# Interpreting Maxwell

1. Apr 6, 2010

### 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?

2. Apr 6, 2010

### 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.

Last edited: Apr 6, 2010
3. Apr 6, 2010

### Staff: Mentor

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.

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.

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

4. Apr 6, 2010

### mysearch

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

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

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?.

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

5. Apr 6, 2010

### fabpan

An example could be the cosmic background radiation.
I'm always astonished when I think that it has been travelling 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.

Last edited: Apr 6, 2010
6. Apr 8, 2010

### 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 [Broken]

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 [Broken]

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

Last edited by a moderator: May 4, 2017
7. Apr 8, 2010

### PhilDSP

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.

8. Apr 8, 2010

### Staff: Mentor

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.

9. Apr 8, 2010

### mysearch

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:

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.

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?

Last edited: Apr 8, 2010
10. Apr 8, 2010

### 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.

11. Apr 8, 2010

### 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 ?

12. Apr 8, 2010

### Born2bwire

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.

13. Apr 8, 2010

### Staff: Mentor

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.

14. Apr 8, 2010

### mysearch

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

15. Apr 8, 2010

### Born2bwire

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.

16. Apr 8, 2010

### PhilDSP

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.

17. Apr 8, 2010

### Staff: Mentor

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.

Last edited: Apr 8, 2010
18. Apr 9, 2010

### mysearch

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 travelling 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.

Last edited: Apr 9, 2010
19. Apr 9, 2010

### Staff: Mentor

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

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

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.

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.

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

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

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.

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.

Last edited: Apr 9, 2010
20. Apr 10, 2010

### bjacoby

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.