[lightarrow]

I think I can prove that E and B are in quadrature using Maxwell's Theory of Electromagnetism:  A changing electric field generates a magnetic field of a strength that is proportional to the change in the E field.(Ampere's Law).  A changing magnetic field generates an electric field of a strength that is proportional to the change in the B field. (Faraday's Law)..

Not exactly.

Faraday's Law:

rotE = -∂B/∂t

--> the line integral of E along a closed loop is equal to minus the time variation of the magnetic flux through that loop.

Ampere's Law in the void:

rotB = (1/c²)∂E/∂t

--> the line integral of B along a closed loop is equal to (1/c²) the time variation of the electric flux through that loop.

[syhprum]

I think we must hold a vote on this matter I have been converted I side with lightarrow

[lyner]

Lightarrow's sums must be right. (He IS pretty reliable in this direction)
SO there is a problem with my interpretation of the energy flow in a wave.
No one has argued with my idea that all other waves consist of PE and KE variations which are in phase quadrature. So put me straight: is not the E field a potential form of energy and is not the B field a dynamic / kinetic form of energy? The fact that they are transverse and spacially in quadrature makes the em different from other waves, I admit but em waves follow the general principle of phase quadrature when guided by a wire / wires / waveguide so what is the difference, once they get launched into space (or a medium; it all can't change just because you have the occasional molecule of air in the way)?
Help me with the physical interpretation of this - there must be one which can reconcile my misgivings - unless the in-phase idea is wrong (god I would so like it to be wrong! Just think of all the  experts being gutted! Even Lightarrow!!!)

[syhprum]

I think the difference lies in currents in conductors dragging electrons around and currents in the void being free of that in-pediment.
In my original question I spoke of the B & H phase difference in the currents in the antenna and them falling into phase when they got a quarter wavelength into the void.
I seem to recall that is what Sterling was telling us in his 1930's textbook

[lightarrow]

Lightarrow's sums must be right. (He IS pretty reliable in this direction)
SO there is a problem with my interpretation of the energy flow in a wave.
No one has argued with my idea that all other waves consist of PE and KE variations which are in phase quadrature. So put me straight: is not the E field a potential form of energy and is not the B field a dynamic / kinetic form of energy?

Sincerely I have never heard anything like that, in an EM radiation, but, who knows; what I know is that energy in an EM wave can be computed from the Poynting vector S = EXH where X means vectorial product: if N is the versor normal to a certain surface then S•N; • = scalar product, is the power per unit area of the EM wave flowing through that surface.

The fact that they are transverse and spacially in quadrature makes the em different from other waves, I admit but em waves follow the general principle of phase quadrature when guided by a wire / wires / waveguide so what is the difference, once they get launched into space (or a medium; it all can't change just because you have the occasional molecule of air in the way)?
Help me with the physical interpretation of this - there must be one which can reconcile my misgivings - unless the in-phase idea is wrong (god I would so like it to be wrong! Just think of all the  experts being gutted! Even Lightarrow!!!)

However let's remember that near a cicuit E and B can be independent one of the other: E comes from charges and B comes from currents and you can make a circuit that will generate the E that you want and the B that you want; you can't do it in an EM wave: E and B are one *dependent* on the other, infact one is generated from the other and not from specific charges/currents configurations.
Anyway, let someone more expert come here and tell us the truth...

[Mr Andrew]

lightarrow, I am aware of the mathematical formulation of the two laws that I referred to.  They mean exactly what I said, except that I didn't mention that the fields that these changes in E or B create are in a direction perpindicular to the direction of the wave's propagation (if the left was nabla•E or nabla•B then the fields generated by the changing fluxes would be in the same direction as c).  I took it as a given that the fields were perpindicular to the direction of propagation as light is a transverse wave.  I know that your calculations are correct (they appear many different places and I have taken the time to check the math, assuming that rot(rotV))=grad(divV)-nabla²V is true, they work out), my problem is that this solution seems to be just as infallible as yours, yet only one can be right.

Does anybody see what's wrong here?

[Soul Surfer]

I think that I have the solution to the problem of phase. Light arrow has not quite completed the analysis.  He has shown that the electric fields and magnetic fields stay in the same phase with respect ot each other but he has not proved exactly what that phase is  (There is always a constant in the integration ) to do that you have to go back to the original equations to determine the constant.  If you do that you will find that is where the phase difference lies.

[lightarrow]

I think that I have the solution to the problem of phase. Light arrow has not quite completed the analysis.  He has shown that the electric fields and magnetic fields stay in the same phase with respect ot each other but he has not proved exactly what that phase is  (There is always a constant in the integration ) to do that you have to go back to the original equations to determine the constant.  If you do that you will find that is where the phase difference lies.

But what you say it's not possible: to have a phase difference Φ between E and B, you should have:

E_y = E₀e^{i(kx - ωt)}

B_z = (1/c)E₀e^{i(kx - ωt + Φ)} = (1/c)e^iΦE₀e^{i(kx - ωt)}

so the phase factor would become a multiplicative constant, not an additive one.

[lightarrow]

lightarrow, I am aware of the mathematical formulation of the two laws that I referred to.  They mean exactly what I said, except that I didn't mention that the fields that these changes in E or B create are in a direction perpindicular to the direction of the wave's propagation (if the left was nabla•E or nabla•B then the fields generated by the changing fluxes would be in the same direction as c).  I took it as a given that the fields were perpindicular to the direction of propagation as light is a transverse wave. I know that your calculations are correct (they appear many different places and I have taken the time to check the math, assuming that rot(rotV))=grad(divV)-nabla²V is true, they work out), my problem is that this solution seems to be just as infallible as yours, yet only one can be right.

Does anybody see what's wrong here?

Let's take Faraday's Law:

rotE = -∂B/∂t

integrating on a surface σ:

∫rotE•dσ = -(∂/∂t)∫B•dσ

using Stokes theorem:

∫E•dl = -(∂/∂t)Φ_B

where: the first integral is computed on the edge of the surface σ, dσ = Ndσ where N is the versor of the element of area dσ and dl is the line element of that edge; Φ_B is the total flux of B through σ.

Now let's take as σ a disk with radius r, so the edge is the circle of radius r; if B is uniform through the disk and perpendicular to it, for cylindrical symmetry the field E in the circle is tangent to it and with uniform intensity E and we can write:

2πrE = -πr²∂B/∂t  -->

--> E = -(r/2)∂B/∂t

For a sinusoidal time-varying B, that is, B = B₀sin(ωt) we have:

E = -(r/2)ωB₀cos(ωt)

and so E is 90° out of phase in time, with respect to B, as you say.

Note however the key hypotesis: if B is uniform through the disk; in our case instead, B is not uniform (it varies sinusoidally also with space and not only with time), so we have to use Faraday's Law in the differential form:

rotE = - ∂B/∂t

and so we have to compute also the spatial derivatives of E; for this reason the "i" factor comes in both members of that equation and so the phase is the same (I remind you that a factor "i" equals a 90° phase difference: i = e^iπ/2).

However, in the post where I computed E and B for an EM wave, I should have said that I took a plane polarized wave (because that's more simple); I'm not sure of what could come out if the wave wouldn't be that way, actually.


About:

rot(rotV))=grad(divV)-nabla²V

it's not very simple to prove it, but it's a well known mathematical rule; you can find it in (good) electrodynamics books.

[lightarrow]

Lightarrow
...
What is your reaction to the 'conservation of energy flow' idea? To me, that sounds like a clincher. It applies to all other sorts of wave, so why not em waves?

I have thought about it but I don't know how to solve this; I don't know if this flow of energy does really have to be constant or not and what it could mean. Interesting question however!

[JP]

There's no reason why the Poynting vector should be constant.  In fact, for a plane wave, it has a sinusoidal form: Re{S}~Cos[2(kx-ωt)].  Time-averaging it gets rid of this time dependence, and that's the form that's usually used.

I think the KE/PE in quadrature is going to arise from the KE/PE relations of the propagation medium itself because you're treating the propagation medium as a set of simple harmonic oscillators.  In such propagation, describing the waves in position and in momentum will lead to two expressions out of phase by Pi/2.  You can do an analogous thing with the electromagnetic field, but it's a relation of the field to its derivative, rather than between the E & B fields.

[lyner]

Does the solution to this conundrum lie in confining the operators to the real parts at each stage in the chain of calculations?
After each 'calculus' operation there are other  products which are, perhaps, 'not really there'. So the E field, generated by the varying B field has to be real - etc.
My maths is so rusty that I couldn't rely on my calculations to prove or disprove anything but perhaps someone could go through the steps, eliminating non-real bits at each stage and see what comes out of it?
I know we blindly accept the complex form of wave representation  in calculations and then say 'just take the real part' at the end. Is it really justifiable? I may be very naive in asking that question.

Else it may be reconcilable by jpetruccelli's ideas (above). It would imply that the higher the refractive index / density of the medium, the more in quadrature the fields would be. Does Maxwell produce that result?

[syhprum]

In the real world linear polarized EM waves are a rarity we only meet them when they are generated by lasers or electronics.
When I want to generate circular polarized waves I feed them into a helical antenna with a spacing of 1/4 wavelength between the turns and a circumference of 1 wavelength.
It strikes me that the rate of rotation is very high.
Do these calculations deal with circular polarized waves or only linear.

[Soul Surfer]

Light arrow you surprise me.  You seem to be so familiar with the mathematics yet seem to forget the cyclic properties of the imaginary exponent.  Remember 

exp^iw = cos w +i sin w  if you put your equations in that form it looks perfectly sensible

There is absolutely no problem with the multiplication.

[Soul Surfer]

Going back to the original equations you will see that  the magnitude of the curl of the magnetic vector  is proportional to the rate of change of the electric vector which for a sinusoidal waveform implies quadrature this is your constant of integration.

similarly  the magnitude if the curl of the electric vector is proportional to the (the negative) of the rate of change of the magnetic vector. 

It is important to understand the physics as well as the mathematics it is the changing electrical fields that create the magnetic fields and the changing magnetic fields that create the electrical ones. 

Alternatively  at a peak point in the electrical or magnetic fields there is a momentary point where the partial derivative with respect to time of the relevant electric or magnetic field is zero. it follows directly from the equations that the value of the corresponding value of the magnetic or electric field must be zero.  This again implies that for a simple propagating plane polarised wave the relative magnitudes of the fields are in quadrature.

[lyner]

n the real world linear polarized EM waves are a rarity we only meet them when they are generated by lasers or electronics.

What about every mf radio and uhf TV broadcast, then?
No only are they common but their phases can be measured a lot easier than light waves!

[lyner]

Do these calculations deal with circular polarized waves or only linear.

A circularly / elliptically polarised wave can be regarded as a pair of plane polarised  waves with E vectors at right angles and with their phases in quadrature then exactly the same calculations can be done on each component, independently. Yes, the rate of rotation is high - it's  2pi radians of rotation per cycle.
You can generate good circular polarised waves (on one axis) using crossed dipoles with a 1/4 wavelength delay in the feed to one of them. The off axis polarisation of a helical antenna also goes elliptical.

[Mr Andrew]

Soul Surfer, that's exactly what I said!

lightarrow, I don't understand how i represents a phase difference of 90 degrees.

[syhprum]

n the real world linear polarized EM waves are a rarity we only meet them when they are generated by lasers or electronics.

What about every mf radio and uhf TV broadcast, then?
No only are they common but their phases can be measured a lot easier than light waves!

These are of course the EM waves I had in mind as being generated by electronic devices (Vacuum tubes, Klystrons, Transistors etc) but in the wider universe most of the radiation is generated by thermal sources with ill defined frequency and polarisation

[lyner]

I realised that, afterwards, Syphrum but they are so very measurable that they are well worth discussing and measuring for verification. I did all my electromag learning with radio waves in mind and see it as the most relevant - just a personal view.
Incidentally - a ferrite rod would not be a good measuring tool because of its enormous inductance. To measure E and B fields one would need a very short dipole. feeding a high impedance  and a very small search coil feeding a low impedance. These would not alter the measured phase of any signal, appreciably.