# Unserstanding Larmor’s Equation?

1. May 17, 2010

### mysearch

On the basis that Larmor’s formula specifies the energy/power radiated from an accelerated charge, provided v<<c, I was assuming that this equation must also characterised the energy power transferred to the EM waves/photons emitted by the accelerated charge. I know Larmor’s work predates the idea of a photon and quantised energy, but looking at the derivation of Larmor’s formula raised some questions, which I was hoping somebody might be able to clarify.

First, the derivation of Larmor’s formula I have been studying exists within section 4.2/page 31 of Daniel Schroeder’s pdf document, which can be access via the following link: http://physics.weber.edu/schroeder/mrr/MRRnotes.pdf . While I am not totally sure whether this approach is generally accepted, it seems to be based on the idea that an accelerating charge causes a ‘kink’ in the electric field lines, such that the net electric field (E) has both a radial (Er) and transverse (Et) component. The strength of these components is given on page 33 as follows:

[1] $$Er=\frac{q}{ 4 \pi \epsilon_0 R^2}$$

[2] $$Et=\frac{qa*sin \theta}{ 4 \pi \epsilon_0 c^2R}$$

These equations show different proportionality with respect to the radius [R] such that (Et) starts to dominate as [R] increases. So my first question is whether these equations are reflective of the idea of ‘near’ and ‘far’ field definitions?

Over the next few page (p34/35) of the Daniel Schroeder’s document, he derives Larmor’s formula starting with:

[3] $$Energy-per-unit-volume=\frac{\epsilon_0 E^2}{2}$$

Equation [2] is substituted into [3] to give:

[4] $$Energy-per-unit-volume=\frac{q^2a^2 sin^2 \theta}{32 \pi \epsilon_0 c^4 R^2}$$

This equation is only accounting for the electric field, i.e. no initial consideration of the magnetic field, which eventually doubles the energy. However, the point I want to clarify is that the energy/unit volume appears to be still subject to the inverse square law. Only after multiplying equation [4] by the volume of the spherical shell, i.e. $$4 \pi R^2 * thickness$$, in which (Et) exists, do we eventually arrive at Larmor’s formula.

[5] $$Power-radiated=\frac{q^2a^2}{ 6 \pi \epsilon_0 c^3}$$

What is unclear to me is whether this equation is representative of energy/power radiated by EM wave(s) in all directions, while noting the directionality associated with the $$sin^2 \theta$$ term in [4]. For while the total radiated energy in the spherical shell remains constant, the energy density associated with (Et), as defined by [4], would seem to diminish as the (Et) kink propagates outwards to a larger radius [R]. As such, I don’t see how this equation really reflects the energy of a EM wave or photon propagating without loss in a vacuum.

A photon as a discrete quantum of energy [E=hf] could define a lossless container of propagating energy, but presumably requires the distribution spread of the EM wave to be aggregated by a statistical distribution, which I am assuming is the subject of quantum theory rather than classical fields. So how did classical theory explain how light travels from the stars without loss?

Thanks

2. May 17, 2010

### GRDixon

In my opinion a resolution of the differences between the Larmor formula and a driving agent's counteraction to the radiation reaction force of Abraham/Lorentz is one of the great, unfinished businesses of classical theory. On the one hand, Larmor gives a correct formula for radiated power (at least for periodically moving point charges) and gives the expected results at adequately great distances from the oscillation site. But Abraham/Lorentz can be used to explain the fact that a point charge does not radiate when subjected to a constant external force. Then there is the problem that Larmor says an oscillating charge radiates at maximum power when it is at a turning point (x=A), but at that moment the charge's velocity (and hence Fv) is zero! An article entitled "Larmor and Abraham/Lorentz Both Right???" examines such disconnects and tentatively suggests a resolution. The article can be found by clicking on the article's title, on the home page of www.maxwellsociety.net.

3. May 18, 2010

### mysearch

Re: Understanding Larmor’s Equation?

Sorry about the typo in the title, but by the time I noticed, it was too late to edit.

Thanks for raising some interesting points. I have only had time to quickly review your link and it seems that there are other articles to read. I guess there are a number of ways of looking at electromagnetism, even within the confines of classical physics, i.e. wave/field versus corpuscular/photon. From an oscillating charge perspective, it would seem that some attributes of Simple Harmonic Motion (SHM) might appear to apply in this case, i.e. maximum acceleration = minimum velocity and vice versa. However, I haven’t yet resolve the kinetic energy of the charge and energy associated with the electric field, see earlier thread; https://www.physicsforums.com/showthread.php?t=402163. As you point out, Larmor’s formula suggests energy/power is zero when acceleration (a) equals zero at the origin, i.e. max velocity. In the case of a mechanical wave, the energy at this point has been transferred to the medium as kinetic energy/momentum, but presumably we cannot even hint at an EM wave being associated with a medium when travelling through a vacuum

From my perspective, I would still be interested in any further insights to the following questions raised in post #1:

- Are equations [1] and [2] reflective of the idea of ‘near’ and ‘far’ field definitions?

- How does the Larmor formula reflect the energy of a EM wave or photon propagating without loss in a vacuum over infinite distance.

- How did classical theory explain how light travels from the stars without loss, ignoring opaque dust clouds and Doppler shifting?

4. May 18, 2010

### mysearch

Re: Understanding Larmor’s Equation?

In a much wider context, the following article also raises a number of interesting issues surrounding the whole idea of acceleration being the source of radiation:
http://www.mathpages.com/home/kmath528/kmath528.htm

By way of a number of cross-references to the Abraham–Lorentz force, as raised by GRDixon:

http://www.maxwellsociety.net/LarmorProCon.html

PF discussion:

Another point for clarification that seems to arise from the Abraham–Lorentz force is the issue of effective mass that these equations may be suggesting.

$$Force =\frac{q^2}{ 6 \pi \epsilon_0 c^3}da/dt =ma$$

Does this equation suggest an effective mass associated with the concept of charge?

$$m =\frac{q^2}{ 6 \pi \epsilon_0 c^3}$$

If so, the effective mass of a unit charge particle, e.g. electron, would seem to be in the order of $$5.69*10^{-54}kg$$ in comparison to an gravitational electron mass of $$9.1*10^{-31}kg$$.

Returning to my original questions, which were rooted in the wave equation derived from Maxwell’s equations; there seems to be a suggestion that the propagation velocity of an EM wave is predicated on both the 2nd order derivative of the E & B fields with respect to time and space. This suggested that the rate of change of these fields has to be subject to acceleration in both time and space. However, based on the Larmor formula, I still do not understand how the energy of an EM wave, in a small region of space, is not subject to the inverse square law given the dependency shown in equation [4] post #1, but reproduced below:

$$Energy-per-unit-volume=\frac{q^2a^2 sin^2 \theta}{32 \pi \epsilon_0 c^4 R^2}$$

As such, I remain unsure how to explain the apparent infinite ability of light to propagate without loss in vacuum unless quantised into a particle, e.g. photon, which post-dates the description of classical physics at the time of Larmor and Maxwell. Therefore, I would really appreciate any insights from other PF members who have already resolved these issues. Thanks

Last edited: May 19, 2010
5. May 19, 2010

### mysearch

I realise that there may not be much interest in this thread at this time, but wanted to table some additional issues for future reference. Some of the material referenced in post #2 and #4 not only expanded the original scope of this thread, but seem to challenge some of the conclusions of classical/particle physics. As such, I accept that some of the issues being raised may extend beyond the remit of this sub-forum.

The following link is another that claims that ‘accelerated electrons don’t radiate’:
http://www.physics.tyrannosaur.eu/Accelerating_electrons_don_t_r/accelerating_electrons_don_t_r.html [Broken]

However, I would like to question the description of the operation of a radio dipole aerial in Scenario 4. This description suggests that “Electrons from the top rush down the aerial, and the holes run up, passing at the middle”. In contrast, calculations suggest that an electric current of 1 ampere flowing through a 1mm^2 copper wire appears to correspond to an electron velocity of 0.075 mm/s, i.e. electrons move slower than snails. On the assumption that radio frequencies extend up to 300 GHz, which I am also assuming would have to be reflected in the alternating voltage applied to the antenna, how far would the electrons be able to travel within the antenna in time t=1/frequency? Would any movement be restricted to a microscopic scale?

However, the general position of this article seems to be supported by other references in #4, although up until this point I had not come across one standard text that highlighted any of these issues. So is it’s the case that I just haven’t been reading the right material or that mainstream science disagrees?

Last edited by a moderator: May 4, 2017
6. May 19, 2010

### PhilDSP

Several researchers have found different criteria for radiation/non-radiation of a moving electron rather than merely the presence or absence of acceleration. One of the most interesting is a paper by Dr. Herman Haus which is called "On the radiation of point charges" where he uses a Fourier Transform extensively to discover that if the transform of current density contains a frequency component equivalent to the speed of light, the particle will radiate but not otherwise.

He shows that this clarifies how Cherenkov radiation arises. There are others who have also investigated along those lines: Philip Pearle, G. H. Geodecke, Tyler Abbott and David Griffiths.

7. May 19, 2010

### GRDixon

Re: Understanding Larmor’s Equation?

I have delineated the 'near' and 'far' fields as follows:

If the past motion of a point charge is known, then at any moment the Poynting vector (S) can be computed at any point other than that occupied by the charge itself. (See Griffiths, 2nd Edition, "Introduction to Electrodynamics," Sect. 9.2.2, "The Fields of a Point Charge in Motion.") For example, given a charge oscillating on the x-axis, S can be computed at all y>0.

When y<$$\lambda$$/4, S appears to flux inward and outward practically equally. (There is a slight excess of energy flow outward in any given cycle time.) But for y>$$\lambda$$, S consistently pulses outward. It might be said that the inward/outward range of y defines an "inertial" or "near" zone, whereas the pulsed outward range defines a "radiation" or "far" zone. The nominal inward/outward energy flux in the "inertial" zone closely tracks mav, where m=q$$^{2}$$/6$$\pi$$$$\epsilon_{}$$$$_{}o$$Rc$$^{}2$$ is the electromagnetic mass in the case of a spherical shell of charge. The outward, pulsed energy flux in the radiation zone tracks Fv, where F=-q$$^{}2$$/6$$\pi$$$$\epsilon$$$$_{}o$$c$$^{}3$$(da/dt) is the driving agent's counteraction to the radiation reaction force (first suggested by Abraham and Lorentz).

8. May 19, 2010

### mysearch

Conservation laws associated with an accelerated charge

Many thanks for the informed feedback in both #6 & #7. However, based on the nature of the references cited by PhilDSP, there seems to be genuine ambiguity surrounding the issue as to whether accelerated charge is the root cause of the energy radiated, which was a surprise to me. On this basis, can we really state with any confidence that Larmor’s formula is descriptively accurate, although it presumably aligns with experimental measurements?

Equally, even if Larmor’s formula is numerically correct, it still seems to suggest, as per [4] in post #1, that the total energy per unit volume is subject to the inverse square law. As such, this would seem to suggest that even if discrete photons do not lose energy, as per E=hf, the idea of a EM wave would suggest a dissipation of its energy over an expanding surface area?

Therefore, I was wondering whether it would help to construct a simple, albeit conceptual model in which to establish which conservation laws, outlined below, are applicable to an accelerated charge:

• Can we assume a model to exist in the vacuum of space far from any other influences, e.g. gravitation, which only consists of a single isolated electron with mass (m) and charge (e)?
• Can we also assume that the accelerated charge, said to be radiating energy conforms to simple harmonic motion (SHM) generated by a sinusoidal voltage?
• Is it correct to assume that this system is not closed, as the voltage, i.e. E/d, is constantly providing input energy?
• Can the charged electron be made to physically oscillate with a frequency-period of 300Ghz, such that it has a maximum velocity (v) with minimum acceleration (a) at some centre point (0) and minimum velocity (v) with maximum acceleration (a) at some maximum offset (A) described by Asin(wt)?

Energy Conservation:
Within an open system, energy is assumed to be ‘lost’ in the form of EM radiation. Presumably, this energy is sourced from the potential energy of the voltage field, which initially drives the kinetic energy of the electron’s acceleration, which in turn is said to produce the E-M energy/wave/radiation/photon which can propagate forever?

Momentum Conservation:
The conservation of momentum seems more difficult to reconcile in that the momentum associated with the electron is constantly changing between some maximum and zero. Momentum, unlike energy, is a vector quantity but I cannot see how momentum is conserved other than within the context of energy conservation, i.e. the kinetic energy of the electron being returned to the potential energy within the voltage field?

Charge Conservation:
I am assuming that the strength of the electron charge remains unaffected in this model?

Mass Conservation:
Within the constraints of the Larmor formula there is no consideration of relativistic effects. As such, can we assume that the electron mass is unaffected? However, there was an implication within the Abraham–Lorentz equation of a form of effective mass being associated with the charge itself:

$$Force =\frac{q^2}{ 6 \pi \epsilon_0 c^3}da/dt =ma$$

I have only forwarded this model for my own education in order to try and clarify what assumptions are made by mainstream science and whether any of these assumptions are contradicted by the articles that claim charged electrons don’t radiate. Thanks

9. May 20, 2010

### mysearch

Devil appears to be in the detail!

I am primarily trying to summarise and attach some additional links to this thread for future reference, but would welcome any further feedback on any of the positions/issues raised throughout the thread.

The following article is provided by Dr James F. Woodward at the department of physics at the California State University. I am assuming that this position is representative of ‘mainstream science’, although possibly not readily understandable by ‘mainstream society’:

Here are 2 quotes from this article that seem to highlight the position being taken:

“A fair amount has been written on radiation reaction since then, but the difficulties he (Feynman) described remain. Chief among those difficulties are problems with "causality" (causes always preceding effects) and seeming transient violations of the conservation of energy and momentum.”

“It is known as a matter of fact that electric charges subjected to constant accelerations radiate electromagnetic waves, and the energy they carry away from their source charges is proportional to the square of the charges' acceleration…..Nonetheless, they radiate energy too. We seem to be faced with an obvious violation of the conservation of energy here.”

However, I think GRDixon’s article is a very concise introduction of some of the maths for anybody wishing to get an introduction, like me, to the issues surrounding radiation from accelerated charges. http://www.maxwellsociety.net/LarmorProCon.html
In addition, there appears to be wealth of information provided on his website linked below, although I have not had time to do anything more than quickly scan a few of the many articles: http://www.maxwellsociety.net/contents.htm [Broken]

So while the complexity of some of the issues raised might take me a long time to review, there was one issue I was hoping to get some general resolution, which is linked to the duality problem, i.e. wave or particle. The Larmor formula seemed to suggest that the energy per unit volume is still subject to the inverse square law. This made sense to me as the energy is being dissipated over an expanding volume of space as defined by the spherical shell in Daniel Schroeder’s derivation, see post #1. As such, any notion of an EM wave propagating this radiated energy any great distance seems problematic. In contrast, the idea of a spectrum of photons being propagated without loss that statistically ’peppers’ a region of space in-line with the Larmor formula seems more reasonable, but post-dates the accepted science of the Maxwell/Larmor era. So how did classical physics think light waves from the stars reached us?

Last edited by a moderator: May 4, 2017
10. May 20, 2010

### PhilDSP

Re: Conservation laws associated with an accelerated charge

Entirely intuitively, what you say in the first sentence sparks the idea that quantization conditions would have it that the energy per unit volume would not always reflect the inverse square law. In those instances the propagation of the fields to a state of equilibrium is incomplete - not enough time has passed to come to such a state. Seemingly ripples or kinks or anomalies in the distribution of fields could develop that might conceivably lead to photon creation. Maybe that can be coupled to Dirac's idea of magnetic monopoles.

Your second sentence, which I agree is valid, indicates clearly that the photon is not the EM wave regardless of whether the photon contains some sort of wave itself.

Not if you want to be rigorous IMO. For an electron orbital you would have to account for the field produced by the nucleus. Even for synchotron radiation an external magnetic field is required to accelerate the electron. Once you have an EM field space no longer behaves as a vacuum. The effective permittivity and permeability changes and is highly variable depending on the strength of the field at any particular point.

In the 20's J. J. Thomson developed a number of equations that model the electron in response to the fields along with counter equations that show the subsequent path of near field propagation - sort of a more modern version of Lorentz' and Abraham's model that didn't assume any particular form for the electron. I'll try to dig them up.

Probably not because even though in large part it may be harmonic, the part that isn't harmonic is likely to be of most interest in contributing to or preventing motion that causes radiation.

Could you explain where you're headed there?

Last edited: May 20, 2010
11. May 20, 2010

### mysearch

Re: Understanding Larmor’s Equation?

Hi,

Thanks for the comments. As indicated, the only purpose in forwarding this hypothetical model to was try to understand the simplest model that could be made to continuously accelerate a charge particle, as least, in theory. I have inserted some quick comments below before trying to answer the query in your last comments

I was only attempting to try to isolate the discussion of an EM wave to a single instance of an accelerated charge electron in order to ask whether classical physics believe it was emitting energy as a continuous wave, which was then superseded by a statistical distribution of photons? Personally, both of these models seem to have major problems in this situation.

Appreciate the insight and would like to see the material. Thanks. Not sure a single charged particle makes any sense either. Can a charge really be thought of as an attribute of a single particle, but that is probably a subject of another thread!

Again, the adoption of a SHM model was only introduced because it seemed the simplest way of producing a continuous accelerated charge and to see whether it helped isolate what conservation laws which were applicable.

Sorry. This was not well explained. It was linked to the notion that the electrons in a ‘real’ dipole antenna move very, very, very slowly based on earlier calculations. As such, a radio frequency of 300Ghz would require a SHM cycle in ~3us. Therefore, started to wonder how this oscillation rate, which encompasses the sequence of max-v/ min-a followed by max-a/min-v could be generated inside an antenna given the electron velocity calculated for copper. In part, it was why I switched the model to a vacuum analogy. In retrospect it was probably not a central issue.

Hope this goes someway to explain all the crazy questions.

12. May 21, 2010

### Cleonis

Re: Devil appears to be in the detail!

I noticed that in his writing Dr Woodward refers to the Feynman/Wheeler collaboration on Absorber theory. It is my understanding that eventually Feynman abandoned Absorber theory.

If I remember correctly Feynman adressed the topic in his Nobel prize acceptence speech. He sketched the developments that ultimately led to the work that he was awarded the Nobel prize for. He describes how he was captivated by the underlying idea of absorber theory, and how he came to abandon it eventually. Still, it had taught him a lot and he was able to apply the lessons learned later.

I find it strange that Dr Woodward does not mention that absorber theory was abandoned. Surely that's relevant.

Quite possibly Dr Woodward's ideas are maverick rather than mainstream.

13. May 22, 2010

### mysearch

So how do EM waves really propagate energy?

This seems to be a very valid point. However, I don't have the background to know whether this idea is generally accepted by his peers.

-----------------------​

In this thread, I have been trying to get to the bottom of how EM waves are thought to propagate, at least, within the confines of classical physics, i.e. Maxwell’s equations. According to Maxwell's equations, a spatially-varying electric field generates a time-varying magnetic field and vice versa, as reflected in the following equations:

$$\nabla \times E = -\frac {\partial B}{\partial t}$$

$$\nabla \times B = \frac {1}{c^2} \frac {\partial E}{\partial t}$$

By solving these equations in terms of the 2nd derivatives leads to an EM wave equation, which has a generic form, as shown below, where (A) can be substituted to reflect either the strength of the electric (E) or magnetic (B) fields in space [r] or time [t]:

$$\frac {\partial ^2 A}{\partial r^2} = \frac {1}{c^2} \frac {\partial ^2 A}{\partial t^2}$$

However, it is unclear what this equation really tells you about the physical mechanism of propagation. The following animation seems to visually represent the accepted description of the E and B fields in phase, but perpendicular to each other, with the wave propagating perpendicular to both.

Although it is not explicitly stated, this animation seems to represent the conceptual propagation of a single EM wave associated with just 1 frequency, i.e. reflective of a single charge in oscillation? It also seems to imply that the oscillation of this charge is subject to acceleration. While this ties in with the dependency on acceleration in the Larmor formula, it still appears to be a point of debate in some quarters, as cited in earlier posts. This said; the following reference seems to be a very good summary, although it cites charge acceleration as the source of radiated power:
http://www.phy.duke.edu/~lee/P54/energy.pdf [Broken]

- Both E and B decrease with distance from the source as 1/r.
- E and B are mutually perpendicular.
- Energy flows in the direction of E x B.
- The magnitudes are related by E = cB.
- The intensity is $$I = c \epsilon_0 E$$

However, the first bullet needs to be qualified in terms of the derivation of the Larmor formula as cited by the Schroeder reference in post #1. In this context, there are 2 field components, i.e. the radial and transverse. The radial component reduces by [1/r^2] and is believed to be confined to the ‘near field’. Only the transverse component reduces by [1/r] and linked to the ‘far field’ that becomes the dominate factor in the calculation of power radiated in the Larmor formula.

This article also defines the energy flow, as shown in quotes below, as well as citing the relationship c=E/B, which supports the constant propagation velocity [c] required provided the ratio of E/B remains constant, which is assumed to be the case for E and B oscillating in phase as shown by the animation cited above.

“Intensity is the amount of power (or energy/sec) crossing unit area perpendicular to the direction of the flow…… The intensity of an e-m wave is given by $$I=c \epsilon_0 E^2$$ that oscillates between zero and its maximum twice per cycle. It is common to use the average over a cycle.”

While in practice, most EM sources will be the aggregate of ‘N’ radiating charges, can we still consider the conceptual case of a single oscillating charge? In this context, there would seem to be an instantaneous point in the in-phase E-B cycle where E and B are both zero. As such, it has led me to ask whether even a wave model requires some degree of minimum package size, i.e. 1 cycle. There also seems to be a problem that this model cannot propagate EM energy forever because the energy transfer implicit in the Larmor formula is still subject to the inverse square law.

Of course, as previously speculated, if this energy were quantised within the idea of a lossless photon [E=hf], it could propagate forever, but this idea post-dates both Larmor and Maxwell. As such, I still question how classical physics explained the light from the stars.

As always, would welcome any knowledgeable input that explains where this summary falls down. Thanks

Last edited by a moderator: May 4, 2017
14. May 25, 2010

### PhilDSP

Re: So how do EM waves really propagate energy?

It may be slightly controversial to say this, but that's known now to be a fallacy and wasn't ever part of Maxwell's theory. Jefimenko has analyzed this (his analysis is also derived in a subchapter of Jackson's EM textbook)

Both E and B fields are generated by varying currents according to Jefimenko's equations which are refinements of the Maxwell Equations - strange as that may seem at first.

15. May 29, 2010

### mysearch

How do EM waves propagate energy?

Does this help explain how an EM wave self-propagates, without energy loss, in vacuum?

I also had some direct feedback from another source outside the PF that is related to this thread and felt that it raised some interesting issues. It also left me wondering where classical electrodynamics has to give way to quantum electrodynamics and whether the latter really has any better answers?

16. May 31, 2010

### PhilDSP

Re: How do EM waves propagate energy?

Apparently so. Jefimenko goes on with his analysis and derives something he calls the electrokinetic field which is crudely equivalent to the vector potential with its time constant value stripped away. The E field is then equal to the negative of the time derivative of the electrokinetic field which gives the relationship one looks for for a traveling wave.

As far as energy loss, in a perfect vacuum there is no dispersion. So there isn't a cause for energy to leak away.

Though the links you've posted look very interesting, I haven't had the time to look into them but will at some point. I don't think I'd come to all of the conclusions you posted about the text from utoronto. But the author is right in saying:

"the Maxwellian fields are devoid of any frame information, and are seen as static in all inertial reference frames, although the frequency and energy density changes for differing reference frames (simply examining Maxwell's equations demonstrate this); this leads ultimately to Einstein's work on Relativity"

But that is because Heaviside ignored a few critical facits of Maxwell's theory in generating his interpretation of Maxwell's work. It is Heavyside's equations you posted earlier, not Maxwell's. Maxwell died unfortunately before others really started to understand his work. For one thing, Maxwell used almost exclusively full derivatives with respect to time. He also gave separate equations for use with moving media (relativistic situations). Neither did he get confused about an ether or medium that moved with respect to physical particles - it's such a confusion that generates the apparent need for a Lorentz Transformation.

Last edited: May 31, 2010
17. May 31, 2010

### mysearch

Clarification of model: wave or photon

What model are you assuming, continuously expanding wavefront or discrete photons?

From the photon perspective [E=hf] the vacuum is non-dispersive to photons of different frequency, but the same could still be said of a continuous wave model. The problem I was struggling to resolve in my mind was the how energy is distributed in these 2 models. If Larmor is generally accepted to be right, the energy from an oscillating charge appears to be dissipated over an expanding EM wavefront in accordance to the inverse square law, albeit in a doughnut shape. This would seem to suggest that the energy density must also fall as a function of the inverse square law and given that the electric field and magnetic field strengths are proportional to the energy density, it would also suggest that they must also reduce in accordance, at least, within the confines of a continuously expanding wavefront model.

Of course, the current model now appears to be based on a discrete photon model that doesn’t lose energy, but presumably adheres to the inverse square law distribution of energy by the statistical distribution of discrete lossless energy photons, less energy density by numbers of photons. However, this model was not known to either Maxwell or Larmor, so how did they tie theory to observation?

18. May 31, 2010

### Phrak

I don't know as it makes a difference, though

$$\frac {\partial ^2 A}{\partial r^2} = \frac {1}{c^2} \frac {\partial ^2 A}{\partial t^2}$$

is not the wave equation of light in a vacuum. The solution will be second order in E and B, or third order in A and phi.

Last edited: May 31, 2010
19. Jun 1, 2010

### mysearch

Hi Phrak,
Could I try to clarify the equation above and then confirm whether the following equations are generally accepted? First, the equation above was only intended to represent the general form of the wave equation, which I had seen described in the following terms:
As such (A) above simply represents (E) or (B), while [r] is simply representative of the spatial coordinates. In the original context this form was used, I was struck by the similarity of this equation to a mechanical wave. The following equations are believed to be wave solutions that follow on from Maxwell’s equations:

$$\nabla^2 E= \frac {1}{c^2} \frac {\partial ^2 E}{\partial t^2}$$

$$\nabla^2 B= \frac {1}{c^2} \frac {\partial ^2 B}{\partial t^2}$$

The derivation of these equations is cited in the following link:
http://physics.info/em-waves/

Would appreciate any confirmation of the validity of the assumptions that underpin these equations and any further insights as to the wider issues raised in this thread. Many thanks.

Last edited: Jun 1, 2010
20. Jun 1, 2010

### PhilDSP

Re: Clarification of model: wave or photon

Though Maxwell had already determined the E and B field fluctuations in a ray of light, and those could be experimentally confirmed in experiments involving polarization, the electron was not discovered for another 30 years. Maxwell spoke very little about radiation and not in the context of it arising from particle motion. I'm less familiar with Larmor's work.

That seems to be one of the essential mysteries of QM: how energy or radiation can be guided and compacted into the form of a photon rather than dispersing as an expanding spherical wavefront. I don't know of any theories that have been proposed but suspect that certain motions can pinch or kick the particle's motion in a way that causes some type of shock wave with a component of angular momentum. It may be that some additional type of energy is involved that we don't yet recognize. Having a workable model of an electron would be a preliminary step.

For far field radiation: photons. For near field: wavefront (taking into consideration the effect of the particle's motion prior to radiation).

Last edited: Jun 1, 2010