Solving a Perplexing Thought Experiment: A DC Electrical Circuit

[sophiecentaur]

Very little difference unless the wavelength involved were very short. More power would be radiated on the way.

 

[Grinkle]

1 more question for anyone, what (if anything) would be different if the battery were replaced by an A/C power source?

Imagine the very long wire is now a very long rope or string or slinky (whichever is easier for you to picture) and it is completely lossless. If you whip the front end of the string you will cause a wave to propagate along the string (or rope or slinky). If there is no loss of energy in the wave, it will go all the way to the end of the string and if it is not dissipated somehow at the far end of the string it will then reflect back towards you. If you continuously whip the front end, you will send a continuous wave towards the far end. This is sort of what A/C voltage is like along the lossless / zero resistance wire. Lossless means, among other things (like heat dissipation due to resistance) I am sure, no radiation.

I struggle to come up with a similar analogy for continuous DC current. Strings / ropes / slinkys don't displace due to force in any way that I think of as similar to DC voltage in a wire.

 

[sophiecentaur]

I struggle to come up with a similar analogy for continuous DC current. Strings / ropes / slinkys don't displace due to force in any way that I think of as similar to DC voltage in a wire.

A mechanical analogy to a DC signal, after it has been switched on and connected to a transmission line, could be to suspend a stiff rope between two points and move one of the points suddenly sideways. The description of both those would be a Step Function. A step function will propagate along both.

 

[Grinkle]

@sophiecentaur The transverse wave on the string is perhaps a bit subtle/deceptive because the analogy to current in a wire is the transverse velocity of the string, not the displacement of the string. I think in your DC proposal you are creating a displacement step, not a velocity step.

Is the below better?

@FreeForAll The below image has the major flaw of not separating the conductor from the current - in the below image the conductor is the current and that is dangerously incorrect in terms of forming early intuitions about electricity. In an actual electric circuit, the picture of a transverse wave moving along a string is much closer to current moving through a wire.

Imagine a long zero-mass rigid cord - this is a zero-inductance long wire. Now let the mechanism that moves the cord be zero-loss, a zero resistance wire.

Instead of the cord moving in a transverse way, it moves along the direction of its own axis. For an A/C analogy, the cord reciprocates. For D/C, the cord moves in one direction at a constant velocity. The D/C analogy requires a reservoir of cord to draw on for a single path conductor analogy. To provide an intuition of the difference between and open circuit and a closed circuit, one needs to bend the rigid cord with a rigid-cord bending device at the far end and continue the path of the still-feeding cord back to the source. A closed circuit will take in the returning cord at the source and let that cord return to the cord reservoir. An open circuit stops moving when the the return cord hits the destination point and is blocked there. The entire loop is present at zero velocity before the "experiment" starts. The rigid cord will have very little compliance in it (capacitance in our electrical wire) so one can intuit that the open circuit will mostly result in a force along the rigid cord with very little actual motion of the cord, and one can see that if the ideal wire has zero capacitance in addition to zero R and zero L, there will be no movement at all.

 

[sophiecentaur]

@Grinkle . There are many possible analogies and many of them fit particular situations better than others. I would say that a step function of position is a good analogy to the step function of Voltage but it's a matter of taste. I just tried to make an analogy with an experiment that we could all do. In fact, a longitudinal wave on a rod would be better because you can get a displacement of the whole length of rod. A rope fails in many ways to fit the requirement; it's just familiar.

 

[Joseph M. Zias]

Keep in mind that the power actually flows outside (albeit very close) to the wire. This is determined by calculation of what is known as the Poynting vector.
There are quite a few references to this on the web, the earliest I know of is 1963. John Kraus in his book "Electrodynamics" - 4th edition illustrates this on page 578.