It is known that the Lienard-Wiechert potentials cannot be derived from the wave equation if a radius R of the classical charge is initially assumed to be equal to zero. In original works of both authors, the radius of the charge is assumed to be finite and after calculation of the potentials, R -> 0 (according to Schott, 'the point laws of Lienard and Wiechert).
But the EM fields are calculated from the potentials under assumption that the charge 'is treated as if concentrated at a point'. In the other words, R = 0.
So my question is: what is a reason that we must assume R = 0 but not R -> 0 after calculation of the fields?
I am concerned in it because the procedure with R -> 0 gives the solution of the wave equation corresponding to so called 'longitudinal EM waves (E_{||} ~ 1/distance).
I add that the existence of such a solution (E_{||} ~ 1/distance) formally doesn't contradict to the Maxwell equation div E = 4\pi\rho because this equation forbids the existence of irrotational component E_{irr} ~ 1/distance. One can see it by solving the Maxwell equations in the gauge which Maxwell itself used, in the Coulomb gauge. The irrotational component isn't identical to the longitudinal component.
The proof of the absence of E_{||} ~ 1/distance follows from Lienard's expression for the EM field (the book of Schott, Sec. 13).
I specify my question.
The L-W potentials are derived for R =/= 0 and at the final stage of computation, R -> 0 , that is correct because the result doesn't depend on the parameter R.
But the EM fields are calculated from the expressions
$$
\phi = \frac{q}{r - (vec{r}\cdot\vec{v})/c}
$$
where r and v are calculated at the retarded instant of time. In this expression, the size of the classical charge is equal to zero (there is no parameter R) in the expression. I am interested why we must calculate the EM fields from this expression but not from the integral representation of the potentials before calculation R-> 0?
The calculation of the EMfield of a classical charged particle can be expressed significantly more simply than appears in traditional presentations. See the formula at the end of http://users.rcn.com/eslowry/elmag.htm . It is more fully explained in Am J of Physics pg 871, 1963 or email me at [email protected] .
We can solve Maxwell's equations with any charge/current, satisfying charge conservation. Mathematically we can use Green function to integrate it with Jmu to get full solution (Jmu can be any distribution, not only ordinary function). The solution, you are talking about with R=0, is just Green function and has some singularities, which is a problem if we want to find field in point charge location. For R=/=0 some non-EM forces must be added to charge equations of motion.
It isn't true. I made calculations of the E field produced by the spherical source. The source of such symmetry should radiate the longitudinal component. 'The spherical source' means a thin charged layer, having a shape of the sphere, expands. In case this layer expands with v = const, div E = 0. If not, div E =/= 0.
Can you specify what isn't true? As far as I understood, in your model you want to get expansion law, taking into account self-action on distributed charge.
In fact, general task can be solved: charge density ro is given as function of "r" for t=0. Find ro(r,t) for t>0. It can be proof, that H=0 for this case. And E(r,t) can be found via ro. Then you can make any limits on ro to get solution for sphere.
No, I didn't consider self-action on the distributed charge. I considered a model of the distributed charge originally proposed by Wesley some tens years ago - it is shortly described in the paper of Monstein and Wesley on detection of the longitudinal EM waves (although their interpretation of the experimental data are very questionable). Wesley stated that the source of the longitudinal waves is the 'charged ball' which is expanding. In the paper of M&W some references are reports on testing the nuclear bombs, that are not published. But in russsian JETP there are two papers of Kompaneetz with similar model of the source (JETP, v 35, p. 1538, 1958).
I calcualted the E field from such a source assuming that distribution of the charges in the layer is perfectly symmetric (Kompaneetz developed the concept that the EM pulse produced by asymmetric component of the charge distribution). Because the charged layer is formed by the Compton electrons, every electron creates the E field which satisfies div E = 0 outside the layer. But the total E field deviates thiss law.
This result seems to be strange but if the layer expands with v = const, the total E field satisfies div E = 0. If the layer expands with v =/= const, div E =/= 0.
I suggest the problem is in the order of calculation of the potentials/fields and going to the limit of the point charge. For example, if one initially treats the charge as the point object, it is impossible to derive the LW potentials. The correct derivation is possible only if the charge is treated as some body having finite sizes, and after calcualtion of the potentials one is able to go to the limit of the point charge.
As I know the derivation of the LW expression with initially point charge is given in the textbooks of Sommerfeld, Electrodynamics, Sec. 29B, and of Landau-Lifshitz, theory of fields, Sec. 63.
Vladimir@
You wrote. "The correct derivation is possible only if the charge is treated as some body having finite sizes, and after calcualtion of the potentials one is able to go to the limit of the point charge.
As I know the derivation of the LW expression with initially point charge is given in the textbooks of Sommerfeld, Electrodynamics, Sec. 29B, and of Landau-Lifshitz, theory of fields, Sec. 63."
Do you mean, that the Landau-Lifschitz derivation of LW potential for a point wise charge is fake? In which place it fails to give the true potentials?
Concerning the longitudinal solufions to the Maxwell equations, they really exist and were known in physics for ages. Even more, there exist so special their solutions, whose velocity is either less or greater of the standard light velocity constant c=300.000 km/sec... So, the problem lies completely in different aspects of the notion "radiation", which is far from being so elementary!
Regards!
To Anatolij
'Do you mean, that the Landau-Lifschitz derivation of LW potential for a point wuse charge is fake? In which place it fails to give the true potentials?'
Actually, Landau-Lifshitz 'derivation' is given for the students.
If one look at their R - the distance, this person can find that after the Lorentz transformations R = r_{observer} - r_{source} - depending on the current (present) time transforms to
R = r_{observer}(t_{peresent} - r_{source}_{retarded}
Do you know how the present time to transform to the retarded time?
I suggest nobody know how to do it because the retarded time is the solution of the transcendental equation.
Landau-Lifshitz transformations look like 'beautiful procedure' having no connection with the L-W potentials.
Dear Vladimir,
your arguing is really not convincing, as the L-L derivation is, in fact, completely equivalent to that given in his manual by Jackson and recdently by Berthold-Georg Englert in his manual Lectures on classical electrodynamics" (World Scientific,, 2014). The same identiacl derivation is also available from the manual by A.M. Fedorchenko (in Russian).
Yet not this is under regard, the problem as such one really lies in the charge singularity for the corresponding field equations and the related infinite internal charge potential energy. The latter is in strong contradiction to the electron stability etc. To the regret, up to date there exists no plausible spatially distributed electron model for which one can construct a safisfactory both quantum and quantum electrodynamics. Moreover, it is assumed ad hoc that electromagnetic field is generated by electric charges in the world, yet up to date there is presented no reasonable derivation of the Maxwell field equations from the basic, for instance, quantum Dirac equation, naturally considered as an electromagnetic field source...
Regards!
Dear Anatolij,
Can you say is the four-vector (63.3) of L-L
A^i=eu^i/(R_k u^k)
Lorentz-covariant?
If not what is the worth of their 'derivation'?
@Vladimir
You wrote: "Do you know how the present time to transform to the retarded time?
I suggest nobody know how to do it because the retarded time is the solution of the transcendental equation."
It is not correct: a not complicated homework allows to show that the retarded time satisfies the d"Alambert equation, thereby defining a Lorentz invariant scalar parameter. Its exact form (exact!) is easily given by means of well known formula for the solution of the corresponding Cauchy data problem for this d"Alambert equation. So, the problem, as it on the whole exists, lies in a completely different place!
So, you can see that there exists no problem with this retarded time! (see attachment)
Sincerely regards!
Dear Anatolij,
It is peculiar the fact that the d'Alembertian is used to define the Harmonic coordinates the set of coordinates recommended by Vladimir Fock (I think the only possible) treating General Relativity in his famous book, which are in agreement with the Minkowskian boundary conditions at infinite distance from the masses . These should manage to take care about gravitational energy issue which haunts GR.
Stefano,
yes, you are right, it is essentially this one which was used by Prof. L. Faddeev in his famous and fundamental article devoted to GR!
Faddeev L D “The energy problem in Einstein’s theory of gravitation (Dedicated to the memory of V. A. Fock)” Sov. Phys. Usp. 25 130–142 (1982); DOI: 10.1070/PU1982v025n03ABEH004517
To Anatolij
You wrote: 'a not complicated homework allows to show that the retarded time satisfies the d"Alambert equation, thereby defining a Lorentz invariant scalar parameter.'
I wish to see the expressions which transform the present time into the retarded time. Well known expressions of Lorentz are written for the 'present times' in both frames. As I understand you are able to derive similar expressions that connect the 'present time' coordinates in one frame with the 'retarded time' coordinates in the other frame.
I think it is impossible to derive such expressions. Simple example:
Assuming a single charge moves in the circular orbit of radius R with constant velocity v around the fixed charge of opposite sign (Bohr's model of the hydrogen atom) in counter-clockwise direction. Let's assume at t = 0 the charge has the coordinates x = R, y = 0.
Where the charge has been located if we measure its E field at a point X, y = 0?
It is typical task to find the EM fields of the rotating charge.
So if you don't know the closed form expression for the retarded time, how can you verify your expressions in the case you obtain them?
And how about my question on Lorentz-covariance of L-L expression (63.3)?
Dear Vladimir,
you did not read carefully:
"Its exact form (exact!) is easily given by means of well known formula for the solution of the corresponding Cauchy data problem for this d"Alambert equation.
Nothing more one can say at present. As you can observe, the retarded time function is nothing else as a special composition of infinitesimal Lorentz invariant mappings.
Regards!
To Anatolij
I have read carefully enough.
You wrote: '"Its exact form (exact!) is easily given by means of well known formula'.
Please, write such a formula allowing to make transformation of the present time variables in one frame to the retarded variables in the other frame.
And how about my second question?
Dear Anatolij,
"yes, you are right, it is essentially this one which was used by Prof. L. Faddeev in his famous and fundamental article devoted to GR!"
I've taken a look at the paper in English and I noticed that the compliance to the conservations in GR, which Fadeev spoke about, is accomplished only for Isolated systems. Hulse and Taylor using the Einstein's field equations gave a consistent result for the gravitational radiation of the Binary pulsars according to their observed reciprocal motion, (under the approximation of isolation). GR with Harmonic coordinates are not still sufficient though to explain the Cavendish experience of approaching gravitational bodies. The system of the two approaching bodies alters its mass/energy configuration and this feature cannot be modeled in GR, we need to go back to the Newtonian's one or find another gravitational theory which also gives a satisfactory model for free falling bodies.
Vladimir,
yet this classical formula, you ask, is available and present in any student manual on mathematical physics! Like Tikhonov-Samarski, V.S. Vladimirov, S.L. Sobolev and in all other ones. It is hard to catch your problem with this.
Regards!
Stefano,
you wrote a very strange sentence: "...which Fadeev spoke about, is accomplished only for Isolated systems" -
The gravity, being theory based on the Lagrangian least action principle, is BY DEFINITION applied only to the conservative and isolated systems, which is our Universe, since in contrast, the theory would have free undetermenined and hiddenn parameters and no prediction could be done on the whole.
Regards!
Dear Anatolij,
yes for sure energy is conserved for isolated systems and GR satisfies such condition. Though GR does not consider gravitational energy and for this it is possible only to treat succesfully orbital motion, photons trajectories and oscillators retardation. The problem of approaching bodies is not treatable.
Anatolij,
Regarding your words, 'yet this classical formula, you ask, is available and present in any student manual on mathematical physics! '
can you write this formula that transforms the four-vector (in Landau-Lifshitz form) from one inertial frame to the other frame?
If the quantity A given by Eq. 63.3 of LL is covariant vector, it can be transformed. How?
Second very simple question: can you compute the E fields from this four-vector (in Landau-Lifshitz form)?
@Stefano Quattrini
The Faddeev's approach is a Hamiltonian one, so all dynamical problems which allow their Hamiltonian description like interaction etc can be treatable. The unique problem which is hiddenn within his approach is the existence of a so called asymptotic Minkowski space-time metric... Yet, an equivalent problem is present also in all other GR models.
Regards!
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