Why Do EM Waves Travel Differently in Good Conductors?

[yungman]

I made an error, using B instead of the time derivative of B, so that most everything I had to say in that last post was wrong.

I better leave it alone before I throw you off any further.

In fact I think you help a lot. The idea you suggested of looking at the wire as coax is a light bulb moment! I was thinking about the secondary effect and this solidified the idea.

The wire can be treated as either coax or micro-strip as shown

That solidify the idea of launching the signal onto the transmission line. That answer the question on the speed of the wave travel on the wire.

Thank you very much.

 

[Phrak]

You're quite welcome. You're helping me as well.

Now got a new question. How is the secondary EM wave propagate?! Now that it looks like we got a EM wave going from the current along the wire! It seem like this EM propagate along the wire! Is this true?? If you look at the coax field pattern, H is circling the inner conductor, E is radiating out normal to the outer surface of the inner conductor. so E=rE and H = -\phiH. r X -\phi = -z which is along the wire!

I don't know why you call it the 'secondary' wave. But I know what you mean. That's what I come up with (at this point) too. the E and B fields of electromagnetic waves in free space are in phase, and the H and E fields around the coaxial cable or stip line also seem to be in phase. (This was the part where I was going wrong, it seems---the phase.) How the EM wave propagates in a coax seems to be the same as a free wave, but where the displacement current terminates on conductors and the loop is competed by the currents in the core, and shield.

I'm very suprised too on the propagating fields around a conductor. All this time working with electronincs and conductors, and I had no idea.

You can always think of the voltage and current in the wire as being primary effects, and the fields as a resultant effect, but I think it all needs to work together.

There are a lot of interesting questions to ask about coaxial cables. Does the dielectric constant change the propagation speed, and if so, for a vacuum dialectric, is the impedence of the cable 377 ohms?

 

[yungman]

You're quite welcome. You're helping me as well.



I don't know why you call it the 'secondary' wave.
For the lack of any good description.

But I know what you mean. That's what I come up with (at this point) too. the E and B fields of electromagnetic waves in free space are in phase, and the H and E fields around the coaxial cable or stip line also seem to be in phase.
If the dielectric of coax, in this case is air and is lossless, E and H are in phase because \eta is real.

(This was the part where I was going wrong, it seems---the phase.) How the EM wave propagates in a coax seems to be the same as a free wave, but where the displacement current terminates on conductors and the loop is competed by the currents in the core, and shield.
That is the part after I read the transmission line theory: There is no EM wave travel in the Tx line, it is lanched in form of Voltage and Current phasor.

I'm very suprised too on the propagating fields around a conductor. All this time working with electronincs and conductors, and I had no idea.
This is my assumption for the moment: If you think of this in term of real circuit. The generator generate an electrical signal that drive into the long wire( which we agreed that it is a coax) and traveled down the wire as Voltage and Current phasor. It never turn into EM wave.
1) The current phasor along the wire create an E field along the wire that in turn cause M field circling around the wire. Together form the poynting vector that point towards the center of the wire. This is inducing the skin effect. This is the primary EM wave that I refer to.

2) At the same time, consider the wire is a coax where the Earth serve as the ground. When current travel down the wire, the E field generated normal to the wire and end at the ground through the space. This radiate into air. Together with the M field that circle the wire, poynting vector show that this EM wave propagate along the wire in reverse direction.


You can always think of the voltage and current in the wire as being primary effects, and the fields as a resultant effect, but I think it all needs to work together.



There are a lot of interesting questions to ask about coaxial cables. Does the dielectric constant change the propagation speed, and if so, for a vacuum dialectric, is the impedence of the cable 377 ohms?
The dielectric constant do change the propagation speed. Speed is ( speed of light)/(square root of \epsilonr). And impedance is close to 377 ohm. This I am pretty sure.

The above is where I am at. I am not saying I am right. I am still waiting for someone here to validate or dis validate my theory. Please if anyone here have something to say, I am all ears!

I think my original question has been answered. The speed of current traveling down the wire is light speed in air. This is because if you consider the wire is a coax with air as dielectric, relative permeativity is 1 and same as traveling in air.

 

[Phrak]

The above is where I am at. I am not saying I am right. I am still waiting for someone here to validate or dis validate my theory. Please if anyone here have something to say, I am all ears!

I think my original question has been answered. The speed of current traveling down the wire is light speed in air. This is because if you consider the wire is a coax with air as dielectric, relative permeativity is 1 and same as traveling in air.

I have an idea, that might work. This thread has become very long, so anyone checking-in
for the first time will probably give up after the first page or two. Also, when you posted
your drawing, the long width confuzzled the software so that the text stretchs too wide.
Now that you know (and me too) you could crop the drawing width before posting.
So you might re-post your theory, as in introductory question, in the electrical engineering
thread and hope for the best. If that fails, we can always continue here I hope.

I've re-read all your post and I'm not sure what your theory is, at this point.
Sorry if I'm obtuse.


Next, I have made some progress. To start out as simple as possible, I consider an ideally
conductive coaxial cable with vacuum dialectric. The coordinates are z, r and theta in
cylindrical coordinates. For a first-pass analysis I'm essentially ignoring negative signs
sometimes, and the values of epsilon, mu and even pi I set equal to 1 sometimes. I'm just
using portionals.

The whole idea of this is to eventually get to resistive conductors (good conductors, as
your text calls them) and the primary wave that moves into the conductor , but I have to
start with the ideal conductor case. So this is what I have so far.

The electric field surrounding the center wire is directed radially, only (this is because it's a
perfect conductor). The strength of E_z drops off inversely with the radius; Gauss's Law for
electric charge.

\textbf{E} = E_r \propto \frac{1}{r} \hat{r}​

The z dependence is intuitive when we assume a propagating waves at one frequency in
the positive z direction. It turns out to work correctly. I'm ignoring the velocity, the ratio
of k/omega for now.

E_r \propto cos (kz - \omega t)​

The electric field doen't vary with theta because it's directed radially.

Using the Maxwell Faraday equation in integral form, the rate change in magnetic flux,
with respect to time can be found around rectangular loops in a plane of constant z
in the vacuum between the wire and the shield. The magnetic field stength results from
differentiating over the area of the loop. The magnetic field connsists of circular loops at
constant z.

\textbf{B} = B_\theta \propto \frac{1}{r} \hat{r}​

This makes things easy. No nasty Bessel functions will get in the way. When B
is proportional to 1/r the integral of B around every loop is the same. Applying Ampere's
curcuital law, give the current, J in the conductor. There can't be any dE_z/dt or in the
space between the wire and shield or the integral of B around inner conductor would be
different in each loop. This is what makes things simple. (Now that I think about it, there
could still be an axial electric field inside the wire. I'll have to think about that one further.)

With all this, it turns out that the magnetic field is in phase with the electric field.

B_\theta \propto cos(kx-\omega t)​

The current at each station in z is proportional to the magnetic field looping around it
at any given radius.

\textbf{J} = J_z \hat{z} \propto dB​

The current is in phase with the magnetic field stength.

\textbf{J} = J_z \propto cos(kz-\omega t) \hat{z}​

The electric field has to terminate on charge. The electric field is radiating perpendicularly
outwared from the inner conductor. The total electric field radiating outward in a unit length
of wire is proportional the total charge per unit length of wire. Using Gauss's law of electric
charge, again:

\rho \propto E_r​

With the charge proportional the electric field strength, the charge density and electric
field stength are proportional at each station along the wire. So they are in phase.

\rho \propto cos(kz- \omega t)​

The voltage at each station along the inner conductor is obtained from integrating the
electric field stength along the wire where it contacts the surface of the wire. I haven't
given the voltage a great deal of thought.

 

[yungman]

I have an idea, that might work. This thread has become very long, so anyone checking-in
for the first time will probably give up after the first page or two. Also, when you posted
your drawing, the long width confuzzled the software so that the text stretchs too wide.
Now that you know (and me too) you could crop the drawing width before posting.
So you might re-post your theory, as in introductory question, in the electrical engineering
thread and hope for the best. If that fails, we can always continue here I hope.
I don't think the forum want me to repost on other sub forum. I think this is getting to the end already. I think I need to study antenna and the answer should be there.
I've re-read all your post and I'm not sure what your theory is, at this point.
Sorry if I'm obtuse.


Next, I have made some progress. To start out as simple as possible, I consider an ideally
conductive coaxial cable with vacuum dialectric. The coordinates are z, r and theta in
cylindrical coordinates. For a first-pass analysis I'm essentially ignoring negative signs
sometimes, and the values of epsilon, mu and even pi I set equal to 1 sometimes. I'm just
using portionals.
I want to verify the coordinates. This is my intepritation:

The whole idea of this is to eventually get to resistive conductors (good conductors, as
your text calls them) and the primary wave that moves into the conductor , but I have to
start with the ideal conductor case. So this is what I have so far.

The electric field surrounding the center wire is directed radially, only (this is because it's a
perfect conductor). The strength of E_z Do you mean E_r?drops off inversely with the radius; Gauss's Law for
electric charge.

\textbf{E} = E_r \propto \frac{1}{r} \hat{r}​

The z dependence is intuitive when we assume a propagating waves at one frequency in
the positive z direction. It turns out to work correctly. I'm ignoring the velocity, the ratio
of k/omega for now.

E_r \propto cos (kz - \omega t)​

This is how I picture it:

The Er radiate normal to surface of the wire and Br circle around the wire. Both on r\theta plane where propagation is in direction normal to the plane.
I envision the fields are like ballons along the wire with the wave length shown.

The electric field doen't vary with theta because it's directed radially.

Using the Maxwell Faraday equation in integral form, the rate change in magnetic flux,
with respect to time can be found around rectangular loops in a plane of constant z
in the vacuum between the wire and the shield. The magnetic field stength results from
differentiating over the area of the loop. The magnetic field connsists of circular loops at
constant z.

\textbf{B} = B_\theta \propto \frac{1}{r} \hat{r}​

This is how I read it:

This makes things easy. No nasty Bessel functions will get in the way. When B
is proportional to 1/r the integral of B around every loop is the same. Applying Ampere's
curcuital law, give the current, J in the conductor. There can't be any dE_z/dt or in the
space between the wire and shield or the integral of B around inner conductor would be
different in each loop. This is what makes things simple. (Now that I think about it, there
could still be an axial electric field inside the wire. I'll have to think about that one further.)

With all this, it turns out that the magnetic field is in phase with the electric field.

B_\theta \propto cos(kx-\omega t)​

The current at each station in z is proportional to the magnetic field looping around it
at any given radius.

\textbf{J} = J_z \hat{z} \propto dB​

The current is in phase with the magnetic field stength.

\textbf{J} = J_z \propto cos(kz-\omega t) \hat{z}​

The electric field has to terminate on charge. The electric field is radiating perpendicularly
outwared from the inner conductor. The total electric field radiating outward in a unit length
of wire is proportional the total charge per unit length of wire. Using Gauss's law of electric
charge, again:

\rho \propto E_r​

With the charge proportional the electric field strength, the charge density and electric
field stength are proportional at each station along the wire. So they are in phase.

\rho \propto cos(kz- \omega t)​

The voltage at each station along the inner conductor is obtained from integrating the
electric field stength along the wire where it contacts the surface of the wire. I haven't
given the voltage a great deal of thought.
This is where I have a different view:
I see the signal launch into the wire as traveling waves:

The wave travel down along the wire which is the z direction. The current generate the E and B fields as you described.

Thanks for taking the time to answer the question.

What you described is what I listed as the the secondary field that radiate out to the air. The big difference is I believe the signal from generator drive into the wire as voltage and current traveling waves only. There is no EM wave at this point. EM wave generated only by the traveling voltage and current waves, not the other way around like you said.

I also talk about the E field that is along the wire ( z direction) which contribute to the skin effect.

 

[Phrak]

I think your coordinate interpretation is as I intended. I can't read the subscrips for the B field.

Yes, where I said E_z, I should have said E_r.

You're picture of how you envision the fields looks good for the E field if the arrow length are intended to mean field strength. The E fields should extend until they find ground return somewhere in the enviroment. The B fields look a little funny as an envelope for the E fields stengths.

After all this I don't understand the ve+ and ve- notaton, sorry. But the instantanious voltage along the wire length is bothering me. It's the integral of the electric field, I think, so that the voltage would be out of phase with everything else. That doesn't seem right. Your equations have them in phase. I'm perplexed.

As you say, this is all about the secondary wave. The primary wave is the interesting part of it all, where the E field apparently has an axial component as you've stessed.

You probably recall that skin effect means that as the frequency increases the current is forced out of the center of the wire so that the bulk resistivity of the wire becomes more of an effect as frequency increase. So it will turn-out that finding how the secondary wave works will require the internal solution of one or both of the electric and magnetic fields inside the wire. If I don't miss my guess, this will lead to a nonanalytical equation, so your texts probably take the course of action of making approximations and dividing the full solution into parts: the primary and secondary waves, as you've called them.

 

[yungman]

I think your coordinate interpretation is as I intended. I can't read the subscrips for the B field.

Yes, where I said E_z, I should have said E_r.

You're picture of how you envision the fields looks good for the E field if the arrow length are intended to mean field strength. The E fields should extend until they find ground return somewhere in the enviroment. The B fields look a little funny as an envelope for the E fields stengths.
Yes the E arrows represent field strength, the field should reach the ground return. The M field is the typical circles of magnetic fields as shown:

After all this I don't understand the ve+ and ve- notaton, sorry. But the instantanious voltage along the wire length is bothering me.
I can't find where are the ve+ and ve- notation. The diagram was wrong, I corrected it already on the original post. It should read:

It's the integral of the electric field, I think, so that the voltage would be out of phase with everything else. That doesn't seem right. Your equations have them in phase. I'm perplexed.

As you say, this is all about the secondary wave. The primary wave is the interesting part of it all, where the E field apparently has an axial component as you've stessed.

You probably recall that skin effect means that as the frequency increases the current is forced out of the center of the wire so that the bulk resistivity of the wire becomes more of an effect as frequency increase. So it will turn-out that finding how the secondary wave works will require the internal solution of one or both of the electric and magnetic fields inside the wire. If I don't miss my guess, this will lead to a nonanalytical equation, so your texts probably take the course of action of making approximations and dividing the full solution into parts: the primary and secondary waves, as you've called them.

Do you agree the point I made that the secondary EM wave that radiate out of the wire are from the Voltage and Current traveling waves that travel along the wire? Actually it is only the Current traveling waves that generate the E field which in turn generate the M field.

After I verify with you on this, I think I better stop this thread because after doing all this digging, This really become an antenna question which I don't have any idea and I am waisting people's time here. I just order 2 books on antenna and I better do some serious study before I asked any more of this kind of question. I am on the the last chapter of electro magnetic...Boundary problems. After the EM, I'll be diving into antenna next.

Thanks for taking the time to work with me.

 

[Phrak]

Do you agree the point I made that the secondary EM wave that radiate out of the wire are from the Voltage and Current traveling waves that travel along the wire? Actually it is only the Current traveling waves that generate the E field which in turn generate the M field.

Yes, definitely, we're in agreement. You're actually approching the problem normally, beginning with current. I started guessing at the electric field and working backwards. It's a matter of taste, but I think it usually works better to assume current and charge distributions and solve for the fields as you are doing.

I'm struck with an idea. I think the primary wave can be derived by assuming the current decays exponentially over distance.

I = I_0 cos(kz - \omega t) exp(-frac{z}{L} )​

Where L is just some length along the wire where the signal strength has dropped to 36%. Either this yields a constant bulk resistivity of the wire, or it's wrong.

Thanks for taking the time to work with me.

And you, as well. I'm learning too.

What units are you using? There are several standards: SI, CGS, Gaussian... It would be good to be both working in the same system of units. Whatever units you are using I will adopt.

 

[yungman]

Yes, definitely, we're in agreement. You're actually approching the problem normally, beginning with current. I started guessing at the electric field and working backwards. It's a matter of taste, but I think it usually works better to assume current and charge distributions and solve for the fields as you are doing.

I'm struck with an idea. I think the primary wave can be derived by assuming the current decays exponentially over distance.

I = I_0 cos(kz - \omega t) exp(-frac{z}{L} )​

Where L is just some length along the wire where the signal strength has dropped to 36%. Either this yields a constant bulk resistivity of the wire, or it's wrong.

I went back and read up on this. This is what I came up with:

I don't know how to find C because of the wire is in the air. But I think you get the point. My experience is if the wire is less than 1m, I don't think you lost too much signal. Mostly all loss from dielectric loss and non in this case. If you use steel, then it might be different because \mu is large and skin depth is much thinner than coper.


And you, as well. I'm learning too.
Still it is good that you spent the time. There is no easy quick answer. Like what you suggested here, I have to go back and read up before I come back.

What units are you using? There are several standards: SI, CGS, Gaussian... It would be good to be both working in the same system of units. Whatever units you are using I will adopt.
I use meter, kg, deg C. I guess is SI?

...

 

[Phrak]

There are some units of measurement where Maxwell's equations will pick-up a factor of pi. I'll assume you're using SI units.

The reason I brought up resistive losses, not because of signal decay, but because this is the origin of the 'primary waves'--or whatever it it that has a Poynting vector directed into the wire.

It's going to take me some time to accumulate references, as I don't have any more than an intermediate text on electromagnetism. More, I'm back working full time.