential in free space is a mathematical representation of the electric and magnetic fields that are generated by moving charges. It is a vector function that describes the electromagnetic field in terms of both its magnitude and direction.

2. How is the electrodynamic vector potential related to Maxwell's equations?

The electrodynamic vector potential is an important component in Maxwell's equations, which are a set of fundamental equations that describe the behavior of electromagnetic fields. It is used to derive the electric and magnetic fields in free space, and is closely related to the electric and magnetic charges and currents that are present in the system.

3. What are the applications of the electrodynamic vector potential?

The electrodynamic vector potential has many applications in various fields of science and engineering. It is used in the study of electromagnetic fields, such as in the design of antennas, electromagnetic wave propagation, and electromagnetic compatibility. It is also used in the development of technologies such as MRI machines and particle accelerators.

4. How is the electrodynamic vector potential calculated?

The electrodynamic vector potential is calculated using the vector potential wave equations, which are a set of partial differential equations that describe the behavior of the electromagnetic field in free space. These equations take into account the properties of the medium, such as its conductivity and permittivity, as well as the boundary conditions of the system.

5. Can the electrodynamic vector potential be measured experimentally?

While the electrodynamic vector potential cannot be measured directly, its effects can be observed and measured through various experiments. For example, its influence on the electric and magnetic fields can be observed using a device such as a magnetometer. Additionally, its effects can also be indirectly measured through the behavior of charged particles in an electromagnetic field.

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