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.
