How does a current carrying wire create circular magnetic fields around it? I know that when a wire has a current flowing through it a circular magnetic field is induced around it, sometimes called the right thumb rule. How exactly does this occur?
I have found many resources showing this and what it is but I cannot find anywhere why this happens. Everywhere I look electromagnetism examples stem from this fact, is this just the known and observed relationship between electric and magnetic fields without a specific known cause?
My knowledge level is A level physics (UK) if that is any help.
 A: Physics, as a science, serves an important purpose by suggesting models and equations in order to describe natural phenomena as accurately as possible.
So, while we may never know why electrical current creates a magnetic field around it, or perhaps, why the speed of light is constant in all reference frames - we are aware of these phenomena due to various observations, experiments and deduction - respectively, Hans Christian Ørsted's observations and the theoretical results and empirical findings by Albert Einstein, Max Planck and many others.
In short - we don't know why electrical current creates a magnetic field around it, but we have observed it occurring many times in order to make laws and equations around it, and take it as a given when performing more complex calculations and deductions.
A: It is very difficult to answer a “why” question when you have not specified what is allowed to be assumed or what is understood. I think that the minimum assumption is conservation of charge and the Hamiltonian formulation of the laws of physics.
From the conservation of charge we can apply Noether’s theorem to find a corresponding symmetry in the laws of physics. In this case the corresponding symmetry is the U(1) gauge invariance of electromagnetism.
From the U(1) gauge invariance it is possible to derive Maxwell’s equations as shown here. https://aapt.scitation.org/doi/10.1119/1.11327
Finally, once you have Maxwell’s equations Ampere’s law says that when you have a current you will get a magnetic field curling around it.
A: You recognized well, that (since 1900) there isn’t any progress in the research of the induction of magnetic fields.

Physics, as a science, serves an important purpose by suggesting models and equations in order to describe natural phenomena as accurately as possible. (uriyaba)

Here is one possible model, which has to do with the cylindrical symmetry (see the comment from PrawwarP):
At the beginning some needed basics:

*

*The electron is a particle with its own electric field. It is less common that the electron is also a particle with its own magnetic dipole (and a fundamental physical constant).

*The electric field of electrons is observable by the separation of charges (with another electric field, for example from an electric generator). The magnetic field is observable by the alignment of the magnetic dipoles of electrons by another magnetic field, for example a permanent magnet).

*Furthermore moving electrons inside a conductor get deflected under the influence of a magnetic field (Hall effect or more broad Lorentz force). And - not surprising now - an external changing magnetic field deflects electrons (by the interaction with their magnetic dipoles) and by this induces an electric current.

At this point - not a suggestion of a physical theory (a thesis), but and only - the conclusion of the known phenomena described above. The electrons, moved (by an electrical potential difference) within a conductor, align the magnetic dipoles of these electrons and due to PrawwarP the common magnetic field must be cylindrically symmetric. This is the circular magnetic field whose nature you are asking about.

One remark. Without an approach (Ansatz) like the above description, and ignoring the intrinsic (i.e., existing independently of any circumstances) nature of the magnetic dipole of electrons (I must repeat myself by mentioning the basic physical constant of the $g_e$ factor), the description is as follows: For each moving - in relation to the own position - charge a magnetic field is created. Please keep this in mind for all examinations.
