Electrodynamics: questions about vector and scalar potentials....

[Aleberto69]

and Lorentz Gauge.

Manipulating Maxwell equations and introducing $\vec A$ vector and $Φ$ scalar potentials the following equations are obtained:

$\nabla^2 \vec A+k^2 \vec A=-μ\vec J+\nabla(\nabla⋅\vec A+jωεμΦ) ~~~~~~~~~~(1)$

$\nabla^2 Φ+k^2 Φ=- \frac ρ ε -jω(\nabla⋅\vec A+jωεμΦ) ~~~~~~~~~~(2)$

The Lorenz Gauge consists of using the degree fo freedom on the choice of $\vec A$, $Φ$ and hence look for solutions with the constrain of

$\nabla⋅\vec A+jωεμΦ=0~~~~~~~~~(3)$

which results in the two independent equations

$\nabla^2 \vec A+k^2 \vec A=-μ\vec J ~~~~~~~~~~(1a)$

$\nabla^2 Φ+k^2 Φ=- \frac ρ ε ~~~~~~~~~~(2a)$

It is then usual practice addressing electrodynamics probelms (especially antenna radiation problems) just solving 1a and 2a.
Often just using 1a, like for calculating the radiation from an infinitesimal electric dipole.
Furthermore equation 3 is no more included in the constrain to which $\vec A$ and $Φ$ need to comply.

The argument ( J.D Jackson " Classical Electrodynamics" 3rd Ed at page240 and others) is the following:
Let assume to have found a solution to 1 and 2 being the couple

$\vec A_1$ and $Φ_1$

but that

$\nabla⋅\vec A_1+jωεμΦ_1=χ≠0$

Furthermore it is easy to prove that any couple

$\vec A_2=\vec A_1+\nabla Λ$ and $Φ_2=Φ_1-jωΛ$

is still providing the same electromagnetic fields and hence is solution of 1 and 2 as well.
If furthermore $Λ$ is found satisfying the equation

$\nabla^2 Λ+k^2 Λ= χ ~~~~~~~~~(4)$

then $\vec A_2$ and $Φ_2$ are solution of 1, 2 and 3 and hence of 1a and 2a too.
The above proof seems resonable to the authors (I have read) for saying that "finding solution to 1a and 2a is finding valid potentials from which calculating the solution for the fields".

Eventually find below, in bullet points, my questions/argument against the above proof.
I would be greatful if somebody could clarify and better explain.

1. The above argument prove that having a solution to 1 and 2 we can always find other solution to them ($\vec A_2$ and $Φ_2$) which satisfy 1,2 3 and hence 1a, 2a. However it is my conviction that this is a correct proof if and only if we can demonstrate that 1a, 2a have only one solution (which therefore needs to coincide with $\vec A_2$ and $Φ_2$). If otherwise 1a and 2a have more solutions, then there is no proof that all of them are solution to 1, 2 and 3 too. In other words, finding one of all the possible solutions to 1a and 2a, it is not obvious that we will end up to $\vec A_2$ and $Φ_2$.
2. It is then also not obvious that finding $Λ$ solution of 4 is always possible. Jackson himself says "provided that a solution to 4 is founded..."

In the same context I have the the following other unsolved questions:

1. Equation 1 and 2 are obtained in the hipothesys of homogeneous medium ( ε,μ constant). Jakson himself says " At this stage it is convenient to restrict our consideration to the vacuum form of Maxwell equations". Is it therefore legittimate that the result obtained solving them can be used for addressing problems where the homogeneity is not verified ( i.e. radiation from a complex metallic structure/antenna)?

For example, using 1 and 2 ( really using 1a and 2a) many authors calculate the radiation of the elementary electric dipole (Green function for the elementary electric dipole) and for proceeding with the solution, they take advantage of the cilindrical symmetry obtaining a function that has a cylindrical dependency.
Furthermore the authors use the solution of the elementary dielettric dipole (together with contur conditions at the surfece of perfect conductors) for obtaining the integral equation for caluculating unknonw currents density addressing the case of more complex metallic (PEC) antennas.
It is my opinion that the correctness of the approch is not prooved for two reasons:

1. Each elementary electric current dipole is in the presence of the conductor (the whole structure) and therefore the hypothesy of homogeneity (that legitimated the calculations for obtaining the formulas for the potentials) is not verified
2. Each elementary electric current dipole is in presence of the conductor (the whole structure) and therefore the hypothesy of cylindrical symmetry which leaded to a cylindrical symmetry for the elementary contribution to the potentials is not verified as well.

I have a possible argument in feavour of the correctenss of the approach for the case of perfect coductors. being based on the equivalence theorem. Using the equivalence theorem we can immagine an equivalent solution where the unknown current density are placed on the geometrical surface that was limiting the conductors. If those current density are supposed radiating in the absence of the conductor but also resulting in a total null field in the volume where the conductors were, then the solution found is also solution of the problem with the conductors returned to their position.
I'm not sure that this argument is valid, and anyway no authors that I read is providing this clarification.
I would be greatful if somebody could clarify and better explain.

Thanks in advance to anyone who would like spending time and sharing arguments around the matter.

Cheers

Aleberto69

 

[Aleberto69]

Hi All,
I'm sad that nobody answered the thread and honeslty I do not know if it was becose the question it is not clear, but I didn't receive a comment asking clarification...
However the mai point is that I do not find satisfactory the demenstration about Lorenz Gouge on EM potentials that many author support.
I still hope somebody would like to help me convincing that the demonstration is righ ..
Thanks anyway
Aleberto69