Would we need a battery if there was no resistance?

I'm struggling to wrap my head around this idea.

Imagine that we have a wire with no resistance ($$R=0$$). According to Ohm's law $$V=IR$$, the voltage drop ($$V$$) between any two points would be 0, and from my understanding, voltage drop is how much work is needed to move a unit of charge between the two points,

$$W=Fd$$

And we know there is definitely a distance between the two points,

$$\because V=0$$

$$\therefore W=0$$

$$\because d > 0$$

$$\therefore F=0$$

So the electrons will be moving at a constant velocity.

Now, for the part that is confusing to me, if the electrons will continue to move at a constant velocity with no force, what would we want a battery for?

Secondly, if we connected a battery, the battery will create a voltage difference between two points of the wire, precisely two points directly touching the terminals of the battery, doesn't that break Ohm's law?

• Energy must still be conserved, so in the case of no resistance in the wires, any work done by a device that is connected to the battery by those zero resistance wires, is ultimately provided by that battery. – David White Jul 24 '19 at 15:29

Now, for the part that is confusing to me, if the electrons will continue to move at a constant velocity with no force, what would we want a battery for?

For one thing, it's not really correct to think of the electrons moving at that velocity. A charge moving through the circuit isn't really best described by electrons moving at that speed. Here's the first example I found of where that is discussed on Electronics Stack Exchange.

Putting that aside, moving with no force isn't particularly useful to us. Sure, the charge would theoretically move through a wire with no resistance indefinitely (regular wires have resistance anyways) but if you actually wanted that charge to do anything, you would need to extract energy from it. Extracting this energy would require the charge velocity to be reduced, which can only be done so much until the charges are no longer moving. We want the battery because energy conservation still holds, so we want to be able to maintain that charge velocity while also extracting useful work from the electromotive force.

Secondly, if we connected a battery, the battery will create a voltage difference between two points of the wire, precisely two points directly touching the terminals of the battery, doesn't that break Ohm's law?

No. Real batteries have some resistance in them, and thus Ohm's law can be applied when those properties of a real battery are accounted for. Ideal voltage sources (how we often model batteries in simple circuits) have no resistance by definition, and thus Ohm's law isn't even applicable. See for example this answer or the Wikipedia page on ideal voltage sources.

If you had a superconducting wire, you could deliver a charge to the other end of it without providing any energy input (thus in some sense "not needing a battery")

But normally when we send current through a wire, we want it to do something useful at the other end of the wire. For example, we might want to produce light from a lightbulb, or calculate something in a computer. Those results still require power, even if the wires connecting the power source to the device doesn't waste any power. So we'd still need a battery (or other power source) to power whatever it is on the other end of the wire.

Even sending data, for example in an Ethernet network, requires delivering energy to the receiver, even if the wires used are lossless.

In general, you will find the "ideal" cases simply do not exist. You cannot make a wire of zero resistance, so all of the nice easy simple equations you are learning will not apply. This is similar to the argument made when you construct an "ideal" circuit short circuiting a battery. The simplified laws just fall short.

However, in this particular case, there is a practical example of this: superconducting electromagnets. Superconducting materials do indeed have exactly zero resistance. Superconducting electromagnets can be incredibly powerful. They don't waste any energy on resistance of the wires in the magnet's coils, so you can reach more extreme magnetic fields.

If you hook a battery up to one of these coils, it basically will act like a short. You'll see the internal resistance of the battery, resistance of the feeder wires, and that's it. The superconducting section simply wont add any resistance to the story.

So when they charge up one of these superconducting magnets, what they end up doing is first creating a loop through the power source. They cool down the magnet until the material superconducts, and then they start driving it with the power source. This current is limited by something called "inductance," which you will learn about. Eventually the power source will have built up the magnetic field to the desired point. At this point, a superconducting shunt is placed across where the power source is feeding the superconductor. Once this is done, you can remove the power source, and you have a superconducting loop, which goes on forever, because it has zero resistence.

Well, almost forever Eventually magnetic coupling with the rest of the world will transfer energy out of the coil, or the coil may "quench," which causes it to cease to superconduct. But for the purposes you are looking at, the electromagnetic fields will simply propagate energy forever around the loop.