# Orbital magnetic moment versus Biot-Savart law

In atomic physics, the fine structure of spectral lines assigned to atomic hydrogen has always been explained by considering an orbital magnetic moment of atomic electron. Still this concept is inconsistent with the basic laws of electromagnetism.

Consider a circular electric current of radius $r$, whose constant intensity is $I = nev/2πr$, where $n$ is the number of conduction electrons, $e$ is their elementary electric charge, and $v$ is their constant linear velocity. This circular electric current originates in surrounding space a dipole magnetic field measured by a magnetic moment $M = IS$, where $S = 4πr^2$. As all magnetic fields are generated by elementary electric charges in motion, this dipole magnetic field is evidently the vector sum of all magnetic fields originated simultaneously by all conduction electrons in the considered circular current. Conforming to Biot-Savart law reduced to the peculiar case of a single electron, the force lines of magnetic field generated by an electron in motion are at every moment concentric circles placed in planes perpendicular to its momentary linear velocity vector. Obviously, such a Biot-Savart magnetic field cannot be defined by a magnetic moment.

Now consider a single electron in uniformly circular motion on the same trajectory of radius $r$. Formally, we can assimilate this electron to a circular electric current with constant intensity $I = ev/2πr$, but evidently this single electron cannot originate a dipole magnetic field measured by a magnetic moment. The shape of all subsequent Biot-Savart magnetic field generated moment by moment by this single electron remains the same all the time, but their orientation in space changes every moment concurrently with the change of motion direction of the electron. And these subsequent Biot-Savart magnetic fields are not cumulative in time. It is very clear, a single electron in motion on a closed plane trajectory definitely cannot originate a dipole magnetic field defined by a magnetic moment. Therefore, the Biot-Savart law proves the falsity of the orbital magnetic moment invented for justifying the fine structure of spectral lines assigned to atomic hydrogen.

However, the fine splitting of these spectral lines is a reality experimentally noticed, so that the dipole magnetic field with magnetic moment about one Bohr magneton has to be found as responsible for this spectral splitting effect.

What is the only possible solution of this apparently unsolvable issue? Please use your logic to answer.

• I did not downvote, but here is the beginning of the third sentence from the first reference on the Biot-Savart law that I checked out: “The law is valid in the magnetostatic approximation”. I.e., this is not a “universal law”. In addition to research effort, the question is not very easy to understand. (E.g., what are “continuously successive magnetic fields”?). If I may make a suggestion, try to write simple sentences. – xebtl Jan 15 '15 at 8:13
• Not my downvote, but there are numerous Q/As on the site explaining that the motion of electrons in atoms cannot be explained by classical laws. The correct solution to this dilemma is that you can't apply classical laws like the Bio-Savart law to atoms. I would guess the downvote is because this is such an obvious point it should have occurred to you. – John Rennie Jan 15 '15 at 8:25
• I think your question question is abstruse, even after several readings. Moreover, it is usually difficult to answer a question starting with statements that look questionable at best. (didn't downvote though I hesitated to do so after reading your comment ^^) – TZDZ Jan 15 '15 at 8:29
• @Sava , nobody takes pleasure in giving minuses, nobody rushes to do it. First of all people try to answer. Now, it is not clear where you find a contradiction. Perhaps you know that even a single electron in movement creates a magnetic field. You can look at the question . Once making this clear, what else do you find problematic? I subscribe to what says John Rennie, but let me hear from you what is the next unclear thing from your point of view. – Sofia Jan 15 '15 at 17:46
• @xebtl - The Biot-Savart law is universal in the sense of its applicabilty to all charged elementary particles in motion (electrons, protons, etc.). The shape of magnetic field generated by their motion is established by this law even in the case of a single electron. – Sava Jan 19 '15 at 8:34