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I have in my mind a thought experiment which I cannot figure out.

Imagine a voltage source working on let's say 1 GHz. One end is connected to a ground, which is very small (we can abstractly assume it is just a point in space) and keeps constant voltage, 0V. The other end is connected to infinitely long single wire.

I wonder how different parameters like input impedance, radiation, voltage and current propagation are gonna look like.

If there were two paralell wires, there would be an electromagnetic wave travelling between them, no mistery about that.

Is there gonna be any propagation in a single wire case? If yes, I would be pretty sure that it would include a lot of radiation and, at some point, all the energy would be radiated off. The question is, how far can the signal go? Also, I'm wondering what would be input impedance in such a case? It would have to be resistive if radiation occurs, am I right?

I'm getting really mixed up about this because I know that propagation over a single wire is impossible. Is it really? Or is it impossible because of radiation? What I mean is that maybe (that's what I don't understand) voltage can propagate over a single wire but at some point it would radiate all the energy off so effectively, propagation over single wire can be regarded as "impossible", because we have very strong limitation - radiation.

I get even more confused when I think about it from the perspective of Maxwell equations. I know that close to the source, there would be E vector between two ends of a source. How would E vector look if the voltage propagated over a single wire far away (like 1 km)?

I hope some of you could clear things up for me :)

Andrzej Dąbrowski
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4 Answers4

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What you are really getting at is: "What is the impedance of a wire in free space". That's simple: 376.73031 ohms (approximately).

To put it differently, a single wire (no ground wire or shield) of infinite length will appear as a 376-ish ohm load to whatever is driving the signal down that wire. This assumes that the copper in the wire has a 0 ohm resistance.

Here is the Wikipedial page for Impedance of Free Space.

  • Thank you for your answer. I know the concept of impedance of free space but I've always thought of it simply as an impedance of electromagnetic wave (E / H) running in space. So, I understand that my voltage source will see 376 ohms? That's resistive so there is some energy out. Should I suspect that voltage will travel along the wire and this energy will be radiated off as an electromaginetic wave? – Andrzej Dąbrowski May 25 '13 at 18:02
  • In a purely academic point of view, that energy is lost to space. Of course, it is more complicated than that when talking about the real world. –  May 25 '13 at 18:54
  • I'm pretty sure that the impedance of a wire in free space is not the same as the impedance of free space. The latter assumes perfectly uniform E and H fields, and around a wire, they are anything but. In fact, I'm pretty sure the answer depends on the diameter of the wire. – Dave Tweed May 25 '13 at 20:15
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Remember, "ground" is not an absolute reference, it's just whatever circuit node you choose to measure your voltages against.

Your infintesimal "ground point" cannot pass any current, and therefore, neither can your voltage source. More importantly, no current will flow into or out of your (semi-)infinite wire.

With respect to everything else in the universe, the long wire will keep the same voltage at all times; while the 1 GHz voltage source is madly wiggling your "ground point" up and down to no effect.

One way to think of it is to realize that your long wire has an infinite amount of capacitance to the rest of the universe, while your ground point has none.

Dave Tweed
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  • I should have written that my inifinitesimal "ground point" can pass any current it wants. The assumption is that it has an infinite capacitance, like earth. – Andrzej Dąbrowski May 25 '13 at 17:57
  • In that case, you're living in a different universe than I am, and I can't answer the question. Where does the current go? – Dave Tweed May 25 '13 at 18:28
  • Hmm, let's remove infinitesimal "ground point" and connect source to the Earth instead. What would happen then? – Andrzej Dąbrowski May 25 '13 at 18:36
  • The Earth's capacitance, while large, is still finite, as compared to your infinite wire. You're basically turning the Earth into a radiator, part of a dipole working in conjunction with the near end of the wire. Perhaps your actual theoretical question is about a semi-infinite wire and an infinite conductive plane. Or maybe two wires with the voltage source between them. (Same answer, actually.) – Dave Tweed May 25 '13 at 19:59
  • Thanks Dave, actually, you're right. The concepts of semi-infinite wire and infinite conductive plane or infinitely-long dipole connected directly to a voltage source would explain everything I want to understand (this hugely simplifies my question :-)). Could you tell me what are the answers to my questions in case of, let's say, infinitely long dipole? – Andrzej Dąbrowski May 25 '13 at 21:28
  • No, I can't. As I alluded in my response to David Kessner, I think it depends very strongly on the diameter of the wire. $Z_0 = \sqrt{L/C}$ Capacitance per unit length goes up with the diameter, while inductance goes down. – Dave Tweed May 25 '13 at 22:12
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It's a weird thing but everything has capacitance. Even if you have a single point in an ideal empty space that point will have a capacitance that is equivalent to the electric field lines terminating at infinity. It's just that this capacitance is so very small that it is swamped by the relative coupling capacitance to other equipotentials in normal scenarios.

So in this case (as Dave Tweed said) your wire will have infinite capacitance, your "ground" will be wiggling up and down ... but it will have a small effect.

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The voltage along the wire will be described in both time and space by cable theory, first applied by Lord Kelvin to model decay in underwater cables. Today, the most common application is in biophysics. See http://en.m.wikipedia.org/wiki/Cable_theory

Scott Seidman
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