Hello again,
That's pretty much it yes
There's one tiny detail about Ampere's we can take a quick look at. Ampere's works in three dimensions so it describes what the field should be for any circumstance. However, the way we "usually" apply Ampere's is where we assume that B is not really a vector, and this limits the geometry scenarios where we can apply Ampere's. Well, we assume it is a vector but has only one component and that the curl of B is also a vector with only one non zero component. What this means for a straight wire is that we have to make sure that the curl of B only has one constant component in the direction related to B itself, and the only way we can get that is with certain circumstances like with an infinite wire. An infinite wire works because everywhere the curl of B has just one non zero component. With a finite length wire however, the curl has at least two components (really three but two would be the same) and that means we'd have to find a function that describes the behavior of B as we move out from the center of the wire. Normally all we have to do is find what B is wrapping around the wire (infinite wire), which is only one dimension. Funny, we could probably do this using Biot-Savart but that might be cheating
Perhaps there is a more elementary theory we can use to do this. What i think we would end up with is a rule like Biot-Savart where we relate the B at every point out from the wire to the length and radius. That would make sense because that's what BS does already
Maybe we can find some history on Biot-Savart to find out how they did it (experimentally or functionally starting with Ampere's).
I think i've explained this correctly but keep in mind it has been some years since i actually did a lot of calculations like this. Biot-Savart seems simple enough so that's my favorite
Maybe if we started from BS and worked backwards using vectors we might find Ampere's at the heart
Here's a view of the magnetic field 'equipotential' lines out from a wire looking in one plane. Note that they do curve. With an infinite wire we'd see just straight horizontal lines.
These points were calculated using Biot-Savart. The points to the far left and right are just beyond the ends of the wire, the wire is red.