What would happen if a neutral wire and a hot wire were vastly different lengths?
In a 2 wire system the only thing you can do is make the total wire so long that it either acts inductively or resistively against your system. In the case of resistive it makes absolutely no difference where the wire is and in the case of inductive the total length or discrepancy therein is of much smaller interest than the path those wires take. For there to be a mile-long difference, you're likely coiling one up, which is much more interesting in that respect.

For three phase systems that directly use the phase synchronicity, such as fixed armature three phase motors, you can disrupt it all if you can delay one of the phases by enough.

Again, however, the speed of light being 3*10^8 m/s and the speed of electrons electricity [credit:Andrew] in free hanging copper being approximately 95% of that, you would need a huge deal of length to get enough displacement, say 10%-ish (rough guesstimate - too lazy to really think heavily on that number) of the 60Hz wave (or in my case 50Hz), to start noticing it. 10% of a 60Hz wave would take about 1.68&mu;s 1.68ms [credit: Jon].

In 1.68&mu;s 1.68ms the wave in copper would have travelled (approximate speed of electricity in copper) * (time) = (3*10^8 * 0.95)m/s * 1.68ms = 478m 478km.

Seems feasible, right? quite ridiculous already...

Well, to go on... Let's, for the ease of it, assume that my 10% was a correct guesstimate and it isn't 30%. Let's also assume that the other wires are all connected directly to the source and thus have no losses at all (good luck with that one), then still:

A 3 phase system usually needs at least 10A per phase (and that's tiny!), if you'd want to limit the total resistive losses to 5%, you'd need a resistance smaller than, if we assume an RMS voltage of 240V for your 120VAC 60Hz situation, of less than 12V/10A = 1.2Ohm. And losing 12V of your RMS is quite a lot to power engineers! To get 1.2Ohm at 478km of wire, and I won't throw around another formula, as when looking for the exact resistivity of copper I found this, which shows me for 47897000cm (= the value I got a bit more exactly, which I rounded to 478km) I'd need a wire diameter of 92.335mm. That's an area of (0.92335 / 2)^2 * pi = 0.67 dm2. I'm using dm, or 1/10th of a meter, for a reason. 47897000cm = 4789700dm. So the volume of copper you need is 0.67 dm2 * 4789700dm ~= 3207235 dm3.
1dm3 is the same as 1 liter. So that's 3207235 liters of copper. Copper weighs about 8.93kg per liter, so that's about 28640616 kg of copper. At currently about €5.30 per kg, that comes to € 152 million!! With only about €10 you can buy an inductor to screw things up just as bad.

Which brings me to the next problem. 478m of copper wire, diameter 2.9201mm, laid out in a perfect loop, without coiling it, from a nice site with a calculator which I did not verify as I am still lazy, is close to 1mH. So in the most efficient way you could place that wire you already get 1mH, assuming the calculator is right. Placed on a normal type of floor which is sort-of-grounded I suspect it to be higher. This may, depending on the motor, already be enough of an inductor to put things out of whack. But if you then want to put that wire in an orientation so that you don't need a HUGE amount of space to make a perfect circle, the inductance will increase significantly.

The calculator stops working at about 5km, but with the numbers I entered I'd say it's safe to say we get in the range of actual Henris, not mili or micro.



At the time of writing €1 is approximately $1.08



Note: I made a mistake in my simplification steps to get 5% of RMS 240V in a three phase means 1.2 Ohm, but since it's all blue-sky I'm not going to re-do it.



Another note: Just calculated "back-of-envelope"-style, because my brain wouldn't let it go, that if you'd want the wire to be coiled such that it could only just stand up in my living room you'd need 77 windings, making an inductor of 74mH, assuming no metals interfere. (And it's a largely concrete living room). you'd need many of my living rooms for the immense length the coil would take, making that a futile exercise.