# Homework Help: Speed of EM in good conductors

1. Dec 26, 2008

### yungman

Speed of EM travel through the good conductor is $$\omega$$/$$\beta$$

I know the speed of signal travel in stripline is c/$$\sqrt{}\epsilon$$$$_{}r$$_{}

My symbols don't look very good but I think you get what I mean. Obvious they are different.

I am confuse because in both case EM wave travel in good conductor, why are they different?

I know the first one is "propagation velocity" the second is "phase velocity". What is the difference?

Thanks

Last edited: Dec 26, 2008
2. Dec 26, 2008

### Defennder

What's the last term supposed to be?

3. Dec 26, 2008

### Staff: Mentor

In general, an EM wave does not travel "in a conductor". What does it travel in instead? Start with free space (vacuum, eh?), then the atmosphere, then glass, then water, then a stripline, etc.

What is the EM wave oscillating/travelling in? What determines its speed and loss?

4. Dec 27, 2008

### yungman

It is the speed of light divided by the square root of relative permeativity. eg. If permeativity is 4, then phase velocity is 3EE8/2=1.5EE8

Thanks

5. Dec 27, 2008

### yungman

Speed and loss depend on $$\gamma$$ which equal to $$\alpha$$ +j$$\beta$$ . $$\alpha$$ is attenuation constant and is real. $$\beta$$ is propagation constant which is imiginary part. Both can be determined depend on classify of the dielectric medium.

EM wave do travel in sea water which is relative good conductor. Just with attenuation. The speed still approx. to $$\omega$$ / $$\beta$$ . which is very different from phase constant.

I guess my question: What is the difference between EM wave travel in stripline and EM wave travel in a media. Is it the first one is conduction current and the second is displacement current.

6. Dec 27, 2008

### Staff: Mentor

Interesting. This actually has distilled down to a very good question, and to be honest, I haven't thought about it from that angle before. Is this the question:

"What is the difference in propagation equations for an EM wave in free space, versus one confined to a transmission line? And why does there appear to be a fundamental difference in the equation for the propagation velocity for these two cases?"

I'll have to think about that some before I can try to give a useful answer. My intuition says that the difference lies in the directivity of the transmission line (TL) case -- that is, the EM wave that propagates along a TL is directed down the TL by the conductors of the TL (coaxial, or twisted pair, or stripline, etc.), so the loss is dominated by the skin effect resistive loss on the conductors, and the parasitic conductance loss in the dielectric of the TL. The only "loss" in a free-space EM wave propagation (not in a conductive media like seawater obviously) is the growing surface area of the wavefront, which gives you a power loss over distance with a 1/r^2 coefficient.

I'll PM a couple other PF'ers to try to get them to address your question, assuming I'm interpreting it correctly. BTW, since this is in the homework/coursework area of the PF, we can't do your work for you (per the PF Rules link at the top of the page). You have asked a very fundamental and interesting question, though, and have shown a lot of your own work. Can you take what I've said about the constrained nature of the Poynting vector and different attenuation mechanisms for a TL versus free space, and see if that can explain the equation differences that you are encountering?

7. Dec 27, 2008

### yungman

This morning I thought of another example. Current travel on the surface of a block of metal where the surface is the xy plane and the depth is +ve z direction. Current is travel in +ve x direction. I sure I know how to copy a drawing onto the post without doing an attachment that require a day for approval!!

This is regarding to current density in a infinite block of metal in text book. The current mainly travel on the surface. The book mainly talk about the attenuation in z direction where it follow the attenuation constant and propagation for the current density at z direction and propagation velocity that is calculate at z direction. The surface current is traveling at higher speed closer to the speed of light.

I also re-read the materials. Propagation of EM wave in stripline is the speed of wave travel through the dielectric that make up the stripline. Which is 1/(square root of permeability X permeativity).............I know it is confusing!!! I can't do symbols and can't upload a drawing!!!

Last edited: Dec 28, 2008
8. Dec 29, 2008

### Defennder

Well I'm not an expert here, but this is what I think: From what I can tell the telegrapher's equation which you can derived from applying KVL to a differential length segment dz is essentially a wave equation. From this, I believe one can infer that the solution of the wave equation applies to t-lines as well; ie. voltage and current can be considered as waves on transmission lines. With that in mind, note that the equations given which seemingly appear different for both phase velocity and propagation velocity for EM waves/voltage-current waves are actually the same. The former can easily be shown to reduce to $$u_p = \lambda f$$ and the latter I believe I've seen a short derivation in a freshman physics textbook. The formulae are equivalent. They hold for waves and they are waves because in both cases, EM waves and t-lines, they satisfy the wave equation.

9. Dec 29, 2008

### yungman

I have been reading also, The equation both from the same wave equation like you said. I have not study the guided wave yet which I believe applies to stripline.

Further. My original question has a lot to do with the skin effect Where the current is in direction of the E field. and the EM wave propagate in direction normal to the current direction. The velocity of current is not the same as the propagation velocity of the EM wave. I wish I know how to put a diagram on the post other than as an attachment which require a day to clear. This is how I read it so far on skin effect:

Consider a long bar with x be the in direction of the length. Width be the y direction and the thickness in z direction. Let current flow in +ve x direction from one end of the bar to the other end.

1) Since the current from in +ve x direction. THis mean E wave is in direction of x. ( E wave is parallel to the length of the bar )

2) E wave induce H wave which is in y direction. ( parallel to the width of the bar)

3) The direction of propagation is normal to both E and H wave which is (x X y = z) So the EM wave actually propagate down in direction of the thickness of the bar.

Therefore the velocity of current travel along the bar is DIFFERENT from the propagation velocity of EM wave in good conductor.

THis is my understanding so far. I still studying. Please give me comments.

Thanks

10. Dec 29, 2008

### yungman

I think my original question might be wrong. Current conduction is not the same as propagate of EM wave in good conductor like coper. From the skin effect explaination, E wave is the same direction of current and propagation is normal to current. I attach the copy of one page of the book. Please tell me your thoughts. That still the question, what is the velocity of the current in the good conductor? Also I want to know why the speed of the stripline is higher and it is not frequency dependent.

#### Attached Files:

• ###### Skin effect.pdf
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11. Dec 30, 2008

### yungman

I found out already. Stripline is TEM mode. Still can't find the speed of the current density on the surface of a good conductor.

12. Dec 30, 2008

### Phrak

You should know that the velocity of a sinusoidal wave and a modulation of a sinusoidal wave will not, in general, be the same. In electromagnetic theory the first is called phase velocity, and the second is called group velocity. This terminology is used in wave guides. I don't claim to be up on the propagation of signals in coaxial wire, or strip line, or just a simple wire in free space, so the terminology may vary, although phase velocity will always mean the velocity of a sinusoidal wave.

Last edited: Dec 30, 2008
13. Dec 30, 2008

### yungman

I don't think so, group velocity is defined for a group of frequencies in the signal going through the medium. The velocities differ with difference frequency.

What we are talking about can be consider a single frequency, not a modulated signal. I believe Phase velocity IS the same as Propagation velocity which is the velocity of the EM wave travel through a medium. Which is $$\omega$$ / $$\beta$$.

In a good conductor like sea water which is a lossy medium, the velocity is only in 10EE7 range compare to vacuum of 3EE8 m/sec. My confusion can be seen in the skin effect paragraph that I attached. The direction of the conduction current is the direction of the E wave which is along the length of the bar. The EM wave is propagate in direction of the thickness which is perpendicular to the conduction current and is much slower than speed of light and suffer great attenuation. The conduction current only suffer from ohmic loss of the surface resistance along the length of the bar and travel at close to speed of light. My question what is the speed of the conduction current.

14. Dec 31, 2008

### Phrak

That's right, the group velocity is a group of frequencies. A modulated sinusoidal wave results in a group of frequencies. Our two points of view are the same. See Fourier transforms--or see Fourier sums; their easier to grog.

I re-read all of your posts, and the others, so I'm up to speed. Propagation of a signal down a conductor is a very differerent phenomena than sending an electromagnetic wave into a conductive media.

There is a hierachy one can make from free electromagnetic waves to a single wire conductor: free EM waves, to wave guide, to differential pair (like the old 75 ohm TV antenna wire), to coaxial, to single conductor. These are all dominated by inductance and capacitance. A pair of wires--like the 75 ohm TV wires--can be modeled as a ladder of infintesimal capacitors and inductors. The capacitors form the steps and the inductors the supports between each pair of steps. Coaxial cables use the same model.

The dominant effect in penetrating a conductive media is resistance such as you would be interested in with stealth technology and ultralow frequency submarine communications. This, I'm not at all familiar with.

You are correct in your assesment to distinguish the two.

Last edited: Dec 31, 2008
15. Dec 31, 2008

### Phrak

You may know this, but it's a common misconception that the velocity of current in a conductor is near the speed of light. Electrons propagate at their "drift velocity" in a conductive media.

http://en.wikipedia.org/wiki/Electric_current" [Broken]

It doen't look like Wikipedia will suppy us with a typical drift velocity. But for a 100 Watt load such as a light bulb powered by 120 VDC instead of volts AC, it's something like one centimeter an hour in an 18 gauge conductor, give or take.

But in an attenuated wave in a conductive media, such as you are interested in, there are transverse current waves. At least I think this is the current you are interested in. The current waves have a velocity. The current oscillates back and forth in phase with the electric field. The electrons slosh back and forth at the drift veleocity. The waves themselves travel at the speed of the electric field in the media.

Last edited by a moderator: May 3, 2017
16. Dec 31, 2008

### yungman

Thank you for taking the time. I should have known that, I just did not relate the two. I have been reading up a little, got more question than when I first posted, but I guess it is progress, at least I know enough to be confused!!! I'll think and study more before I post on this one. It is new year's eve and I have my grandson with me over night. I'll come back after tomorrow.

Happy new year to you all and thank you all for helping.

Last edited by a moderator: May 3, 2017
17. Dec 31, 2008

### Staff: Mentor

Happy New Year, yungman. You've been asking some good questions, and doing good learning on your own. Enjoy the kids!

18. Dec 31, 2008

### Phrak

Happy newyear, yungman!

I'm kicking myself over a statement I made about resistance being a dominate a effect in a penetrating wave. It's meaningless. A completely reflectived wave will always have a penetration depth (exponentially decreasing), even where there is no loss. But any penetration of a conductor will suffer losses due to the resistivity of the material. Sea water is especially interesting because it has both resistive losses, and additionally has a very high dielectric constant.

In any case, I've about exhusted everything I know on the topic, and at this point you should probably be instructing me in short order.

19. Jan 1, 2009

### yungman

Hey we are all leaning. The more we share, right or wrong, everyone learn. We all have our moments. I take the comfort that we learn beyond just getting the correct answer to homework and test when we start questioning more, whether we make correct assumption or not. This is the third time I read and work over the topic and all of the sudden, I stop and questioning all this.

Actually you gave me a lot of fruitful thoughs. Since your last post, I was kicking myself, I study those before, just not enough to become a common sense to apply freely!!! I print out your comments and the link and study it. Here is more question from thinking in the last two days.

From part A of the drawing, I proved that EM-wave propagate towards the center of the thick wire. I show in instanteneous form using sine wave in +ve and -ve direction separately to prove both case, no EM-wave propagate into space(to the surrounding medium). E wave is always in direction of current which is just AC current of cos($$\omega$$t). That means E is along the length of the wire. Question is what is the speed of the signal through the wire even though you show me clearly that the electron move very very slow in the wire. Somehow the signal do travel close to speed of light as we all know.

Further in Part B I show a loop antenna make of the same thick wire and driven by signal generator as shown. If my intupretation is correct, no EM-wave emmited from the loop directly. Only M wave propagate out as shown. E wave can only from the M wave that propagate out, not from the original current through the loop.

This is what I think so far, please give me more feedback.

Happy new year.

20. Jan 1, 2009

### Phrak

I haven't looked into the details of a conductor as much as you are now. There are some
interesting points. The negative of the integral(PdA) is quntity of energy. In the DC form
your drawing is commonly used to model cylindrical resistors, where the energy moves
into the resistive material, then is dissipated as heat. But, then, this model shouldn't work
for a superconductor. I don't see why it doesn't.

An infinitely long conductor in empty space should be the same as a coaxial cable
with the outer conductor at infinity; R-->infinity. The Poynting vector at the surface
of the outer conductor has it's own Poynting vector. This is a conserved field in empty
space ('cause energy is conserved), so somehow all of these Poynting vectors have to
origintate at the source of EMF.

Your second drawing is a loop antenna. It has the same radiation pattern as a dipole
antenna at large distances. Where the distance R>>wave_length, the space around the
loop is dominated by propagating fields. So there is an accompanying electric field, and
the ratio of the electric to the magnetic field strengths is that of radiation in a vacuum.

The near fields are not the same as the far fields, but I only know the rough details.
If we put a receiver antenna at one loop radius distance, we're more likely to call the
arrangement a transformer and the field coupling is not same as the coupling from