Lienard Wiechert potential of a point charge ?

In summary, there is confusion about the validity of the e/r equation as a solution for the Lienard-Wiechert potential of a point charge. It is considered "guessed" and cannot be derived from more basic equations. The correct solution involves an infinite series of higher order dipole moments and differentials of Dirac impulse functions. The technique for obtaining this solution is not straightforward and involves taking derivatives from a retarded potential. Further explanation can be found in Chapter 2 of a book by Hans de Vries.
  • #1
snapback
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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  • #2
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.
 
  • #3
snapback said:
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_Chapter_EM_LorentzContr.pdf


Regards, Hans
 
  • #4
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
 
  • #5
snapback said:
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.

snapback said:
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_Chapter_EM_LorentzContr.pdfRegards, Hans
 
  • #6
Hans de Vries said:
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

Hans de Vries said:
an infinite series of higher order dipole moments,
described by an infinite series of higher order differentials of Dirac impulse functions.
,

so that the
Hans de Vries said:
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
 

Related to Lienard Wiechert potential of a point charge ?

1. What is the Lienard Wiechert potential of a point charge?

The Lienard Wiechert potential of a point charge is a mathematical representation of the electromagnetic field produced by a moving point charge. It takes into account both the electric and magnetic components of the field and is used to calculate the force and energy associated with the moving charge.

2. How is the Lienard Wiechert potential derived?

The Lienard Wiechert potential is derived from Maxwell's equations, which describe the fundamental laws of electromagnetism. It involves solving differential equations and integrating over space and time to obtain the potential at any given point in space.

3. What is the significance of the Lienard Wiechert potential?

The Lienard Wiechert potential is significant because it allows us to understand and predict the behavior of electromagnetic fields produced by moving charges. It is also used in various applications, such as in antenna design and in the development of electromagnetic wave theories.

4. How does the Lienard Wiechert potential differ from the Coulomb potential?

The Lienard Wiechert potential takes into account the motion of the point charge, whereas the Coulomb potential only considers the charge at rest. This means that the Lienard Wiechert potential can describe effects such as time-varying fields and the emission of electromagnetic radiation, which the Coulomb potential cannot.

5. Can the Lienard Wiechert potential be applied to systems with multiple charges?

Yes, the Lienard Wiechert potential can be extended to systems with multiple charges by summing the potentials of each individual charge. However, this can become very complex and is often simplified by using more advanced mathematical techniques or numerical simulations.

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