# Voltage and Current along a Wire

At first I was looking for a physical explanation of the fact that energy or the signal, respectively, travels much faster through a wire than the electrons inside of it do (drift velocity).

After reading the article, I understand that neither the amount of current flowing around, nor the voltage between two points is responsible for the speed of the energy transfer. It’s rather the material surrounding the conductor that’s affecting the propagation speed with its permittivity and permeability. Is this correct up to here?

Figure 4, however, makes me wonder two things:

1. Of course, every point of a conductor has the same potential and inside of conductors, there won't be an electrical field. But why is that? You learn that since we're inside a conductor, a current will work against every potential difference that might exist between two connected points, thereby compensating it. But isn't current (the actual flow of charge carriers) incredibly slow? And doesn't this make an immediate compensation impossible?

2. Now let's say two connected points will always have the same potential. When talking about RF engineering, you learn that as soon as the wavelength becomes of the order of the physical wire's length, you will measure different voltages across one single wire. How does this correspond to the idea of an immediate response of the charge carriers?

What am I missing?

• One thing you should understand is that there is an important difference between an ideal conductor and a physical (real) conductor. The voltage at every point on an ideal conductor is, by definition, the same but this is not necessarily true of a real, physical conductor. Commented Apr 23, 2014 at 16:46
• @JoeHass, even with ideal conductors, it takes time for a signal to progagate along a transmission line. And you measure different voltages at different points on the line at any given point in time. Commented Apr 23, 2014 at 16:55
• @ThePhoton In my experience, an ideal conductor and a transmission line are two very different things. In SPICE, for example, a transmission line is a specific circuit element which is not simply constructed from ideal wires. Ideal wires have no properties or parameters, transmission lines do. Commented Apr 23, 2014 at 17:08
• @JoeHass, the basic model of a transmission line assumes the wires that make it up are ideal (0 resistivity). My point is that OP is asking about the difference between distributed circuits and lumped circuits. This distinction is orthogonal to the difference between ideal conductors and physical conductors. Distributed circuits behave differently from lumped circuits, even if they are made from ideal conductors. Commented Apr 23, 2014 at 18:13
• @ThePhoton I didn't try to answer the question, I just added a comment to make sure the OP understood the difference between an ideal conductor and a physical conductor. In the same paragraph the OP says "every point has the same potential" and "every potential difference that might exist between points". Orthogonality between my point and yours doesn't make mine irrelevant. I didn't downvote your answer or say that it was incorrect. Commented Apr 23, 2014 at 19:17

1 - Current is not "the actual flow of charge carriers" any more than sound is the actual movement of air molecules. To equalize a potential difference between points A and B, it is not necessary for any of the charge carriers originally at point A to show up at point B (just as none of the air molecules at a sound source actually make it to your ear). Maybe a better analogy is turning on the faucet at the kitchen sink. When you do this, the pressure at the spout is less than the pressure in the water main and this forces water to come out. But, in the time it takes to fill a glass, no water that was previously in the main will make it out of the spout.

2 - There is no such thing as an "immediate response of the charge carriers". Nothing "real" moves from place to place faster than the speed of light. Thus, when you consider frequencies with wavelengths comparable to the physical size of your conductor, potentials and currents will be different at different places. Don't let anyone tell you otherwise.

What am I missing?

When you step into a full bathtub, the water level rises instantly everywhere. Yet your foot-meat didn't leave your bones and fly outwards to all points! The water only moves slowly. How could the water level rise instantly? You're missing the concept that wires are not like hollow pipes. Instead they're already full of mobile charges; typically ten thousand coulombs per cc.

Wires act like bathtubs, or like long canals, where if you stick your foot in (or pour more water in,) the water-level rises everywhere, seemingly instantly. (Go fill your tub and try this!) Yet the water itself barely moves, and some blue-dyed water which you added stays where you placed it.

You'll discover that with very long canals, the distant water-level doesn't rise instantly. Measure the delay, and you find that it's caused by the speed of sound in water. So, when you step into a bathtub, your foot must be sending out a high-velocity pressure-wave which pushes the water outwards, and forces the water-level upwards. The water is the medium, and only moves slow, while the wave propagates at enormous speed. It works similarly for wires. The charge is the medium and the energy is the wave. Coulombs and amperes describe the medium, while joules and watts are measurements of the waves.

The second thing you're missing: even with zero-ohm superconductors, if you wind a coil, the inductance won't be zero. This means that any sudden changes in potential cannot spread instantly to the whole wire. True, for zero-ohm conductors there cannot be DC potential differences anywhere. But AC is a very different matter, because impedance (reactance) then plays a role.

Make a shorted one-turn inductor using zero-ohm wire. Now stretch it out, make it into a narrow loop 300,000KM long. It won't be a simple inductor anymore, because the two sides of the circle are spaced closely, and have significant capacitance. Next, cut one end of the loop and suddenly insert a battery. The shorted loop will obviously draw a current, but it won't be infinite, even with zero-ohm wire. Instead the loop will act like a resistor of a few hundred ohms. Sound familiar? There's a wattage going outwards from the battery, and joules of battery-energy seem to be vanishing into the zero-ohm wire. The energy is being stored as a magnetic field surrounding the two halves of the long loop, and also as an e-field between the two halves. A long, long "sausage" of EM field-energy is spreading out along your narrow loop, and since the loop provides a load on your battery, the loop must act as a resistor. Weird: make your loop much shorter or much longer, and this "resistance" remains the same!

If there is a light bulb connected into the far end of your 300,000KM loop, the battery will light it up, but after a delay of one second. The EM fields spreading along the loop were moving at the speed of light in air. Try adding heavy plastic insulation to your loop. Yep, it now takes longer for the light bulb to flash. Your loop is a waveguide for EM. But not for microwaves, it's working at DC! The name for all this stuff is "transmission line." A transmission line is an EM waveguide, but unlike hollow microwave waveguides and optical fibers, it operates independent of frequency. It works fine all the way down to 0Hz.

The math behind all of this was worked out by Oliver Heaviside, who was trying to figure out why Morse-code dots on long telegraph lines were getting stretched out and blurred as they traveled. He discovered that, if we add the wire-resistance back in, then waves of different frequencies travel at different speeds, all a bit slower than "c". Lol, conspiracy theory, he was attacked in print and nearly silenced by W. Preece who ran the British telegraph system. Then he had his ridiculed ideas stolen by M. Pupin of Columbia U., who became a multi-millionaire by patenting them and selling them to Bell Telephone. This cured the unexplained distortion which made voices unintelligible on phone lines longer than miles. So, essentially Heaviside invented the long-distance telephone network. Look up Heaviside's "Telegrapher's Equation." And as with all of these stories, after his enemies got rich off stolen ideas, Heaviside died penniless, and didn't become world-famous until later.

I think your main missing link is that the concept of electrical potential, as it's initially taught, is a concept of electrostatics, and doesn't apply in dynamic (ac) situations. From Wikipedia,

When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply in terms of a scalar potential V because the electric field is no longer conservative.

The concept of potential can be extended to ac and distributed circuits, as outlined in the Wiki article. However this requires first defining a magnetic vector potential, which requires first choosing a gauge. As far as I know the ac potential concept is not used in much circuit design work, but is used regularly by antenna designers.

1. I think you have misunderstood the meaning of current. The current is the consequence of a flow of electrons (charges), but the speed of the current has not sense, the speed of the charges is proportional to the current, but both concepts dimensional are different ( Amperes and m/s are not the same). The propagation of the current and voltage variations in a wire is quasi immediate ( light speed in ideal conductors), the propagation act like an electromagnetic wave and follow the Maxwell equations. In DC the charges move very slow and in AC even slower but the energy transmission into the wire is a electromagnetic wave.
2. Additionally, you say "every point of a conductor has the same potential and inside of conductors there won't be an electrical field" it is not always so, depend of the type of signal due to the Kelvin or Skin Effect, the density of charge into a wire section increase toward the external surface with AC currents.
3. You refer to RF,then think in Maxwell equations to understand the energy flow and not with Newton mechanical concepts due to the movement of the charges. The energy is caused by EM field variations and its propagation (Maxwell), and this variations are caused by the movement/oscillations of the charges, and doesn't depend on the distance that the charges moved in a direction into the wire.