Interpreting Maxwell's Classical Theory of Electromagnetism

[mysearch]

Hi,
Apologises for the length of this post, but I am struggling to resolve some of the physical interpretations of Maxwell’s classical theory of electromagnetism. Therefore, I would appreciate any help on offer from those who have already resolved any of the issues raised. Many thanks.

As the questions are really asking for insights into the physical interpretation of Maxwell’s equations, rather than the maths, I am only providing a Wikipedia link to them:
http://en.wikipedia.org/wiki/Maxwell's_equations

So while I have generally worked through the derivation of Maxwell’s equations from the maths side, I am struggling with some of the physical interpretation of the interplay between electric and magnetic fields, especially when it comes to EM wave propagation. However, I have initially raised some issues for clarification against Maxwell 1 & 2 followed by Maxwell 3 & 4 before outlining the main questions concerning EM wave propagation:

Some issues of interpretation with Maxwell [1] & [2]:

1) Maxwell [1] & [2] are time independent, therefore the charge [q] is considered stationary to some given observer. If [q] is stationary, no magnetic field (B) can be measured. So does the idea of a stationary permanent magnet or electromagnet need to be qualified in terms of Maxwell [4]?

2) For instance, permanent magnets exist only as a special form of electromagnetism, i.e. the magnetic field is a result of moving charges that exist within the structure of the material?

3) What happens if a central stationary charge was surrounded by a neutralising shield, which could be switched on and off. When switched off, would an electric field propagate outwards at [c] without any implied magnetic field or would this propagation act as [dE/dt] with respect to Maxwell-4 and create an EM wave with a magnetic field component?

Some issues of interpretation with Maxwell [3] and [4]:

4) Magnetic fields only exist by virtue of a moving charge, but the perception of the charge velocity [v] is relative to a given observer and therefore the magnetic field (B) is relative to the velocity of the observer and proportional to v*qSin(theta)?

5) I am assuming the relative nature of the magnetic field does not apply within an EM wave because this wave always moves at [c] within all frames of reference?

6) While (E) and (B) are perpendicular to each other, these fields exist in a circular fashion in a plane that is perpendicular to the source in the context of Maxwell 3 or 4. I read that a current of one amp is equivalent to ~10^18 electrons. Therefore, I am not sure whether Maxwell-4 can be applied to a single electron moving at non-relativistic speeds in a vacuum. However, this model would seem to suggest that the electric field (E) would exist in all directions based on the inverse square law, while the magnetic field (B) would be proportional to qvSin(theta) implying a sine shaped magnetic field surrounding the moving charge at a given distance; this field would also be subject to the inverse square law. Presumably, this model reflects the E-M fields surrounding a physical charge [q] moving at a non-relativistic velocity [v] and not the EM wave emanating from the moving charge at velocity [c]?

The EM wave equation of motion: http://physics.info/em-waves/
The maths presented in this link seems as clear a derivation of the EM wave equation as I could find, although I prefer to substitute E=E0 Sin(ky-wt) and B=B0 Sin(kz-wt) to represent the perpendicular plane waves for E & M. I liked the approach as it directly shows the development of the EM wave equation from Maxwell’s equations. However:

8) Should I interpret the picture of an EM wave, as illustrated in the link below, in terms of the Lorentz force equation. For example, as an EM wave passes a point in space occupied by a unit charge, this charge would be subject to the perpendicular component forces defined by F=q(E+vxB)? http://www.montalk.net/emavec/EMtransverse.jpg

9) On the basis that the charge density [I,J] are zero in Maxwell [1] and [4] what is the classical model for the source of an EM wave, i.e. Maxwell’s equation predates the idea of a photon or even the atomic model, so was the classical source of an EM wave always assumed to be an oscillating charge?

10) Standard texts suggest the energy associated with both the electric and magnetic field is proportional to the square of the electric field (E). This seems analogous to a mechanical wave where energy is always proportional to the amplitude squared. Is there a connection between these models and when was it subsequently tied to Planck’s equation Energy=hf?

11) If I assume the EM wave is started by an oscillating charge in free space, which quickly stops, should I also assume that the energy in the EM wave continues to propagate as a finite pulse without any loss forever, i.e. energy conservation applies in the absence of any loss mechanism for EM waves in vacuum?

12) In a mechanical wave, the total energy at any point of the wave can always be approximated as the sum of the potential and kinetic energy. However, because the E and B components of the EM wave are in phase, there seems to be points where E and B are both zero. Is there is some minimum ‘quanta’ of an EM wave over which the energy has to be aggregated, which also leads to the idea of some minimum quanta of momentum = E/c?

13) How did classical physics of Maxwell’s equation explain 12) prior to the idea of photon quanta?

 

[Phrak]

"2) For instance, permanent magnets exist only as a special form of electromagnetism, i.e. the magnetic field is a result of moving charges that exist within the structure of the material?"

Maxwell's equations don't talk about the properties of materials, or how magnetic fields must originate. Only that the total flux leaving a region must be the total flux entering. Since magnetic materials are understood at the atomic level, the origin of the field is treated with quantum mechanics. However, it is possible to model the origin of magnetic fields as current moving in a loop. The modeling method assumes that current is circulating in a circular loop. Then the radius of the loop is shrunk to zero at a point. The thing left at the point is called a magnetic dipole. Disperse a bunch of these dipoles in a volume, with all their axes pointing in the same direction and you have modeled a permanent magnet.

 

[Dale]

4) Magnetic fields only exist by virtue of a moving charge, but the perception of the charge velocity [v] is relative to a given observer and therefore the magnetic field (B) is relative to the velocity of the observer and proportional to v*qSin(theta)?

Yes, the electric and magnetic fields are frame-variant, meaning that they depend on the reference frame used to express them, what is a purely electric field in one frame can be an electric and a magnetic field in another frame. In fact, one of the most interesting aspects of special relativity is the connection formed between different concepts, like the electric and magnetic fields.

5) I am assuming the relative nature of the magnetic field does not apply within an EM wave because this wave always moves at [c] within all frames of reference?

It applies then also. The E and B fields are different in different frames, but they obey Maxwell's equations in all reference frames meaning that they result in EM waves propagating at c in all frames.

11) If I assume the EM wave is started by an oscillating charge in free space, which quickly stops, should I also assume that the energy in the EM wave continues to propagate as a finite pulse without any loss forever, i.e. energy conservation applies in the absence of any loss mechanism for EM waves in vacuum?

Energy conservation applies in all cases, even in the presence of a loss mechanism for EM waves (interaction with matter):
http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

 

[mysearch]

Maxwell's equations don't talk about the properties of materials, or how magnetic fields must originate. Only that the total flux leaving a region must be the total flux entering. Since magnetic materials are understood at the atomic level, the origin of the field is treated with quantum mechanics. However, it is possible to model the origin of magnetic fields as current moving in a loop. The modeling method assumes that current is circulating in a circular loop. Then the radius of the loop is shrunk to zero at a point. The thing left at the point is called a magnetic dipole. Disperse a bunch of these dipoles in a volume, with all their axes pointing in the same direction and you have modeled a permanent magnet.

I agree with your outline. The issue of a permanent magnet raised the question in my mind as to whether all Maxwell’s equations ultimately revert back to a stationary or moving charge. Only in the latter case does a magnetic field exist. Thanks

Yes, the electric and magnetic fields are frame-variant, meaning that they depend on the reference frame used to express them, what is a purely electric field in one frame can be an electric and a magnetic field in another frame. In fact, one of the most interesting aspects of special relativity is the connection formed between different concepts, like the electric and magnetic fields.

I have had a quick look at the Lorentz transforms which support your statement. However, I need to understand the physical effects a little better. Thanks

It applies then also. The E and B fields are different in different frames, but they obey Maxwell's equations in all reference frames meaning that they result in EM waves propagating at c in all frames.

Haven’t had a chance to work through the implications of what you seem to be suggesting. While the propgation velocity [c] is often given in terms of:

$$[c]=1/\sqrt{\mu \varepsilon}$$

It seem more informative to look at c=E/B. As such, it would seem to suggest that this ratio must remain constant in all frames of reference, i.e. B can never be zero for a propagating EM wave?.

Energy conservation applies in all cases, even in the presence of a loss mechanism for EM waves (interaction with matter):
http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

Thanks for the link. I need to look closer at the energy issues. Presumably, in vacuum, no energy loss is assumed?

 

[fabpan]

11) If I assume the EM wave is started by an oscillating charge in free space, which quickly stops, should I also assume that the energy in the EM wave continues to propagate as a finite pulse without any loss forever, i.e. energy conservation applies in the absence of any loss mechanism for EM waves in vacuum?

An example could be the cosmic background radiation.
I'm always astonished when I think that it has been traveling for more then 13 billions of years, even though, in this case, the expansion of the space makes the conservation of energy a difficult topic.

 

[Dunnis]

I'm trying to interpret Maxwell as well, let's exchange information...On Physical Lines of Force: -"The ratio of m to mu varies in different substances; but in a medium whose elasticity depend entirely upon forces acting between pairs of particles, this ratio is that of 6 to 5, and in this case E^2= Pi*m"

Q1: What is this 6:5 ratio and how did he make that conclusion?
On Physical Lines of Force: -"To find the rate of propagation of transverse vibrations through the elastic medium, on the supposition that its elasticity is due entirely to forces acting between pairs of particles

[PLAIN]https://www.physicsforums.com/latex_images/26/2647530-2.png

where 'm' is the coefficient of transverse elasticity, and 'p' is the density."

Q: Where did he get numerical values for this elasticity 'm' and density 'p'?
Did anyone notice Maxwell's original "wave equation" is actually 'wave equation for vibrating string': -"The speed of propagation of a wave in a string (v) is proportional to the square root of the tension of the string (T) and inversely proportional to the square root of the linear mass (μ) of the string:

[PLAIN]https://www.physicsforums.com/latex_images/26/2647530-3.png

". - http://en.wikipedia.org/wiki/Vibrating_string

 

[PhilDSP]

12) In a mechanical wave, the total energy at any point of the wave can always be approximated as the sum of the potential and kinetic energy. However, because the E and B components of the EM wave are in phase, there seems to be points where E and B are both zero. Is there is some minimum ‘quanta’ of an EM wave over which the energy has to be aggregated, which also leads to the idea of some minimum quanta of momentum = E/c?

13) How did classical physics of Maxwell’s equation explain 12) prior to the idea of photon quanta?

The E and B components of the wave are 90 degrees out of phase with each other. If that weren't the case there would be no transfer of energy between them and the wave would not propagate.

There is no concept of a quantum limitation of energy of any sort involved with either the Maxwell Equations or Maxwell's general theory. You could view quantum limitations as belonging strictly to the configuration of energy within certain structures such as an electron or atom - in particular involving the angular momentum of those particles or their components and the interrelationships among the components of angular momentum.

 

[Dale]

The E and B components of the wave are 90 degrees out of phase with each other. If that weren't the case there would be no transfer of energy between them and the wave would not propagate.

This is not correct. You are thinking of a LC circuit where the energy in the capacitor and inductor are 90 deg out of phase. That is not the case for a EM plane wave radiating in free space where the E and B fields are in phase.

 

[mysearch]

I'm trying to interpret Maxwell as well, let's exchange information...’On Physical Lines of Force’

Dunnis: I am assuming that you are making reference to Maxwell’s 1861 paper (?) which I have not read. If so, you might wish to give the page references so that anybody wishing to comment on your issues may make direct reference to the section in detail. While I can’t help you on either Q1 or Q2, the following issues raised by PhilDSP and DaleSpam may be relevant to your 3rd issue:

The E and B components of the wave are 90 degrees out of phase with each other. If that weren't the case there would be no transfer of energy between them and the wave would not propagate.

I agree with DaleSpam's comment. The (E) and (B) fields propagate in phase in an EM wave in vacuum, at least, according to Maxwell.

There is no concept of a quantum limitation of energy of any sort involved with either the Maxwell Equations or Maxwell's general theory. You could view quantum limitations as belonging strictly to the configuration of energy within certain structures such as an electron or atom - in particular involving the angular momentum of those particles or their components and the interrelationships among the components of angular momentum.

Generally, I was trying to avoid mixing the discussion of classical EM wave theory with the idea of photons simply because Maxwell published this theory in 1865. I don’t know how this relates to the paper Dunnis is referring to? However, Dunnis also made reference to a mechanical wave equation, which links the 2nd derivative with respect to space [x] to the 2nd derivative with respect to time [t] via the velocity of propagation [v]. This relationship seems to hold true for both mechanical waves and EM wave. It would seem that EM waves also share some similarities with mechanical waves when it comes to energy, although the following comments are more by way of questions than statement of facts:

1) The energy density of an EM wave is proportional to the square of the electric or magnetic field ‘amplitude’ [E²] or [B²]. The total energy density of the EM wave is also the sum of the electric and magnetic energy density.

2) In this respect., there seems to be some analogy to a mechanical wave in which the potential and kinetic energy of the wave is proportional to the square of the amplitude of mechanical wave.

3) It would also seem that electric field relates to potential energy, while magnetic field is analogous to kinetic energy by virtue of its dependency on velocity [v].

Therefore, I would be interested in any further insights to the way mechanical and EM waves propagate. Going back to the point raised by PhilDSP, Maxwell developed his EM equations 40 years before Planck defined E=hf, which is said to cover all EM waves. Therefore, Maxwell could not have considered the discrete nature of a photon. However, the point that I was trying to raised, in post #1, points 12 & 13, was if E & B are both zero at a given point, because they are in phase, does this mean that an EM wave cannot exist less than a finite length, presumably defined in terms of some integer multiple of its wavelength?

 

[Born2bwire]

The simularities between electromagnetic waves and say mechanical waves lies purely with the fact that the simple wave equations are similar. The same could be said about acoustic waves and probably even some specific cases of fluid dynamics. I would not get too hung up on trying to use mechanical waves as insight into electromagnetic waves. While the classical wave equations are similar, the physics are completely different. So any insight that can be applied is going to be merely a simularity of mathematics. For example, mechanical waves in a medium can allow for transverse and longitudinal waves. However, in a source-free homogeneous medium we can only allow for transverse waves with electromagnetics.

I will briefly harp on question 12 since it was never explicitly answered. The energy density is the sum of E \dot D and H \dot B, or E^2+B^2 and some constants if we have a homogeneous volume. So, yes, since electromagnetic waves are spatially varying and can have nodal points there are regions where this is zero. However, it should be noted that this is an energy density, it is only zero at an infinitesimal point. Realistically we would observe the fields over a volume of non-zero size. Technically then, we would still measure some amount of energy. Still, as previously mentioned, classical electromagnetics does not account for quantum effects like photons. If we were to treat the problem as a quantum mechanical problem when we would see different behavior. But quantum mechanics is only really meaningful when applied to a statistical set of measurements. So if we have an area that should be very low energy density, then we would observe a very small number of photons over a large number of samplings. Some samplings may be zero, others may not. With a statistical set we can get an idea of the energy density.

As to the EM wave not existing less than a finite length. No, again, this is an energy density. So the energy of the wave is non-zero only at infinitesimal points.

 

[mysearch]

Just a quick reply to #10, thanks for the informed comments, I think I am in tune with what you are saying, but need to reflect further. However, I was interested in how Maxwell resolved the problem back in 1865 before quantum mechanics was even thought of. For example, a solution of Maxwell’s equations for an EM wave propagating in free space is:

c=E/B

How did Maxwell and his contemporaries resolve c=E/B=0/0 ?

 

[Born2bwire]

Just a quick reply to #10, thanks for the informed comments, I think I am in tune with what you are saying, but need to reflect further. However, I was interested in how Maxwell resolved the problem back in 1865 before quantum mechanics was even thought of. For example, a solution of Maxwell’s equations for an EM wave propagating in free space is:

c=E/B

How did Maxwell and his contemporaries resolve c=E/B=0/0 ?

Experiment. Maxwell built his equations off of existing equations that were derived from empirical methods. His main contribution was the addition of a displacement current in Ampere's Law. With the addition of the displacement current, you could then work out a wave equation from the four equations. Out of this wave equation is the predicted speed of light in a vacuum (or medium). Around this time there was also ongoing experiments to measure the speed of light as well. So Maxwell would also have had an experimental result for the speed of light to confirm his calculation.

 

[Dale]

How did Maxwell and his contemporaries resolve c=E/B=0/0 ?

It doesn't seem to me to require resolution, and I don't think that QM would be relevant.

If E and B are both 0 over some finite volume then there is no wave and the wave velocity in that region is undefined as it should be. If it is only 0 over some infinitesimal volume then you are dealing with limits and the limit is well defined and equal to c.

 

[mysearch]

Experiment. Maxwell built his equations off of existing equations that were derived from empirical methods. His main contribution was the addition of a displacement current in Ampere's Law. With the addition of the displacement current, you could then work out a wave equation from the four equations. Out of this wave equation is the predicted speed of light in a vacuum (or medium). Around this time there was also ongoing experiments to measure the speed of light as well. So Maxwell would also have had an experimental result for the speed of light to confirm his calculation.

Fair enough, but I would have thought a few people would have puzzled over the anomaly implied by c=E/B=0/0 as not only does seem to raise a question about theory versus empirical verification, but more importantly it seems to question the structure of the wave. For example, does it make sense to ask what is the minimum length of a radio EM pulse?

P.S. This question is raised in the context of 1865, i.e. pre quantum theory

 

[Born2bwire]

Fair enough, but I would have thought a few people would have puzzled over the anomaly implied by c=E/B=0/0 as not only does seem to raise a question about theory versus empirical verification, but more importantly it seems to question the structure of the wave. For example, does it make sense to ask what is the minimum length of a radio EM pulse?

P.S. This question is raised in the context of 1865, i.e. pre quantum theory

Why would E/B = 0/0 ? And as I stated before, there isn't a minimum length. The eletromagnetic wave is a field, it permeates a volume of space. Sure we could confine it temporally (and thus spatially) by pulsing the signal and also confine it spatially via directive sources. But the, let's call it breadth, of the pulse is defined by the bandwidth of the signal generating the pulse and the physical size of the source. In classical electromagnetics, the basic theory does not have an upper limit on frequency and we could always take the limit of our source's aperture to infinity. But of course even before getting to quantum electrodynamic theory our available bandiwidth is already constricted due to the limited range of frequencies over which classical theory is fully valid and the limitations of generating waves.

 

[PhilDSP]

I agree with DaleSpam's comment. The (E) and (B) fields propagate in phase in an EM wave in vacuum, at least, according to Maxwell.

To be more specific: the E and B fields vary in phase with respect to time but 90 degrees with respect to space.

That's a bit unusual, isn't it? Other traveling waves do have a parameter with a 90 degree phase difference in time. We can note that a 90 degree phase shift across all frequencies implements the derivative function if its accompanied by an increase in the value of the parameter with respect to frequency.

 

[Dale]

To be more specific: the E and B fields vary in phase with respect to time but 90 degrees with respect to space.

PhilDSP, you don't know what you are talking about. They are in phase in both time and space. Please stop spreading misinformation.

PS You may be confusing phase with direction. The equation for a linearly polarized plane wave propagating in the z direction is given by:
$$\mathbf{E}=E_0 cos(k z - \omega t) \hat{\mathbf{x}}$$
$$c \mathbf{B}=E_0 cos(k z - \omega t) \hat{\mathbf{y}}$$
The phase is the term inside the cos function, which is equal for both the E and B, the direction is the vector outside the cos function, which is 90º different.

 

[mysearch]

Why would E/B = 0/0 ? And as I stated before, there isn't a minimum length. The eletromagnetic wave is a field, it permeates a volume of space. Sure we could confine it temporally (and thus spatially) by pulsing the signal and also confine it spatially via directive sources. But the, let's call it breadth, of the pulse is defined by the bandwidth of the signal generating the pulse and the physical size of the source. In classical electromagnetics, the basic theory does not have an upper limit on frequency and we could always take the limit of our source's aperture to infinity. But of course even before getting to quantum electrodynamic theory our available bandiwidth is already constricted due to the limited range of frequencies over which classical theory is fully valid and the limitations of generating waves.

Just to clarify one point, I am not challenging accepted theory, only trying to understand it. So maybe I could try to highlight the areas that I am trying to better understand by a conceptual example. Assume 3 frames of reference, i.e. frames A, B & C, separated by some distance in freespace. Let us also assume, and align, a [xyz] coordinate system in each frame. In frame-A, a single charged particle is traveling at velocity [v] along the x-axis with respect to its own frame and that of frame-B, but appears to be stationary with respect to frame-C.

Q1: Is it true to say that this model essentially conforms to Maxwell 4th equation and, as such, produces a circular magnetic field (B) in the yz-plane around the charged particle moving along the x-axis, when observed from frame-B, but not frame-C?

Q2: Is it true to say that an electric field (E) exists in all directions around the charged particle subject to the inverse square law. This field exists with respect to both frame-B and frame-C?

Q3: Does this moving charge results in an EM wave and what direction does this wave travel with respect to the source charge?

Q4: My interpretation of the wave equation derived from Maxwell’s equations was that this wave propagates along a 1-dimensional line in freespace, x or y or z?, and at each point along this line, spearated in both space and time, there is an associated value of (E) and (B), which results in a forward propagation that is perpendicular to both (E) and (B), where c=E/B in the case of freespace when $$\rho=0; J=0 ?$$

Q5: Because the propagation results in a 1-dimensional wave, the energy is not dissipated over an expanding surface and in the absence of any other loss mechanism, this wave would propagate forever?

Q6: To be honest, I don’t yet understand the applicability of an energy density [ $$\epsilon E^2$$] that most texts seem to derived from the field density of a flat-plate capacitor to an EM wave in vacuum, but accept the units are consistent with energy*m³. So is the interpretation that this energy density exists in a very small volume of space along the path the EM wave is travelling?

Q7: Is it correct to say that while the charged particle in frame-A continues to move, the observer in frame-B will continue to detect an EM wave. However, if this charge only moves a very short time, comparable to the wave period [P=1/f], can I still assume it emits an EM wave, i.e. energy propagates at [c], for this period?

Q8: Finally, can I tie some of the issues in Q6 & Q7 to the idea that the energy propagated by an EM wave is ‘packaged’ into a particle-like volume of space, which we might name a photon without necessarily invoking quantum mechanics at this stage. While I accept this line of thought might be completely wrong, it is what led me to my original question about E/B=0/0, because it seemed to suggests that there is no energy at this point and therefore implied that the energy density has to be aggregated over, at least, some portion of a wave cycle?

Sorry for belabouring these points, but they seemed to be fairly fundamental to any physical interpretation or alternatively highlight where I need to correct my present understanding. Thanks.

 

[Dale]

In frame-A, a single charged particle is traveling at velocity [v] along the x-axis with respect to its own frame and that of frame-B, but appears to be stationary with respect to frame-C.

From your description A and B are the same frame since the particle is moving at the same velocity v in both. In any case, the fields for a single charged point particle undergoing arbitrary motion are given by the Lienard Wiechert potentials. Here is a Wikipedia link that is pretty good and also my favorite academic link on the subject:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential
http://fermi.la.asu.edu/PHY531/larmor/index.html

Q1: Is it true to say that this model essentially conforms to Maxwell 4th equation and, as such, produces a circular magnetic field (B) in the yz-plane around the charged particle moving along the x-axis, when observed from frame-B, but not frame-C?

Yes, it conforms to all of Maxwell's equations, not just the 4th.

Q2: Is it true to say that an electric field (E) exists in all directions around the charged particle subject to the inverse square law. This field exists with respect to both frame-B and frame-C?

There is a near-field term which decays as 1/r². It will be Lorentz contracted in frame-B, so it will not be the same as Coulomb's law in that frame, but it still does follow an inverse square law.

Q3: Does this moving charge results in an EM wave and what direction does this wave travel with respect to the source charge?

There is a far-field term which decays as 1/r, however for the specific scenario you described (constant velocity, no acceleration) the far-field term is 0, meaning no EM wave.

Q4: My interpretation of the wave equation derived from Maxwell’s equations was that this wave propagates along a 1-dimensional line in freespace, x or y or z?, and at each point along this line, spearated in both space and time, there is an associated value of (E) and (B), which results in a forward propagation that is perpendicular to both (E) and (B), where c=E/B in the case of freespace when $$\rho=0; J=0 ?$$

Q5: Because the propagation results in a 1-dimensional wave, the energy is not dissipated over an expanding surface and in the absence of any other loss mechanism, this wave would propagate forever?

That is correct for a plane wave. Your scenario here does not correspond to a plane wave.

Q6: To be honest, I don’t yet understand the applicability of an energy density [ $$\epsilon E^2$$] that most texts seem to derived from the field density of a flat-plate capacitor to an EM wave in vacuum, but accept the units are consistent with energy*m³. So is the interpretation that this energy density exists in a very small volume of space along the path the EM wave is travelling?

No, the energy density exists everywhere the fields are non-zero.

Q7: Is it correct to say that while the charged particle in frame-A continues to move, the observer in frame-B will continue to detect an EM wave. However, if this charge only moves a very short time, comparable to the wave period [P=1/f], can I still assume it emits an EM wave, i.e. energy propagates at [c], for this period?

Frame A and B are the same frame and your scenario is for a uniform velocity which contradicts the "only moves a very short time". I don't know what to make of this question. Where is the wave period coming from? If it is coming from the motion of the charge then what you describe is not possible.

Q8: Finally, can I tie some of the issues in Q6 & Q7 to the idea that the energy propagated by an EM wave is ‘packaged’ into a particle-like volume of space, which we might name a photon without necessarily invoking quantum mechanics at this stage. While I accept this line of thought might be completely wrong, it is what led me to my original question about E/B=0/0, because it seemed to suggests that there is no energy at this point and therefore implied that the energy density has to be aggregated over, at least, some portion of a wave cycle?

Obviously you must always aggregate (integrate) the energy density over a finite volume. Energy density is just that, a density, so if you have 0 volume then you have 0 energy, regardless of the density. So the fact that there is an energy density of 0 over some infinitesimal volume is completely unimportant. I don't know why you are so fixated on that, it is really trivial.

 

[bjacoby]

Obviously you must always aggregate (integrate) the energy density over a finite volume. Energy density is just that, a density, so if you have 0 volume then you have 0 energy, regardless of the density. So the fact that there is an energy density of 0 over some infinitesimal volume is completely unimportant. I don't know why you are so fixated on that, it is really trivial.

DaleSpam is correct about this. It is very common for people to interpret the Poynting vector as a power density at a point in space. While such an interpretation is often useful, it is also wrong. It is only the integral of the Poynting vector over a surface that is defined. Which validates DaleSpam's remarks.

I would also point out that one is going to have some serious problems trying to interpret electromagnetics in terms of wave models. Maxwell stated that energy can only be transmitted from A to B in two ways: Kinetically by moving mass, or by waves transmitted in a medium. Today modern physics says that there is a third way: waves in no media whatsoever. EM waves are said to use this third way. (Yeah, I know it makes no sense whatever)

To quote from Resnick and Halliday Vol I P 393. "No medium is required for the transmission of electromagnetic waves, light passing freely, for example, through the vacuum of outer space from the stars." So attempting to create mechanical models to explain electromagnetics is destined for trouble.

 

[mysearch]

DaleSpam,
First and foremost, thanks for the time and trouble taken to help me understand some of the wider issues surrounding the topic of EM wave propagation. I appreciate that it must sometimes be frustrating to answer questions that standard text, and this forum, may have already addressed many times. However, I am starting to realize (honestly) that I have been working on some false assumptions about EM wave propagation, based on your responses, which I really need to address by widening my reading around this topic. So while I should do this BEFORE raising any more questions, I will post some of the issues I am trying to now resolve for future reference.

I was unaware of the significance between a 'moving charge' and an 'accelerating charge', as defined by the following quote taken from another discussion in the forum:

https://www.physicsforums.com/showpost.php?p=1717598&postcount=3
Cepheid: “A time-varying electric field produces a time-varying magnetic field, which in turn produces a time-varying electric field and so on. As a result, the fields are self-sustaining and can continue to propagate independently of the source that produced them. Of course, something has to get the fields propagating in the first place. This requires ‘accelerating’ charge as unmoving charge produces a ‘static’ electric field, which is unvarying with time, while a steadily moving charge produces a ‘static’ magnetic field, which is also unvarying in time. Therefore, in both these cases, there will be no fields that are changing with time, and therefore no electromagnetic wave will be radiated. If the charge is accelerating, however, then EM radiation will be produced. One way to do this is to have an oscillating electric dipole.”However, I am not totally sure how to reconcile the following statement with the previous one.

http://en.wikipedia.org/wiki/Electromagnetic_radiation
“According to Maxwell's equations, a spatially-varying electric field generates a time-varying magnetic field and vice versa. Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on. These oscillating fields together form an electromagnetic wave.”

The first sentence above seems to express Maxwell’s 3rd law, where the spatially-varying electric field is defined by ($$\nabla \times E$$). However, I am not sure where the inference of acceleration is in this expression?

In the broadest sense, a wave is normally described as a mechanism that transports energy. So do only accelerating charges generate an EM wave that transport energy? A couple of points I still need to resolve:

1. If you could just ‘create’ a static charge [q1] at a point in freespace with respect to another charge [q2] at distance [r], it would presumably create an electric field [E], but would this field take a finite time to reach [q2], i.e. t12=r/c, assuming it expands into space at the speed of light [c]?
   
2. Although I am assuming the electric field (E) would be subject to the inverse square law, its theoretical range appears infinite. Therefore, the time [t12] might be quite large. If the static charge [q1] could also be switched off after a time [t11<t12] would the electric field (E), which was originally expanding towards [q2], continue and exist at [q2] for [t11], even though it no longer exists at [q1]?

I have raised this issue because it seems to imply that a force would eventually be exerted on [q2], no matter how small, which would suggest that an energy is associated with this force, although as a static charge there is no obvious inference of a magnetic field or an EM wave being generated to transport this energy?

I will leave it there for now and go away and do some more reading, but thanks again for the helping hand.

I would also point out that one is going to have some serious problems trying to interpret electromagnetics in terms of wave models. Maxwell stated that energy can only be transmitted from A to B in two ways: Kinetically by moving mass, or by waves transmitted in a medium. Today modern physics says that there is a third way: waves in no media whatsoever. EM waves are said to use this third way. (Yeah, I know it makes no sense whatever)

Thanks for this insight as I am somebody who is having some serious problems trying to interpret electromagnetics in terms of wave models

 

[jtbell]

If you could just ‘create’ a static charge [q1] at a point in freespace

Doing so violates Maxwell's equations. From Maxwell's equations, one can derive the continuity equation for electric charge and current:

$$\frac {\partial \rho} {\partial t} = - \nabla \cdot \vec j$$

This basically says that if charge appears at some location, it has to come from somewhere nearby, via a current that "flows into" that location.

 

[mysearch]

This basically says that if charge appears at some location, it has to come from somewhere nearby, via a current that "flows into" that location.

Accepted. Presumably this is also a requirement of the conservation of charge? However, it was essentially just a conceptual idea in which an existing charge in freespace was neutralised by some sort of shield that could be switched off. This might not be possible either, but I was really only interested in whether the electric field itself could be said to also propagate in space with velocity [c]? If so, would the field effectively carry energy via virtue of the force it would eventually exert on another charge?

Sorry, if this conceptual example is impossible, such that the question just can't apply.

 

[Dale]

I was unaware of the significance between a 'moving charge' and an 'accelerating charge', as defined by the following quote taken from another discussion in the forum:

https://www.physicsforums.com/showpost.php?p=1717598&postcount=3
Cepheid: “A time-varying electric field produces a time-varying magnetic field, which in turn produces a time-varying electric field and so on. As a result, the fields are self-sustaining and can continue to propagate independently of the source that produced them. Of course, something has to get the fields propagating in the first place. This requires ‘accelerating’ charge as unmoving charge produces a ‘static’ electric field, which is unvarying with time, while a steadily moving charge produces a ‘static’ magnetic field, which is also unvarying in time. Therefore, in both these cases, there will be no fields that are changing with time, and therefore no electromagnetic wave will be radiated. If the charge is accelerating, however, then EM radiation will be produced. One way to do this is to have an oscillating electric dipole.”However, I am not totally sure how to reconcile the following statement with the previous one.

http://en.wikipedia.org/wiki/Electromagnetic_radiation
“According to Maxwell's equations, a spatially-varying electric field generates a time-varying magnetic field and vice versa. Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on. These oscillating fields together form an electromagnetic wave.”

The first sentence above seems to express Maxwell’s 3rd law, where the spatially-varying electric field is defined by ($$\nabla \times E$$). However, I am not sure where the inference of acceleration is in this expression?

I don't know if this will help, but basically the first quote is primarily talking about the fields' interactions with matter, and the second quote is primarily talking about the fields' interactions with each other. So you can think of the first quote as describing the behavior in a region containing charges and currents, and you can think of the second quote as describing the behavior in a region of free space.

The importance of the acceleration of a charge in causing EM waves is a direct result of solving Maxwell's equations for the special case of an arbitrarily moving point source. When you solve Maxwell's equations for an arbitrarily moving point charge you get a solution known as the Lienard Wiechert potential. Here are my favorite two links on the topic:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential
http://fermi.la.asu.edu/PHY531/larmor/index.html

If you look at equation 19 on the second page you see that there are two terms. The first term is the "near-field" term which decays as 1/r², that term is always non-zero meaning that a point charge always has a field which decays as 1/r². The second term is the "far-field" term which decays as 1/r. That term is only non-zero when the charge is accelerating meaning that an accelerating charge has an additional field which radiates.

In the broadest sense, a wave is normally described as a mechanism that transports energy. So do only accelerating charges generate an EM wave that transport energy?

Basically yes, but I wouldn't say it that strongly. The reason I wouldn't say it that strongly is because you can also say that a spatially and temporally varying EM field generates an EM wave, but then you get into a chicken-and-egg discussion about if the accelerating charge causes the field to change or if the field causes the charge to accelerate, etc.

A couple of points I still need to resolve:

[*]If you could just ‘create’ a static charge [q1] at a point in freespace with respect to another charge [q2] at distance [r], it would presumably create an electric field [E], but would this field take a finite time to reach [q2], i.e. t12=r/c, assuming it expands into space at the speed of light [c]?

As jtbell mentioned you cannot create a static charge in free space, it violates the continuity equation which is derived directly from Maxwell's laws. Violating the continuity equation can lead to nonsense results, particularly when you consider energy.

However, what you can do is to separate two equal and opposite static charges at a point in free space and create what is known as an electric dipole. The field from doing that does indeed radiate out into space at the speed of light and forms the basis of radio transmitters.

[*]Although I am assuming the electric field (E) would be subject to the inverse square law, its theoretical range appears infinite. Therefore, the time [t12] might be quite large. If the static charge [q1] could also be switched off after a time [t11<t12] would the electric field (E), which was originally expanding towards [q2], continue and exist at [q2] for [t11], even though it no longer exists at [q1]?

Yes, but for the dipole instead of an isolated charge. Also, like the Lienard Wiechert fields given above, the dipole radiation solution has a near-field term that decays as 1/r² and a far-field term that decays as 1/r.

I have raised this issue because it seems to imply that a force would eventually be exerted on [q2], no matter how small, which would suggest that an energy is associated with this force, although as a static charge there is no obvious inference of a magnetic field or an EM wave being generated to transport this energy?

That is exactly the energy problem you get from violating the continuity equation. When you use a dipole since you are moving the two charges apart there is a current and therefore there is a magnetic field also, so you do have an EM wave transporting the energy. Also, this energy is finite and equal to the work done in separating the charges and bringing them back together, whereas the energy that you would get from violating the continuity equation would be infinite.

 

[Born2bwire]

Yes, but for the dipole instead of an isolated charge. Also, like the Lienard Wiechert fields given above, the dipole radiation solution has a near-field term that decays as 1/r² and a far-field term that decays as 1/r.

I would also further note that the higher the multipole moment, the shorter the distance that the fields extend over. A single electric charge, a monopole, falls off as 1/r^2. Two charges of equal and opposite charge in close proximity create a dipole which can be approximated as a set of fields that fall off as 1/r^3. If we have two dipoles close to each other then we can have a quadrupole which falls off as 1/r^4. And so each additional increase in the order of the pole adds another factor of 1/r to the field drop off.

So you can quickly see how everyday macroscopic objects have no appreciable electric fields even though they are technically quasi-neutral (that is we can find areas in very small volumes that have net charges since the electrons and protons are not colocated).

When we typically talk about a problem with charges. We generally implicitly imply that the problem was set up by bringing in these charges out from infinity, one by one, and then letting enough time to pass such that any transient behavior from these actions dies out.

 

[mysearch]

Many thanks for all the help and patience. I will now go away and read the references suggested and reflect on all the comments raised. I had already started to read the links concerning the Lienard Wiechert potential, but it will take me some time to work through the maths. As such, I thought it might be useful to summarise some links that other forum members, trying to better understand this subject, might find useful:

Classical electromagnetism
http://en.wikipedia.org/wiki/Electrodynamics

Basic Concepts
http://en.wikipedia.org/wiki/Electric_field
http://en.wikipedia.org/wiki/Magnetic_field
http://en.wikipedia.org/wiki/Lorentz_force

Electromagnetic fields
http://en.wikipedia.org/wiki/Electromagnetic_field

EM radiation
http://en.wikipedia.org/wiki/Electromagnetic_radiation
http://hawkins.pair.com/eRadiation.html

Maxwell’s Equations
http://physics.info/maxwell/
http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/maxeq.html

EM Wave equation
http://physics.info/em-waves/
http://hyperphysics.phy-astr.gsu.edu/HBASE/Waves/emwv.html
http://galileoandeinstein.physics.virginia.edu/more_stuff/Maxwell_Eq.html
http://en.wikipedia.org/wiki/Poynting_vector

Near and far fields
http://en.wikipedia.org/wiki/Near_and_far_field

Energy conservation
http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

Continuity Equation
http://en.wikipedia.org/wiki/Continuity_equation

Liénard–Wiechert potential
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

Liénard–Wiechert potential
http://fermi.la.asu.edu/PHY531/larmor/index.html

QED
http://en.wikipedia.org/wiki/Quantum_electrodynamics

Would appreciate any additional links to good references on any aspect of electromagnetism. Thanks

 

[PhilDSP]

PS You may be confusing phase with direction. The equation for a linearly polarized plane wave propagating in the z direction is given by:
$$\mathbf{E}=E_0 cos(k z - \omega t) \hat{\mathbf{x}}$$
$$c \mathbf{B}=E_0 cos(k z - \omega t) \hat{\mathbf{y}}$$
The phase is the term inside the cos function, which is equal for both the E and B, the direction is the vector outside the cos function, which is 90º different.

Yes, of course. I see that now. And apparently neither do the elliptically polarized equations show a general orthogonal relationship between E and B fluctuations. The reason I think it might be important to focus on that is that it seems to indicate that the relationship between the E and B fields isn't responsible for the propulsion of the wave front (a conclusion which your clarification makes clear).

Since there seemed to be some interest in this, the next relationship to look at if we're seeking to identify a propulsion mechanism might be the vector potential. If we assume that the Lorentz condition holds then the relationship between the E field and the vector potential becomes:

$$E = \frac{1}{c}$$$$\dot {A}$$

Which does indicate an orthogonal relationship which could propel a wavefront. Does this make sense?