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Lienard Wiechert potential of a point charge ?

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#1
Jan22-09, 03:43 PM
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Good day to everybody,

I got stuck at certain (basic) question regarding Lienard Wiechert (LW) potential of a point charge:

In Heitler's great book "Quantum Theory of Radiation" (I've used the edition from 1954), on page 18, there is a familiar statement that:
[tex]\frac{e}{r}|_{t-r/c}[/tex] is not a valid solution of the retarded potential integral. Heitler justification why this result is wrong is somewhat unclear to me: he states that if "the integral [tex]\int \rho(P',t')dr'[/tex] would not represent the total charge". Does anybody has any idea how the explicit mathematical calculation works in this case or where it could be found ?

I consulted few books (e.g. Jackson, Feynman II, Chapter 20) but only found a derivation of LW potentials (but Feynman nevertheless states, that the above given simple equation is worng, but he also does not give any mathematical justification"

thank you for your kind help
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clem
#2
Jan23-09, 08:58 AM
Sci Advisor
P: 1,261
The e/r equation is wrong because it cannot be derived from more basic equations, and it does not equal the L-W potential which is derived.
Hans de Vries
#3
Jan23-09, 10:11 AM
Sci Advisor
P: 1,135
Quote Quote by snapback View Post
In Heitler's great book "Quantum Theory of Radiation" (I've used the edition from 1954), on page 18, there is a familiar statement that:
[tex]\frac{e}{r}|_{t-r/c}[/tex] is not a valid solution of the retarded potential integral. Heitler justification why this result is wrong is somewhat unclear to me: he states that if "the integral [tex]\int \rho(P',t')dr'[/tex] would not represent the total charge". Does anybody has any idea how the explicit mathematical calculation works in this case or where it could be found ?
There is an additional effect, similar to a shockwave. The potential is "compressed"
in front of the moving charge and stretched behind it. It's all explained in detail in
chapter 2 of my book.


http://www.physics-quest.org/Book_Ch...rentzContr.pdf


Regards, Hans

snapback
#4
Jan23-09, 02:58 PM
P: n/a
Lienard Wiechert potential of a point charge ?

glad to see your answers !

I would rather consider the e/r equation as "guessed", but guessing does not automatically means an answer being wrong, since, for example, you can solve integrals by "guessing" (even it is not the usual way of calculating stuff). But when you guess a solution, you of course need to verify, that it is really a solution by doing the calculation "backwards" (e.g. by differentiating the integral).

Coming back to my previous question: do you know where such"backward" calculation which clearly shows that the e/r-solution does not yield the correct charge density can be found ?

Thanks
Hans de Vries
#5
Jan23-09, 04:58 PM
Sci Advisor
P: 1,135
Quote Quote by snapback View Post
glad to see your answers !

I would rather consider the e/r equation as "guessed",
Why consider the wrong answer?? It violates special relativity.
There would be no Lorentz contraction if true.

Quote Quote by snapback View Post
but guessing does not automatically means an answer being wrong, since, for example, you can solve integrals by "guessing" (even it is not the usual way of calculating stuff). But when you guess a solution, you of course need to verify, that it is really a solution by doing the calculation "backwards" (e.g. by differentiating the integral).

Coming back to my previous question: do you know where such"backward" calculation which clearly shows that the e/r-solution does not yield the correct charge density can be found ?

Thanks

That would be a (very) complicated and not straightforward calculation. If you perfectly
understand all the calculations done in the Lienard-Wiechert theory then you might start
to think about doing things like this.

The answer would be a moving point charge density distribution which, when at rest, would
have a non spherical potential field: An ellipsoid.

A point charge which would do this contains an infinite series of higher order dipole moments,
described by an infinite series of higher order differentials of Dirac impulse functions.

You don't get something like that rolling out straightforwardly from taking the d'Alembertian.
Taking derivatives from a retarded potential is trickier then taking the derivatives from a
normal function. Try to understand section 2.10 where the electrical field is derived from
the potential field.

http://www.physics-quest.org/Book_Ch...rentzContr.pdf


Regards, Hans
snapback
#6
Jan24-09, 04:01 AM
P: n/a
Quote Quote by Hans de Vries View Post
Why consider the wrong answer?? It violates special relativity.
There would be no Lorentz contraction if true.
First of all I would like to cite N. Bohr in a fairly sloppy manner:
When it comes to atoms, language can be used only as in poetry.
what I'm searching for is not some sentences in words, that everybody might understand in a different way, NO!, I precisely want to see that thing which you described as

Quote Quote by Hans de Vries View Post
an infinite series of higher order dipole moments,
described by an infinite series of higher order differentials of Dirac impulse functions.
,

so that the
Quote Quote by Hans de Vries View Post
would
will turn into is.

I simply want to see the technique applied, regardless whether the solution is right or wrong and wright. So I would equally be glad to see that the "proper" solutions namely eqn. (2.4) and (2.5) in your chapter 2 really lead to the proper charge and current density (kind of generalized procedure of taking d'Alembertian)

Anyway, I will try to follow the hints in section 2.10 with the "retarded time differentiation" + chain rule, this looks interesting.

have a nice saturday


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