Speed of Electricity in Plate Capacitor & Cable

[Philip Koeck]

Consider a large, charged, plate capacitor with a distance of, let's say, 1 lightsecond between the two parallel plates and also a plain cable of the same length. Everything is in vacuum.
If I first connect one end of the cable to one plate and then connect the other end to the other plate the electric pulse will travel to the other end of the wire in one second.
Is that right so far?

Now if I use a cable that's twice as long, but don't change the distance between the plates, the cable obviously has to follow some curved path.
Would the electric pulse take 2 seconds now to travel through the cable or 1?

In other words: Does the distance between the plates or the length of the cable determine the time?

 

[berkeman]

Interesting problem. You need to figure out what the Transmission Line (TL) characteristics are for the different geometries, IMO. The propagation of an EM wave down a transmission line is determined by the geometry and dielectric materials used in the TL. The "Velocity Factor" which tells you how much slower than $c_0$ the EM propagates down the TL is given by this:

where L' and C' are the inductance and capacitance per unit length of the TL, and $c_0$ is the speed of light in a vacuum.
https://en.wikipedia.org/wiki/Velocity_factor

Your geometry is unusual for a TL, and I need to think more about it. But from a simple perspective (maybe not right yet), you could consider the C to be the capacitance between your large plates, and the L to be the inductance of your wire. For the case with a longer wire, you have to wind it somehow or zig-zag it back and forth to route it between the two plates, and this will alter the L' value to increase it most likely, so that is one of the things that will change (lower) the propagation velocity.

 

[Philip Koeck]

Interesting problem. You need to figure out what the Transmission Line (TL) characteristics are for the different geometries, IMO. The propagation of an EM wave down a transmission line is determined by the geometry and dielectric materials used in the TL. The "Velocity Factor" which tells you how much slower than $c_0$ the EM propagates down the TL is given by this:

View attachment 320775
where L' and C' are the inductance and capacitance per unit length of the TL, and $c_0$ is the speed of light in a vacuum.
https://en.wikipedia.org/wiki/Velocity_factor

Your geometry is unusual for a TL, and I need to think more about it. But from a simple perspective (maybe not right yet), you could consider the C to be the capacitance between your large plates, and the L to be the inductance of your wire. For the case with a longer wire, you have to wind it somehow or zig-zag it back and forth to route it between the two plates, and this will alter the L' value to increase it most likely, so that is one of the things that will change (lower) the propagation velocity.

If we take as an example a tightly wound coil that exactly spans the distance between the plates, with the length of the cable twice as long as the distance between the plates.
What does "propagation along the TL" mean in that case? Does the EM-disturbance travel along the cable (in a spiral) or does it travel in a straight line along the coil?
Would the time it takes for the pulse to reach the plate on the opposite side be between 1 s and 2 s or would it be longer than 2 s?
I'm trying to get a more detailed picture of the usual textbook/Wikipedia description that the cable is like a wave guide for an EM-wave that travels in the medium surrounding the cable (vacuum in my example).

 

[tech99]

All conductors tend to behave as transmission lines because they have distributed capacitance and inductance. If a wire is coiled up to reduce space, for instance, it still tends to be a transmission line, but the electrical length of the line tends to be less than the physical length of wire. I have seen a number of calculations to try and quantify this and it not an easy problem. We may notice that the coil has capacitance between its turns, which might alter the result.
Is your capacitor experiment simply to explore this point or is there something more you are looking for?

 

[Philip Koeck]

All conductors tend to behave as transmission lines because they have distributed capacitance and inductance. If a wire is coiled up to reduce space, for instance, it still tends to be a transmission line, but the electrical length of the line tends to be less than the physical length of wire. I have seen a number of calculations to try and quantify this and it not an easy problem. We may notice that the coil has capacitance between its turns, which might alter the result.
Is your capacitor experiment simply to explore this point or is there something more you are looking for?

Yes, I'm just trying to get a more realistic picture of how electric signals propagate.
Nothing more than that.
It seems it's really not as simple as often stated.

 

[apostolosdt]

Consider a large, charged, plate capacitor with a distance of, let's say, 1 lightsecond between the two parallel plates and also a plain cable of the same length. Everything is in vacuum.
If I first connect one end of the cable to one plate and then connect the other end to the other plate the electric pulse will travel to the other end of the wire in one second.
Is that right so far? [...]

I can't follow your reasoning. A 1 lightsecond distance is 300,000 km. A charged capacitor of such dimensions requires a LOT of charges to behave like an ordinary capacitor so that classical electrodynamics concepts can apply. So, at best, the entire concept is a bit bizarre. OK, but even if one attempted to do calculations on such a setup, one should consider what would happen to the cable inside such an enormous electric field. To go one step further, what about the physical vacuum quantum-field theoretical properties?
The solar radius is roughly 2 lightseconds, so we are talking about QFT, stellar physics, and more... all at once.

 

[tech99]

Yes, I'm just trying to get a more realistic picture of how electric signals propagate.
Nothing more than that.
It seems it's really not as simple as often stated.

A single wire line is a bit different to a pair of wires. I believe the electric field will mainly extend between points on the wire having differing potentials, whilst the magnetic field will circle the wire. With a pair of wires, the electric field is mainly between the two wires. In general I think we would say that both modes exist in differing proportions depending on the geometry.

 

[DaveE]

The formulae related to transmission lines all assume a relatively simple geometry so that solutions to Maxwell's Equations can be approximated accurately. Once you start describing difficult geometries you must reassess the simplifying assumptions or go back to FEA style field calculations. Even the question "speed of electricity" needs to be looked at carefully in weird setups. So, yes, it's really complicated, and no, IDK because I don't want to do the computer simulations required. OTOH, if you can make your problem look sort of like one of the classic scenarios, that would be a good starting point for an approximation.

If you want to investigate solved problems like this you could look into microstrip or stripline PCB trace routing and some of the serpentine geometries used to manage impedance and propagation delay in high speed circuits. Those may be solved problems, although in practice there's a lot of trial and error in the design process.

 

[Philip Koeck]

I can't follow your reasoning. A 1 lightsecond distance is 300,000 km. A charged capacitor of such dimensions requires a LOT of charges to behave like an ordinary capacitor so that classical electrodynamics concepts can apply. So, at best, the entire concept is a bit bizarre. OK, but even if one attempted to do calculations on such a setup, one should consider what would happen to the cable inside such an enormous electric field. To go one step further, what about the physical vacuum quantum-field theoretical properties?
The solar radius is roughly 2 lightseconds, so we are talking about QFT, stellar physics, and more... all at once.

You can replace 1 lightsecond with any distance you want and reformulate the question.
It's just a thought experiment.
My main point was to understand better how one can think of propagation of an EM-puls along a conductor.
The question was triggered by a statement on the internet that one can regard the conductor as a wave guide for a wave that propagates in the surrounding medium and that's why an EM-signal travels with the speed of light in that medium, but this doesn't seem to be completely accurate according to the answers I got.

 

[tech99]

Yes I agree that the line acts as a waveguide. We arrive at the idea that the electrons guide a wave along the wire. In order to move an electron we have to apply an electric field along the wire, and as the electron moves it has to build a magnetic field. This represents stored energy. Therefore we have to do work, and the energy is stored in the wave or pulse and travels away from us. When we drive a transmission line (or a single wire) with a sine wave of voltage (electric field), the generator sees resistance. The resistance arises due to the inertia (inductance) and springiness (capacitance) of the system, and usually amounts to a few hundred Ohms. If we apply a continuous sine wave, before long we experience reflected waves coming back, and that will alter the load which our generator is seeing. The topic of the relationship between fields and electrons has been discussed at length on the forum.

 

[apostolosdt]

You can replace 1 lightsecond with any distance you want and reformulate the question.
It's just a thought experiment. [..]

Sorry, but still confused. Of course, it's a thought experiment, but even such experiments should run within realistic frames. I would be cautious to play between 300,000 km and 1 mm plate distance and accept arguments and conclusions from one distance to the other.

My intention was not to discourage you from doing thought experiments---apologies if I unintentionally did so.

 

[vanhees71]

A pair of wires ("Lecher line") is pretty difficult to analyze. Even Hertz didn't succeed. You can find the proper treatment in A. Sommerfeld, Lectures on Theoretical Physics, vol. 3 (electrodynamics).

A simpler case is the coaxial cable.

 

[Philip Koeck]

Consider a large, charged, plate capacitor with a distance of, let's say, 1 lightsecond between the two parallel plates and also a plain cable of the same length. Everything is in vacuum.
If I first connect one end of the cable to one plate and then connect the other end to the other plate the electric pulse will travel to the other end of the wire in one second.
Is that right so far?

Now if I use a cable that's twice as long, but don't change the distance between the plates, the cable obviously has to follow some curved path.
Would the electric pulse take 2 seconds now to travel through the cable or 1?

In other words: Does the distance between the plates or the length of the cable determine the time?

Reading the older thread https://www.physicsforums.com/threads/speed-of-electricity.996609/, especially post 10, I see that my assumption that a signal would travel with exactly the speed of light through a straight bare wire in vacuum is actually wrong.
I guess that's because even for a single straight wire L' and C' affect the velocity.

 

[Philip Koeck]

Sorry, but still confused. Of course, it's a thought experiment, but even such experiments should run within realistic frames. I would be cautious to play between 300,000 km and 1 mm plate distance and accept arguments and conclusions from one distance to the other.

Good point. I guess when the dimensions of the set up become comparable to some typical wavelength of the EM wave or puls the whole theory should change quite a bit.

 

[vanhees71]

Reading the older thread https://www.physicsforums.com/threads/speed-of-electricity.996609/, especially post 10, I see that my assumption that a signal would travel with exactly the speed of light through a straight bare wire in vacuum is actually wrong.
I guess that's because even for a single straight wire L' and C' affect the velocity.

In a coax cable at least for the TEM mode the waves move with the "speed of light in the dielectric" as in free space, $c_{\text{matter}}=c/\sqrt{\epsilon_r \mu_r}$.

 

[tech99]

I suspect the slight disparity between speed of a wave on a single bare wire in vacuo and c is caused by the mass of the electrons, which add a small inductive component to the system due to inertia.

 

[tech99]

As a matter of interest, the proposed experiment, if scaled down in size to a length of about 2m, is the same as carried out by Heinrich Hertz in 1888 when he demonstrated the existence of radio waves. During his experiments, Hertz also carried out a "race" between waves in free space and those travelling along a wire.

 

[tech99]

Good point. I guess when the dimensions of the set up become comparable to some typical wavelength of the EM wave or puls the whole theory should change quite a bit.

If we want to scale the experiment up or down, at small sizes we will run into the atomic dimensions. But for all larger sizes the EM wave will simply scale up and down, being maybe three times the length of the wire. So I am not sure why the theory should change with size.
I also believe that the capacitance between the plates will be very small, and will be greatly exceeded by the self capacitance of each plate.

 

[hutchphd]

Reading the older thread https://www.physicsforums.com/threads/speed-of-electricity.996609/, especially post 10, I see that my assumption that a signal would travel with exactly the speed of light through a straight bare wire in vacuum is actually wrong.
I guess that's because even for a single straight wire L' and C' affect the velocity.

May I recommend Feynman's Lecture:
https://www.feynmanlectures.caltech.edu/II_23.html#Ch23-F16
I love this illustration. One of my favorite lectures overall. Spend some time