[Physics] Why are magnetic field lines circular around a current carrying wire

Question

I asked this question to several people.
Some say it is found by placing the compass around a current carrying wire. Some say magnets always exist as dipoles so field lines emerge from the north pole and end in the south pole and so it looks circular. And this user on Stack Exchange says:

Rather than constantly transforming back and forth between frames, we invent the magnetic field as a mathematical device that accomplishes the same thing.

And another user:

This electric field in the moving frame clearly exerts a radial force on any test charge originally at rest with respect to the wire in the stationary frame. But, given that there is no radial force or acceleration in the stationary frame, there also cannot be a net radial force on the charge when it is in the moving frame either. The force that counteracts the radial electric field in the moving frame is the Lorentz force due to a mystery (B-)field. As the Lorentz force due to the mysterious (B-)field ….

and I can't understand either.

If I place a proton just as in the diagram:

then from the proton's frame of reference, the protons on the wire appear to be denser than the electrons in the wire, so it must experience an outward force. So its path should be like a parabola. Field lines are found by tracing the path followed by a test charge or test mass or whatever, placing them in a field. So in case of magnetic field lines it should be a weird field line, not circular.

So why are the magnetic field lines circular? Or how is the magnetic field used as a mathematical device as mentioned?

I'm searching for an answer for three months. I got frustrated lot of times and still I am searching. Please help me. Please don't involve too much mathematics (I am just in 12th grade).

Answer 1

I have a simple argument in mind. for an infinitely long wire carrying current in the +X direction, the magnetic field at a given distance r from the wire, has 2 components(suppose) radial and tangential. Consider specifically the radial component and assume it points outwards(or inwards). Now, switch the current direction to -X. The radial component should also switch directions. But notice that this current is no different from what it was before, if you just turn the wire around(i.e) watch the wire from the other side, the situation is exactly similar to what it was before, yet the radial field direction has switched. This is an impossibility, proving that the radial field must be zero, leaving behind only circles, as the possible field shape. EDIT: CASE 1: from side A you see the current moving to the right. and the magnetic field is (suppose) radially outward. Now you reflect yourself about the wire and reach side B. You see the current moving to the left now, but the situation is essentially same(just observing from 2 sides of a wire), so the radial field is still outward. CASE 2:you return to side A, and now reverse the current(it now travels left), so the radial component must switch(inward). But havent you already seen this current? A current flowing left was encountered in CASE 1 from side B!, where you observed the radial field to be essentially outward. But now you observe it to be inward. CONTRADICTION. thus no radial field can exist.