# Which came first, E or B?

1. Jun 6, 2010

### mysearch

What came first E or B?

This question is in the style of the chicken or egg, but I would be interested to hear any views as to whether the electric field is the primary source of energy and whether the magnetic field is essentially just a manifestation of this energy linked to a moving charge?

If we look at the units of energy $$(kg.m^2/s^2)$$ and the electric field $$(kg.m/Cs^2)$$, we have what seems to be a logical association, i.e. the electric field appears to align to force per charge or energy.metre per charge. If you apply the same logic to the magnetic field (B=kg/Cs), the association with energy is less clear until you add velocity by virtue of the relationship E/B=c. Of course, the relativity of a moving frame of reference also seems to indicate that any measure of a magnetic field is also relative.

So what is an EM wave and how do we measure the E and the M strength?

It would seem the Lorentz equation of force $$F=q(E+vB)$$ provides a suggestion that a test charge would be subject to a force if place in the path of an EM wave, but is E driving B or B driving E at all points in space based on Maxwell’s 3rd and 4th equation in vacuum, i.e.

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

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

I have swapped the ordering of these equations, as compared to post #10 in this thread, because I would have thought that the curl of E is the 'effect' resulting from the 'cause' being the rate of change of B with time, at least, in the specific case of a propagating EM wave. A similar argument applying to the 4th equation. However, if E and B are both in-phase in a propagating EM wave, is the rate of change of E with time still the primary physical force/energy at work here?

Would be interested in any clarifications of the points raised and I hope this question is in-line with the purpose of this thread.Thanks

Last edited: Jun 7, 2010
2. Jun 6, 2010

### Staff: Mentor

The components of E and B are simply different components of the relativistic electromagnetic field tensor, and transform among each other between different reference frames according to the Lorentz transformation. So I don't think it's meaningful to say that one "comes first" or is "more important" than the other.

3. Jun 6, 2010

### cabraham

Under time-changing conditions, neither can exist w/o the other. Neither "comes first". You mentioned that this question sounds like the chicken-egg riddle. I agree because that is what it is.

E & B are best described as mutually inclusive under ac conditions. Under static (dc) conditions, a superconducting loop has a B w/ zero E. A perfest dielectric capacitor can have an E w/ zero B. However as soon as one of them is changing w/ time, the other has to be non-zero.

Claude

4. Jun 6, 2010

### Bob S

E came with the first electrons.

B came with the first moving electrons.

So the question might be whether fields or charged particles came first.

Bob S

5. Jun 7, 2010

### mysearch

Revised title: What is the source and nature of EM energy?

Possibly the reference to the ‘chicken and egg’ was not a good idea as I was primarily interested in trying to better understand the source of energy and the nature of its transfer within a system as a whole. As a generalisation of classical physics, energy is said to come in 2 basic forms, i.e. potential and kinetic. The force equations between 2 charges and 2 masses share a very similar form based on the inverse square law. Therefore, if you integrate them with respect to distance, the implication seems to be that gravity and the electric charge field both relate to potential energy. Now within this classical model, we might consider a system that consists of a large central positively charged mass and a much smaller negatively charged mass falling radially towards the centre from a great distance. At the start, it would seem that the negative charge can tap both gravitational and electric field potential energy, but in doing so, it acquires kinetic energy, i.e. velocity, and as a moving charge it would seem to imply that a magnetic field would now surround (axial symmetry) the moving negative charge. Equally, according to many standard texts, this negative charge would also be accelerating and therefore must be emitting EM radiation, which Maxwell’s equations seem to support as a wave, but now also implies the quantum nature of the photon.

So, at one level, we seem to have started out with just the gravitational and electric field potential energy, ignoring the mass energy, but end up with kinetic energy of motion, a magnetic field that can exert a force and therefore presumably has the ability to do work (energy) and energy being radiated as EM waves/ photons.

So can this apparent complexity still be resolved into the 2 basic forms of energy, i.e. potential and kinetic?

Of course, there is the caveat concerning the reference frame of the observer. For example, from the inertial frame of a free-falling observer collocated along side the small negative charge, its velocity and acceleration might be perceived differently, but this is possibly a subject of another thread.

So if we now just focus on the implication of the EM radiation, which was assumed to be the result of acceleration on the charge, it seems that this radiation is energy, either in the form of an EM wave or photons, which is now being transported independent of the source from which it originated. So another question I was trying to address was:

If you could position a series of super-sensitive E-M field detectors along the path of this propagating energy, what would you expect to measure, i.e. how would you expect the E-M field strengths to vary with time and distance along it path?

Thanks

Last edited: Jun 7, 2010
6. Jun 7, 2010

### Tomsk

The energy density of the EM field is $1/2(|E|^2 + |B|^2)$ and the Lagrangian density is $1/2(|E|^2 - |B|^2)$ so in some sense the E field is the kinetic part and the B field is the potential part. I'm not sure that answers you question though?

7. Jun 7, 2010

### mysearch

Thanks for the response. Not sure I understand how you are drawing your conclusion about the E-field being the kinetic part and the B-field the potential. I have not covered the Lagrangian density, so the inference of the minus sign that separate the 2 equations is lost on me. I understand the general concept of the energy density, but would like to clarify the scope of the form you quote based on my current understanding. The following equations are representative of the logic I have been looking at:

[1] $$\eta_E = 1/2 \epsilon_0 E^2 = \frac{B^2}{2 \mu_0} = \eta_B$$

Equation [1] also leading to E/B=c and the total energy density:

[2] $$\eta = \eta_E + \eta_B = 1/2 (\epsilon_0 E^2 + \frac{B^2}{\mu_0})$$

The equality of the E & B contributions in [1] allows [2] to be reduced to:

[3] $$\eta = \epsilon_0 E^2$$

I have understood this to be the instantaneous energy density, which if associated with a sinusoidal waveform would reduce to the average energy, i.e. 1/2:

[4] $$\eta_{AV} = 1/2 \epsilon_0 E^2$$

The energy per second, i.e. the power, crossing some given area perpendicular to the perceived flow of energy is defined by the intensity (I) and the expression:

[5] $$I = v \eta$$ where v=c for an EM wave

I understand that intensity corresponds to the magnitude of the Poynting vector such that:

[6] $$I = S = c \epsilon_0 E^2$$

Would appreciate any corrections to these assumptions. many thanks.

P.S. I am not assuming the EM wave conforms to a sine wave in the example outlined in #5.

8. Jun 7, 2010

9. Jun 7, 2010

### mysearch

Thanks for the link. The maths looks a bit daunting at first glance, so any secondary links that provides a little more detail of the principles would be much appreciated. However, I did find the following statement very interesting:
Is this statement questioning the accepted mechanism of how an EM wave is thought to self-propagate through a vacuum?

10. Jun 8, 2010

### cmos

Not exactly.

A more fundamental elucidation of Jefimenko's argument is the "proper" way of writing the two curl equations. Should it be 1)

curl E = -(1/c) dB/dt
curl B = (1/c) dE/dt + (4*pi/c) J

or should it be 2)

curl E + (1/c) dB/dt = 0
curl B - (1/c) dE/dt = (4*pi/c) J ?

Obviously these are the same mathematically. But physically.....

For simplicity, let's set J=0. So the way it's commonly explained (especially to undergrads) is to take Set 1. Then it follows that time-varying fields act like sources; sources for curling fields. In other words, a time-varying B-field sets up an E-field. Setting up the E-field takes time which thus sets up a B-field and so forth. This self-manifestation is thus manifested as wave propagation. Note that in this case we have two singular entities, the E-field and the B-field.

If you take Set 2 to be the "proper" way of writing Maxwell equations, then the interpretation is as follows (again, let J=0 for simplicity). The electromagnetic field (i.e. a single dual entity) exists in such a way that curl E must cancel dB/dt. Similarly, curl B must cancel -dE/dt. The way in which these quantities cancel themselves when non-zero is manifested as wave propagation.

So the second argument is along the lines of what Jefimenko is stating. The end results, including the propagation of waves in charge/current-free space, is not changed by choosing either of the two interpretation. Note that (when you include Gauss' law) in Set 1, the sources are charge and current while the time derivatives act as quasi-sources. In Set 2, the only sources are charge and current.

11. Jun 9, 2010

### mysearch

Many thanks for the comments in #10, as they have triggered a number of questions in my mind. However, I was first hoping that I might clarify a few issues concerning the system of units used to describe Maxwell’s equations. Basically, we seem to have a number of variant systems of units in use, i.e. CGS, MKS, MKSC, SI. I guess I am ‘old school’ in the sense that I like to understand units in their most basic form, i.e. length, mass, time and charge. Broadly speaking SI was developed from MKSC, but likes to honour famous scientists by naming units after them, which is fine although not particular helpful to basic understanding. I raise this issue because I am getting confused by all the variants of Maxwell’s equation, not just between differential and integral, but the different forms that presumably result from the assumptions of units in use. For example, if we focus on Maxwell’s 3rd and 4th equations in differential form in vacuum, where charge density [p] and current density [J] go to zero, I have seen the following forms:

$$\nabla\times \vec{B} = \frac{1}{c} \frac{\partial\vec{E}}{\partial t}$$
$$\nabla\times \vec{E} = -\frac{1}{c} \frac{\partial\vec{B}}{\partial t}$$

This seems to be the form adopted in post #10, but does not appear to align to the SI units as described in the Wikipedia page on Maxwell’s equation – see link below as reference. In fact, the form appears to align to CGS units as shown in the same link near the end of the page.
http://en.wikipedia.org/wiki/Maxwell's_equations

In SI units, the equations above are presented in the following form:
$$\nabla\times \vec{B} = \mu_0 \epsilon_0 \frac{\partial\vec{E}}{\partial t}$$
$$\nabla\times \vec{E} = -\frac{\partial\vec{B}}{\partial t}$$

However, on the basis that:
$$c= \frac {1}{\sqrt{ \mu_0 \epsilon_0} }$$

I would have thought that the first of the previous equations could be written as:
$$\nabla\times \vec{B}= \frac{1}{c^2} \frac{\partial\vec{E}}{\partial t}$$

The reason for raising this detail is that it is not always easy to understand, whether the presence of the speed of light signifies a ‘variable’ that has some physical inference or is being used as a ‘constant’ to balance the underlying units. Any clarifications welcomed. However, returning to the insights raised in post #10:
I am specifically interested in the physical interpretation of these equations. From a classical perspective, I understood the E-field can be said to exist independently of the B-field when viewed as a static field, i.e. no charge motion. When you set the charge in motion, this E-field must change in time and space and, as such, generate a B-field? Therefore, I would assume this B-field must also being changing and as a result generating a secondary E-field? I don’t know whether this secondary E-field acts in opposition to the primary E-field, due to the conservation of energy?, causing the net E-field to fall and thereby perpetuating the rate of change?

On the basis that E/B=c and that E and B are always in-phase and proportional to each other, then ‘c’ would remain constant for all values, except when E=B=0. I don’t really understand the implications of this crossover point.
Ignoring the ambiguity of relative motion, from a ‘chicken & egg’ perspective, it would seem that charge is a fundamental unit from which the idea of an E-field is extrapolated. As I outlined in #5, the E-field appears to align to a form of potential energy. The movement of a charge due to either gravitational attraction or charge attraction/repulsion would convert the potential energy into kinetic energy and create a magnetic field around the charge moivng relative to a stationary observer. The units of a magnetic field only align to the units of force when multiplied by velocity and, as such, imply an associated energy.

As always, would appreciate any corrections to my current misunderstanding of this subject, which is still a learning process. Many thanks

12. Jun 9, 2010

### phyzguy

The simple answer to your question on units is that the E and B in the CGS or Gaussian units are not the same E and B as the E and B in SI units. In Gaussian units, E and B have the same units, so I can write the energy density, for example, as $$\frac{1}{8\pi}(E^2 + B^2)$$. However, in SI units, E and B are measured in different units, so the energy density is $$\frac{1}{2}(\epsilon_0 E^2 + \frac{B^2}{\mu_0})$$
I know this can be confusing, but you just need to learn it and live with it.

Last edited: Jun 9, 2010
13. Jun 9, 2010

### mysearch

Thanks for the general insight in #12. To be honest, given that SI/MKS was a standard being adopted when I went to school in the 1960’s, it didn’t occur to me that so many variants were still in use 50 years later. I guess the old quote “the nice thing about standard is there so many to choose from” still applies.
Fair enough. The most coherent summary I could find was here:
http://nlpc.stanford.edu/nleht/Science/reference/conversion.pdf
Based on the table on the first page, I am assuming that the Gaussian unit of charge, i.e. the Franklin, differs from SI units by the units of the speed of light. Therefore, if SI units are the Coulomb, the Franklin has units of C*m/s.

The SI units for B are the Teslas, while the Gaussian units are the Gauss. However, the underlying units seem to be equivalent as the conversion factor is just numeric. As such, the units can be equated to kg/Cs.

The fundamental SI units for E are m.kg/Cs^2, which are divided by ‘c’ in the Gaussian system and become equivalent to kg/Cs, which means the E and B have the same units in the Gaussian system.

Presumably, the SI relationship E/B=c reduces to 1 in the Gaussian system?

So to restate the position outlined in post #11. The following equations align to the gaussian/cgs system?
$$\nabla\times \vec{B} = \frac{1}{c} \frac{\partial\vec{E}}{\partial t}$$

$$\nabla\times \vec{E} = -\frac{1}{c} \frac{\partial\vec{B}}{\partial t}$$

The following equations align to the SI/MKS system?
$$\nabla\times \vec{B} = \mu_0 \epsilon_0 \frac{\partial\vec{E}}{\partial t} = \frac{1}{c^2} \frac{\partial\vec{E}}{\partial t}$$

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

If this is correct, could I confirm whether presence or absence of ‘c’ has any real physical significance or is it simply required to rationalise the underlying units?

Any other insights into any of the other issues raised in post #11 or the thread as a whole would also be appreciated. Thanks

14. Jun 9, 2010

### LostConjugate

So if the E and B fields are coupled, and the B field can only interact with a charge, then couldn't the B field just be written as components of the E field and eliminated altogether.

For example you could have a loop current and say there is a B field perpendicular and any electron in the field will precess.

Or you could say there is a loop current and any electron perpendicular to the current will precess because of the accelerating E field emitted by the charge as it loops.

I attached a doodle of what I mean. A movement of a charge can be explained without the B field, only the E field.

If e1 starts to rotate, e2 starts to precess because of the new component in the E field.

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15. Jun 10, 2010

### cmos

Regarding the presence/absence of c in the Maxwell equations:

It's just a matter of the system of units; i.e. where c shows up will make the units work out. Note however that c MUST appear somewhere in the Maxwell equations. It does not have to be explicit; for example, in SI units, the Maxwell equations are written with two other constants, viz. the permittivity and permeability of free space. These two constants together give the speed of light. It is completely allowable to make your own system of units with with n constants. These constants must, in some way, come together to give c.

Note that, regardless of choice of units, the derivation of the wave equation will always yield the same equation. This gives some motivation for the foregoing discussion.

Interesting note: In the Gaussian system, we have zero additional fundamental units added to length, mass, and time. We then have only one physical constant in the Maxwell equations, c. In the SI system, we have one additional fundamental unit, current. When then have two physical constants in the Maxwell equations, permittivity and permeability. I wonder if this trend continues?

16. Jun 10, 2010

### cmos

Regarding implications of the crossover point E=B=0:

You correctly stated that (in SI units) E/B=c when E and B are non-zero. Let's write the left hand side as E(B)/B. Then the limit of this quantity as B tends to zero gives c.

17. Jun 10, 2010

### cmos

Your argument seems flawed to me. Let's say that we place a charge at rest in the same area you draw the precessing charge in your picture. Then, by your argument of an "accelerating E-field," the charge must move. What must happen though is that the charge does not move since v X B = 0.

18. Jun 10, 2010

### mysearch

I appreciate the feedback in #15 & #16. To be honest when the issue of all the variant systems of units came up, my initial reaction was that the last thing that this subject needed was unnecessary complexity in describing the same thing in different ways. However, it does seem to raise a key issue about how ‘we’ come to use composite quantities, such as energy, force, fields etc, without always considering how these composite quantities are specified in the first place in terms of more fundamental quantities. I realise that I am going off topic here, but would be interested in any other perspectives,

While some might argue that length and time are not separate fundamental units in terms of spacetime, most of basic physics seems to accept/require these units. At one level, we might describe these units as the ‘stage’ on which the ‘actors’ perform. The ‘actors’ in this case being the other 2 of 4 fundamental units, i.e. mass and charge. Again, it is possible to question the fundamental nature of these units, for example, if we equate $$E=mc^2=hf$$, it might be argued that ‘h’ and ‘c’ are both constants and therefore the implication is that mass is somehow equivalent to frequency. On a slightly more philosophical note, we might also question the very nature of a particle at the sub-atomic level, where we are unable to name the substance the ‘particle’ it is made of. As such, frequency may be a more fundamental concept than mass. Still, at some level, we have the tangible perception of ‘particles’ and therefore the concept of mass becomes a useful unit of containment. Of course, such ideas would also cause problems for charge, which classically is considered to be an attribute of a mass particle, i.e. electrons and protons. There is also the additional problem that charge cannot really exist in isolation, i.e. it only a concept that exists between 2 ‘charged particles’.

Of course, even if we accept the 4 fundamental units at face value, we still need to introduce the concept of force $$(m.kg/s^2)$$ and energy $$(m^2.kg/s^2)$$ to describe how the action takes place. However, reverting to the fundamental units does not really seem to help describe these composite ‘quantities’. From my perspective, I tend to consider force as something that results from energy and energy as something that ultimately comes in 1 of 2 basic forms, i.e. potential and kinetic. In this context, potential energy might simply represent a system that is not in energy equilibrium and kinetic energy is the manifestation of a system trying to restore equilibrium, either via particle motion or wave motion. Of course, if the particle’s mass is related to frequency, then maybe there is only one real mechanism at work.

In part, I was touching on these points in post #5, when asking about the fundamental nature of the energy driving EM radiation. From the simplicity of a hypothetical model of 2 opposite charge particles being pulled to together from an ‘infinite’ distance, it might be argued that the only energy initially available was the potential energy of the gravitational and the electric charge fields. As the small charge particle ‘falls’ toward the much larger central charged particle, it acquires kinetic energy plus a magnetic field relative to a stationary observer and radiates energy due to acceleration. However, I am still not sure whether this last point is the subject of debate or simply goes beyond the scope of classical physics?

OK, with this said, the reason I started the thread was because I was struggling to understand how Maxwell’s equation really explained the interaction between the E and B fields in a self-propagating EM wave in vacuum, not just as a mathematical equation, but as a physical process. One of my first issues was how to interpret Maxwell’s 3rd and 4th equations, as presented in post #13 in respect to the presence of the speed of light:
The suggestion seems to be that the presence/absence of ‘c’ is not directly implying any propagation velocity, but possibly more in-line with its use in $$E=mc^2$$, it suggests an association with the speed of light required by the fundamental units in use. If so, Maxwell’s 3rd and 4th equations only seem to suggest that a change in the E-field, at some point in space, causes a rotational B-field, associated with with the curl, which then causes a similar effect with respect to the E-field and so on. However, there doesn’t seem to be an explicit description of how this change propagates. The point I am making goes back to the issue that an E-field only exists between 2 charges, which must already have connected field lines.

So can any change to the position of a charge be thought to propagate along ‘pre-existing’ field lines at the speed of light in order to ‘communicate’ the new field strength, which has to fall in-line with the inverse square law?

If this were the case, wouldn’t this correspond to a transfer of potential energy?
I recognise that I gone on for too long, but in some ways I just wanted to table some issues for my own future reference. Thanks

19. Jun 10, 2010

### PhilDSP

It sounds like you would be right at home with J. J. Thomson's later day investigations because he focused intensely on that. But he agreed with de Broglie that the particle is separate from the EM wave - that $$hf$$ is certainly not implicit in the particle but is indicative of the particle's travel through space or the medium.

20. Jun 10, 2010

### LostConjugate

How would it not move? There is an E field from e1 that would give it an acceleration in the upward direction (since electrons repel) and also since there is a change in the E field the electron would begin to follow the field.