A physical interpretation of classical electrodynamics

1. Jun 14, 2010

mysearch

I was wondering whether any members of this PF sub-forum would help me towards some physical interpretation of Maxwell’s time-dependent equations, which must ultimately underpin any classical description of EM wave propagation. I know that some might simply suggest reading a good textbook, but I am not sure that anybody can really resolve all their questions, raised by this subject, from a textbook; believed me I have tried. In this initial post, I will avoid any mention of photons and simply want to try confirming or rejecting some of the basic assumptions that seem to lead from the following equations:

$$\frac{\partial\vec{E}}{\partial t} = c^2 ( \nabla\times \vec{B} )$$

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

Basic assumptions:

I have changed the usual orientation of these equations so that the rate of change in time precedes the change in the curl of the field. For it seem that the rate of change in time is the more likely cause and the change in the curl is the effect – see descriptions below.

These equations are based on SI/MKS units, not Gaussian/CGS, as different unit systems lead to ‘c’ appearing in different places. As far as I can tell, the appearance of ‘c’ in these equations acts more as a conversion factor for changing the unit of time to the unit of space, rather than implying any physical propagation

On this basis, the first equation seems to suggest that a change in the E-field, with respect to time, causes a corresponding change to the rate of rotation, i.e. the curl, of the B-field around a point in space. Physically, this can be visualised in the form of the B-field surrounding a wire carrying a current, although the current is only representative of charge movement through a point in space, which can then be described in terms of a rate of change of the E-field, at that point, with respect to time. Sorry to be pedantic about this point, but it seems important to initially correlate the equation to some physical interpretation.

The second equation suggests a similar process, i.e. the rate of change of the B-field with time causes a corresponding change to the rate of rotation of the E-field around a point in space. Again, the process might be visualised in term of a magnet moving backwards and forwards through a wire loop. Therefore, again, when we talk about the rate of change with respect to time, we are selecting a single point in space through which the magnetic is moving.

Basic Model:

By way of a somewhat contrived and possibly hypothetical model, a large static charge [Q] is initially positioned behind an equally large piece of lead, which effectively shields a distant unit charge [q] detector from measuring any E-field from the static charge [Q]. This lead shielding is then very quickly removed.

Basic questions:

Based on the assumption that the distant detector [q] now starts to measure a change in the E-field strength from the charge [Q], albeit subject to the inverse square law, by what means and speed is the change in the E-field thought to have propagated between Q and q?

If the E-field changes at some point in space, would it not also satisfy the criteria of the first equation above, i.e. E changes with respect to time and therefore implies a corresponding change in the curl of the B-field at this point?

Is the previous suggestion problematic in the sense that a magnetic field has been measured without any physical movement of the charge Q with respect to q?

Of course, if the B-field does change at this point in space, does it not also imply a change of B with respect to time and therefore some secondary change in the curl of E-field at this point in space?

2. Jun 29, 2010

AJ Bentley

The reason those equations are normally written as they are is because usually one is trying to calculate B from a changing E, or E from changing B.
E does not change with time because there is a curl in B, its the other way round. The value of the curl of B comes from the change of E

Why lead? It sounds like you are confusing the radiation stopping power of a dense material with an ability to 'block' the electric field. (nothing can block the E and B fields - they are a part of the fabric of space-time)

You've gone too far too fast and need to back up.
Start with the fact that there is such a thing as charge, it comes in two types (+ve & -ve) and these attract/repel.
The attraction and repulsion is felt throughout all space, falling of as r2 and the influence (we call it 'E') is delayed by r/c - that is it appears to spread through space at a velocity c.
(And just because c happens to be the velocity of light, don't get the idea that E is the same thing as light - it isn't.)
When you've mastered E you can start to look at B.

3. Jun 30, 2010

mysearch

Appreciate the response.
Why lead? It seemed a reasonably dense material, which might therefore affect the measurement of the E-field on the other side. Could I clarify a few of the points raised? I wasn’t actually thinking about EM radiation, as I was interested in the measurement and ‘propagation’ of the E-field in isolation. I assumed, it appears wrongly, that if an electric charge was ‘shielded’ by some suitable dense material then the measurement of the E-field at various distances, on the other side of the shielding, would be reduced.

Are you saying that the strength of the E-field is unaffected, even if the source charge is completed encased in any material?

The reason I raised this issue was because I could find very little discussion of the ‘propagation’ of E-field in isolation. Again, I am not thinking of EM radiation, but rather the rate of change of the E-field as a ‘force-at-a-distance’. As such, I was assuming that 'if' the E-field was blocked by some shield, which could be removed (or switched off), then it would take time for any change in the strength of the E-field to propagate outwards. I realised that, according to relativity, [c] is a fundamental restriction to all propagation. Therefore, as this hypothetical scenario is assumed to be taking place in a static/no velocity frame of reference, there would be no obvious magnetic field, although I was interested as to whether there would be a rate of change of E with respect to time as any change in the E-field strength propagated outwards.

Last edited: Jun 30, 2010
4. Jun 30, 2010

AJ Bentley

The electric field concept is of a static 'thing' that fills the universe. In that field are charges, positive and negative that attract/repel each other.
If a charge is moved by some external force, then all of the other charges feel that movement and they also move. However, they move after a short delay that depends on the distance away. And that distance/time delay can be expressed as a speed - which turns out to be c - the speed of light.
(That shouldn't be a surprise because light is caused by the jiggling of charges - so it inevitably goes at the same speed)

Now, I said that the 'thing' fills the universe - it does, and there is nowhere where it doesn't exist. But there are places where it is zero! That isn't a contradiction.
What happens is that you get a configuration of charges in an area that work together to cancel out the field between them, making it flat.

That is what happens inside a charged conductor. Because the charges are free to move, they repel each other and get as far apart as possible - that is they go to the surface - where they further arrange themselves in the most comfortable (lowest energy) positions they can. When they do that - lo and behold - the field between them inside is perfectly balanced. You can take that as zero.

This is a REALLY simplistic description. I haven't mentioned that it's a vector field, or talked about it's counterpart the magnetic field.
As you go further you'll find this isn't even the only description of how it works, there's an even better system that uses vector and scalar potential fields instead - but that needs some heavy duty math to even start to explain.

5. Jun 30, 2010

mysearch

I openly accept that this is the first time I have really started to look at the subject of electrostatics and electrodynamics, so forgive a degree of naivety in some of my questions.
Is it possible to contrive a situation whereby the E-field of a static charge is cancelled out and then ‘switched’ back on so that it propagates a change in the strength of the E-field at the [c]?

If so, wouldn’t this effectively suggest a momentary rate of change of E with respect to time [t], i.e. [dE/dt], at all points in the surrounding space?

If the momentary effect of [dE/dt] is real, would Maxwell’s equations suggest a corresponding momentary change in [dB/dx]?

6. Jun 30, 2010

AJ Bentley

You either know a great deal more than you are letting on or possess a remarkable degree of insight.
What you are describing is electromagnetic radiation or 'light'.

You don't need to 'switch' the field - simply moving a charge rapidly from A to B will cause a pulse in the field. The pulse induces a corresponding B field, which in turn induces a corresponding E field. The disturbance moves through space at c with the E and B vectors constantly dancing around each other at a frequency determined by how fast you move the original charge.

7. Jun 30, 2010

Staff: Mentor

I have no problem with rewriting the equations in any algebraically consistent form, although I wouldn't make the inference about cause and effect that you do.

However, you cannot stop an e-field with an uncharged shield, not using lead nor any other material. That is Gauss' Law. If there is a net charge inside a surface then there will be a net field across the surface.

The conceptual errors you find in your later analysis are primarily due to this non-physical initial condition.

8. Jun 30, 2010

mysearch

One of the issues I was trying to resolve in my mind was the implication that any change in the E-field must naturally propagate at [c], even when it is not directly linked to EM radiation, i.e. charge moving with constant velocity only. At one level, the E-field appears to correlate to a form of potential energy and is associated with energy density, i.e. proportional to E^2. So if [dE/dt] changes as a result of the constant velocity of the charge only, is there not an implication that energy is being transported via this mechanism?
Thanks for the link to Gauss’ law that clarified the situation. So, is it true to say that the only way the strength of an E-field can vary in time [dE/dt], at a given point in space [x], is if the charge source(s) move in space?

As such, if some motion is always involved, then is there always some corresponding change in the B-field strength in the surrounding space?

If [dE/dt] changes at some arbitrary point [x], does this change also generate its own change [dB/dx] at [x] in-line with Maxwell’s equation’s or is the source of the change in B due entirely to the magnetic field of the source charge in motion, analogous to the Biot-Savart law restricted to single charge?

Again, the questions above are initially assuming that the charge only has constant velocity in the current frame of reference.

Many thanks

9. Jun 30, 2010

AJ Bentley

Constant motion in a straight line is constant in any F.O.R.

A charge moving at constant velocity causes a constant B field (which spreads out at c - obviously there's no such thing as constant - it has to start and stop moving at some point)
The E and B fields do in fact have energy - or at least the disturbances in the field do - that's why EM waves have energy - it/they also have momentum - which serves to allow for Newton's second law (action/reaction), which would otherwise have a hard time of it.
(How does the first charge know the second is there so it can feel it's reaction straight away if it happens c/d seconds later?)

Once a charge has moved and initiated a disturbance, it doesn't stop until it's absorbed by moving another charge (usually far more than one). Space is filled with a constantly rippling E/B field with the basic B=dE/dt and E=dB/dt parts of the M. Equations working together.
The other parts of the equations relate to the interaction with charges (charge density, current density)

You can see from this that E and B are in fact not separate things - they are different views of the same thing. The only reason that we use them in that form is because they were developed historically from experiments with static electricity and magnetism down different routes.

These days we have better formulations for the EM field using scalar and Vector potentials based on quantum mechanical ideas.

10. Jun 30, 2010

mysearch

You raised a number of interesting points, which I would like to expand, if possible
Constant motion in one FoR can be stationary in another. If so, the magnetic field would not exist, as resolved by the Lorentz transforms?
If the E-field is viewed as potential energy, this energy exists between 2 charges even if undisturbed. Any disturbance, e.g. movement, causes a change in the potential energy, which I am assuming propagates at [c]? This seems to be a distinct form of energy than that carried by an EM wave?
I thought the issue of action-at-a-distance was resolved in gravity by the curvature of space, as such, it explains the instantaneous effect. I am not how this is resolved in classical electrodynamics?
The format of Maxell’s time-dependent equations in gaussian units:

$$\nabla\times \vec{E} = \frac{\partial{E_y}}{\partial x} = -\frac {1}{c} \frac{\partial{B_z}}{\partial t}$$

$$\nabla\times \vec{B} = -\frac{\partial{B_z}}{\partial x} = \frac {1}{c} \frac{\partial{E_y}}{\partial t}$$

The questions I have about these equations is first: does the presence of [c] have any other inference beyond the system of units in use? Second: [dE/dt] or [dB/dt] only makes sense to me if it is anchored at some point in space, e.g. [x]. So does the corresponding change [dB/dx] and [dE/dx] take place at [x] or in the direction of propagation, i.e. [x+dx]?
This is indeed confusing, especially for people like me trying to understand the physical interpretation that follows from all the equations. For a start, E and B have different units in SI, but the same in Gaussian. They can still be resolved via the electric and magnetic constants, plus have a common link to energy density. However, the E-field seems to exist in all FoR’s, while the B-field only exists in moving FoRs. At one level, I was wondering whether the E & B fields within an EM wave has to play a similar role to potential and kinetic energy conservation in a mechanical wave?
Can you recommend any accessible starter tutorials?

Last edited: Jun 30, 2010
11. Jun 30, 2010

AJ Bentley

Don't confuse zero with doesn't exist.

Stationary is constant motion, the B field is zero. you're just looking at something that exists in n dimensions from the n+1th, sideways.

Careful with assumptions: The E field is not potential energy, that position is reserved for Voltage, which is the spatial integral of E (it's gradient).

Maxwell's equations are correct for special relativity - the constant speed of light is already built-in. But that doesn't alter the fact that a charge would appear to have no way to know how to react. That's why the transfer of energy and momentum are necessary.

Maxwell's Equations pre-date relativity but because they were formulated in an environment where relativistic effects dominate, they HAD to be relativistically correct.
Newton's theory of gravitation and his other laws OTOH were not and had to be modified.

The c factor leads from the fact that the equations are relativistically correct, where you put the c depends on how you set your unit system - clearly if you set your system so that c=1, the term vanishes.
Epsilon and Mu are there because, historically, different units were chosen in the apparently different fields of Electrostatics and Magnetism.
It's no coincidence that Epsilon0 * Mu0 = 1/c^2

dx is an infinitesimal - in the limit it's zero - and the limit is where physical reality exists.

E and B don't translate or even compare to KE and PE.

Try this
http://www.enigmatic-consulting.com/Communications_articles/EnMnofields/The_AEPhi_of_ENM.html" [Broken]

Last edited by a moderator: May 4, 2017
12. Jun 30, 2010

Staff: Mentor

It can also vary in time due to a changing B-field even in a region with no charges.

It is more in-line with Maxwell's equations. The Biot-Savart law is only an approximation to Maxwell and is only valid in the magnetostatic limit.

You might want to read about the Lienard-Wiechert potentials. Those are the full potentials for a point charge undergoing any arbitrary motion.

13. Jul 1, 2010

mysearch

Many thanks for all the thoughtful insights – it has given me a lot to think about.
I will have to think about that :tongue2:
Agreed. I didn’t mean to imply that E was potential energy, only that it is related via the integral of F/q, in a similar fashion as gravitational potential is related to gravitational force. At this point, could I clarify one additional point that you (#2) and DaleSpam (#7) both raised about the ‘pervasive’ nature of the E-field via Gauss’ law, which I hadn’t really thought about. Is the potential associated the E-field also analogous to gravity in that it can’t be blocked by any material? For example, if an electric charge was sealed inside an insulating material, would the electric flux on the outside be unaffected?
Thanks for the clarification. Need to think about the implications of the sentence in bold.
I assume you mean Maxwell’s equation? Given that these were produced prior to relativity, what specific aspect of EM theory led to this implicit result?
I have tried to resolve SI and Gaussian units via the following relationships:

$$(SI) \epsilon_0 E^2 = \eta_E = E^2 (Gaussian)$$

$$(SI) \frac {B^2}{\mu_0} = \eta_B = B^2 (Gaussian)$$

Of course, you then have to work through all the knock-on effects that stem from the difference in the fundamental units of these 2 systems and cause [c] to pop in different places. While initially very confusing, it does seem to help you focus on the implications.
Wasn’t sure whether this limit is where physical reality ended So does [dE/dt] imply a change to [dB/dx] at the same point in space or some offset [dx] that underpins forward propagation?
As we seemed to have established that E is related to a form of potential energy and B is dependent on motion, i.e. kinetic in nature (?), I was making some analogy between the propagation mechanism between mechanical and EM waves. However, I possibly need to explain why I made this comparison a little better. At a fundamental level, waves seem to be a mechanism for dissipating energy within a system that is not in energy equilibrium. In mechanical waves, energy is proportional to the square of the amplitude (A^2), which seems to have some correspondence to the relationships above between E and B with energy density. Of course, EM waves don’t require a medium, which suggest the E and B are the equivalent mechanisms for energy dissipation in vacuum. In mechanical waves, the net result of the energy transported seems to correlate to the input of excess potential energy input into this system over and above some equilibrium point. In a physical media, the wave propagates by converting potential energy to kinetic energy in the wave cycle. Again, there appears to be an analogy in the EM wave, where E must drive B and B drives E. What confuses me somewhat is that E and B are in phase, which is why I am questioning the implication of [dE/dt] on [dB/dx] and [dB/dt] on [dE/dx] to gain a better idea of how the wave actually propagates as an energy system. Sorry to belabour this point.
Thanks this looks like a very useful link from an initial scan.

Anyway, I really appreciate the help as it is easy to get the wrong idea about certain things when you are self-learning a subject. Thanks

Last edited by a moderator: May 4, 2017
14. Jul 1, 2010

mysearch

I am not sure my basic understanding is up to this level, as yet. Initially, I took what seem to be the easier option to look at the Larmor formula, i.e. v<<c, to get some basic idea of where the near (1/R^2) and far field [1/R] components came from. However, I am still having some problems resolving the rules of how electric fields lines are drawn from a moving charge, as opposed to an accelerating charge. However, this is possibly a subject for another thread; maybe one already exists (?) – I will need to check. Thanks

15. Jul 1, 2010

AJ Bentley

I was hoping you'd ask that

Magnetism is simply the consequence of the Lorentz contraction on moving charges.
The relative motion leads to an apparent charge imbalance causing a force that only appears when there is movement.
That's why you are finding yourself immersed in changing frames of reference and the speed of light.

That's the thumbnail sketch - If you want detail, you can't do worse than get hold of Feynman's books, Vol 2 is electromagnetic theory, but get all three and you have everything you need to keep you happy for a very long time.

16. Jul 1, 2010

AJ Bentley

It's related to momentum.

The trouble with the Maxwell approach is that it IS mathematically complex. You have curls and divergences and vectors in all directions, it gets difficult to create a mental picture of a model that fits. Ultimately, that's why I've abandoned that approach (I'm having a hard time remembering this stuff).
The best way of dealing with it - I think what most people do - is to break it into bits and deal with each practical problem separately using tools like Ampere's Law, Coulomb's Law, Biot-Savart etc.
For example, in thinking about EM waves I think most people rarely consider the B part and concentrate on the E, which is what matters most of the time.
As for them being in phase or out - the d/dt would suggest they are out of phase wouldn't it? I can't think of anything that contradicts that conclusion.

17. Jul 1, 2010

AJ Bentley

The E and B fields (if we must have both!) are utterly pervasive in precisely the same way as gravity.

I like to think of them as actually part of space-time itself. Others point out that that position is reserved for gravity and accuse me of speculation. But to me, since the correspondence appears to be precisely 1:1 (relativity remember?). I maintain that they should reconcile ultimately to aspects of the same thing.

18. Jul 1, 2010

Staff: Mentor

http://physics.weber.edu/schroeder/mrr/MRRtalk.html
and the other associated material at:
http://physics.weber.edu/schroeder/mrr/mrr.html

It gives a good flavor for the subject with lots of pictures and a sprinkling of math.

19. Jul 1, 2010

Staff: Mentor

No, in electromagnetic radiation they are in phase, and their ratio is c.

20. Jul 1, 2010

Staff: Mentor

I will warn you again against speculation. There is no mainstream theory where electromagnetic fields are treated as part of spacetime, so your statements to this effect are speculative.

In addition, contrary to your assertion here there are many differences between electromagnetism and gravity, the two most important being that there are no negative gravitational charges and the lowest order of gravitational radiation is quadrupole. Also, the reason that gravity can be geometrized and associated with spacetime is because it is proportional to mass. Electromagnetism is not proportional to mass, so there is good reason to believe that it cannot be geometrized and thus made part of spacetime. Your statements are speculative, unfounded, non-mainstream, and probably wrong.

This forum is not the appropriate place for speculation. There are plenty of other forums on the internet where it is encouraged, but you agreed to the rules when you signed up. If you did not intend to follow the rules then you should not have signed up for your account.

21. Jul 24, 2010

jason12345

I get the feeling that you're trying to get a feel as to how a changing E causes B, and a changing B causes E.

Even though this view is still promoted in some engineering and physics text books, the modern view is that Maxwell's equations do not express causal relationships between E and B since they are measured at the same time and space point. However, you can solve the equations to get E and B as a function of the velocity and acceleration of a source charge, where they both propagate from it at velocity c, with:

B = [ R / |R| ] X E

where R is the position vector from the retarded position of the charge to the field point.

22. Jul 25, 2010

mysearch

That’s a fairly good way of putting it. Basically, I don’t want to simply learn the equations by rote without some intuitive understanding of the physical process at work. Therefore, in my attempt to self-learn about Maxwell’s equations from scratch, I was puzzled that so few texts discussed the propagation of E and B fields in isolation, as any mention of the word ‘propagation’ seemed to assume EM waves or radiation predicated on the charge particle accelerating in some way. However, a charge particle moving with only constant velocity would also appear to be changing the electric potential in the surrounding space and this change also propagates at [c].
I think I understand the implication of your statement about the causal relationship at a single point in space. However, if you again restrict the discussion to the specific case where a charge is moving with constant velocity [v] in space, which I am assuming causes the E-field to change in the surrounding space tied to a propagation delay, i.e. dt=x/c, then there would appear to be a causal factor between the field at a given point in space and its source.

$$\frac{\partial\vec{E}}{\partial t} = c^2 ( \nabla\times \vec{B} )$$

In the context of the points raised above, I was uncertain how to interpret Maxwell’s time dependent equations. For the suggestion is that any change of E with time, at a given point in space, causes a corresponding change in B in space at that point. However, if this were the case, I would have thought this would cause further field changes. Again, if we ignore self-propagation of EM waves or radiation, as we are only considering constant velocity, the basic classical equations seem to suggest that both the E and B fields, at any given point in space, are defined as a function of distance [R], which is again subject to a finite propagation delay due to [c].

$$E= \frac {Q}{4 \pi \epsilon_0 r^2}$$

$$B= \frac {Q}{4 \pi \epsilon_0 r^2} \frac {v^2}{c^2}$$

This leads into what is really another area of discussion based on the extension of the equations above to consider relativistic effects. However, after looking at the derivation of this equation, it was unclear whether it really accounts for the propagation delay implied above.

$$E= \frac {q}{4 \pi \epsilon_0 r^2}* \frac {1- \beta^2}{(1- \beta^2 sin^2 \theta)^{3/2}}$$

Anyway, I appreciate your feedback. Thanks

Last edited: Jul 25, 2010
23. Jul 25, 2010

Staff Emeritus
That's because they are fields. Fields are defined as having values for all points in space at all times.

24. Jul 25, 2010

Antiphon

Dalespam, I think you're off the mark here. AJ Bently is not speculating about theories but voicing his interpretetation of possible mechanisms of utterly mainstream theories. Furthermore he is right on the money. None other than Einstein himself spent the better part of his professional career persuing the line of research which AJ described. Would you ban professor Einstein from this forum?

The idea that Maxwell's equations are intimately connected to space-time has been around for a long long time. Kaluza-Klein equations are the 5-dimensional formulation of general relativity with an additional space dimension. Guess what? You get the equations of general relativity in 4+1 dimensions together with Maxwells equations. I'd say that's a powerful reason to suggest that some aspect of AJ Bentley's speculations about spacetime and the EM field may be right on.

25. Jul 25, 2010

Antiphon

Dalespam is right, they are in phase by Poynting's theorem. The time derivatives are out of phase with the curls by 90 degrees. The curls are out of phase with the fields by another 90. This gives you a net 180 degrees between the the two fields which just gets lost in the minus signs.