(Alpha) Does an EM-Field contain a vorticity

Are you being serious friend?

The thread is called ''Does an EM-Field contain a vorticity''?

I thought it says it on the tin, so-to-say.

 

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If you want to know if the EM field has vorticity, then just look at Maxwell equations.
The OP has nothing to do with this.
Why did you write the first post which is totally unrelated to the question?

 

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Ah. As for the actual Wikipedia quote we have:

http://en.wikipedia.org/wiki/Momentum#Momentum_in_electromagnetism

The text was added in near its present form by user AndyWall on March 28, 2006, who does not have much physics content.

http://en.wikipedia.org/w/index.php?title=Momentum&action=historysubmit&diff=45866672&oldid=45835019

The text is still not proper English ("The definition canonical momentum " should properly read: "The definition of canonical momentum"), but the phrase "Canonical Momentum" is a term of art within the domain of Lagrangian mechanics and it is not obvious that this is correct or suitable for a general Wikipedia page on momentum, especially as the Lagrangian is not displayed or (as AlphaNumeric pointed out) a covariant one derived from modern treatments of electromagnetism.

See J.D. Jackson Classical Electrodynamics (1962) or similar for

\(p_\mu = \left( { E/c \\ \vec{p} } \right) = \left( { m \gamma c \\ m \gamma \vec{v} } \right) - q \left( { \phi \\ \vec{A} } \right) = m U_\mu - q A_\mu \)
(if I haven't left out any minus signs or factors of c)
http://sciforums.com/

 

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Point in the right direction then.

And the OP is very much to do with this. :bugeye:

 

Oh for god sake, it's absolutely fascinating how I manage to pick out the one equation which does not satisfy needs.

 

I told you: look at Maxwell equation, you know curle(E) = ... etc.

You said yourself

A is the field
By changing v into curl(v) you do not change the field.
Don't you see that this does not change/check the vorticity of the field.

By the way, when you write the cannonical momentum in the Hamiltonian, and writting down the Hamilton's equation of motion, you get the Lorentz force that act on the particle

 

If the equation is true, there may be methods in detecting a vorticity of the EM field - as you will notice, it seems someone has provided evidence of this.

 

But it is not because:
1. The cannonical momentum P = mv + qA in the hamiltonian gives you the correct equations of motion
2. When you take P = mv + qA, this is a charged point particle, its velocity and positions are independent, hence curl(v) = 0.
So changing the particle momentum to qA is just wrong.

Yes I know, his name is James Clerk Maxwell, he gave the evidence more than 100 years ago

 

Well, someone should edit the wiki page if it's incorrect.

 

So, I've been reading more. One can say the momentum density of the electromagnetic field as:

\(\vec{p}= \frac{1}{c^2} \vec{E} \times \vec{H}\)

Where \(p\) is for momentum, E is the electric field and H is the auxilary magnetic field. One can clearly see how this is related to the Poyinting Vector.

 

Now, 1102, you said maxwell has derived the properties I am seeking. If we use the definition of momentum above, how does the vorticity enter the equations?

As I said before, I have searched the net, and I found two equations supposidly describing the vorticity, but both of them looked very dubious.

 

The momentum that you gave in the OP are not the field momentum. They are the momentum of a particle in the field.
The Poyinting vector ExB is the momentum density of the field. They are two different things. One is the momentum of a PARTICLE, the other is the momnentum of a FIELD.
A field has vorticity when its curl is non zero.
Look at Maxwell's equation and you will see that both curl(E) and curl(B) are in general non zero

 

No, I am not referring to that equation any more.

Look to post 30 please.

 

I will add, I recognize that the equation in the OP is incorrect, so, I have studied on it more, hence why I am now referring to the equation in post 30.

 

One can clearly see how this is related to the Poyinting vector since This is the Poyinting vector.

Now, what has it to do with vorticity of the fields?

 

The pedant in me is compelled to point (chuckle) out that the correct spelling is POYNTING vector. It's so called because it's named after John Henry Poynting, not because it points at anything.

 

In fact I knew that the Poynting vector was not named that way because it points to direction of energy flow (even though the name is appropriate), but after the name of the physicist. This is why I wrote it with a "yi", thinking that this is the correct spelling.
Thank you for correcting me.
From now on it will always be "Poynting"

 

I could take the mathematical route and say that on the left-hand side of the equation could be modified to suit the curl of the velocity.

I bet there is a much more substantial and correct answer, which is were you come in.

 

In which equation do you want to take the curl of the velocity?

 

I am reminded of Multitwist Optical Möbius Strips by Isaac Freund (Optics Letters Vol. 35, No. 2 January 15, 2010). A forked fringe is employed, and the vortices are caused by light. Here's a "fair use" extract from it:

The exact r dependence of the number of halftwists and handedness of the Möbius strips requires an analysis that is too complicated to present here. Instead, insight into what happens can be obtained by considering the angle αxy=arctan(αy, αx) shown in Fig. 2, where αx and αy are the x,y components of α. Because α is a line, not a vector, αxy is plotted modulo π. On the small circle at the center of the figure αxy winds around the central C point, which in such plots appears as a vortex, with a winding number IC =−1/2; IC is a conserved property of the C point.

IMHO an EM field does contain a vorticity. Take a look at On physical lines of force and note the title. It's The Theory of Molecular Vortices. I'd have to sit down and read it again to check, but I think Maxwell was suggesting that the electromagnetic field was a sea of vortices, and particles move through it. If so, he got it back to front. We make electrons via pair production, out of light. The particles are the vortexes. And looking at the thread title, don't forget that at the heart of an electromagnetic field, what you'll find is something like an electron.

But sadly, nobody seems to understand displacement current. There was a good article about that in the IoP PhysicsWorld last month, but it isn't online. I'm afraid. I thought this was an interesting paper though: The Maxwell wave function of the photon. It's by Raymer and Smith, and was written for a SPIE Optics and Photonics conference in 2005.