Magnetic field lines can be open
27/11/24 00:30
Yes, you have read correctly. Magnetic field lines are not always closed. If you are surprised about this then you fell prone to a common misconception that is perpetuated by many textbooks (Sommerfeld, Griffiths, and many more) and instructors. This includes me until recently a student asked about this and I actually had a closer look.
The first thing to realize is that closed magnetic field lines are a sufficient condition for the validity of Maxwell's equation div B=0. But closedness is not a necessary condition. A simple example for this observation is a pair of identical Helmholtz coils with a common axis and equal but opposite currents. The magnetic field at the symmetry point between the two coils vanishes. The magnetic field lines approaching this symmetry point along the axis from either of the coils therefore have to terminate at the symmetry point. On the other hand, there are magnetic field lines starting at the symmetry point and running perpendicular to the axis. The overall effect is that the flux through a small sphere around the symmetry point vanishes, consistent with a divergenceless magnetic field. A second more generic situation are magnetic field lines that continue forever without beginning or end, but also without being closed. Imagine the closed field lines around a single Helmholtz coil, which you then perturb by a current running through some wire nearby. For a weak perturbation the field lines still wrap around the Helmholtz coil, but in general they will do so in an incommensurate manner without being closed. As a matter of fact, closed magnetic field lines are a set of measure zero in the space of current geometries. However, many textbook examples consider specific simple geometries leading to closed lines, which one then believes to always be the case. Putting it otherwise: Simply solvable cases can really give you the wrong idea about what is generic.
All of this is perfectly well known since at least 1929 (textbook by I. Y. Tamm), but the wrong picture is prevalent. Just go around and ask your fellow students/colleagues, the reactions can be quite entertaining. For more information including lots of references and the two examples above (plus many other examples) I recommend the following two articles: The Misconception of Closed Magnetic Flux Lines and A taxonomy of magnetostatic field lines (from 2024!).
The first thing to realize is that closed magnetic field lines are a sufficient condition for the validity of Maxwell's equation div B=0. But closedness is not a necessary condition. A simple example for this observation is a pair of identical Helmholtz coils with a common axis and equal but opposite currents. The magnetic field at the symmetry point between the two coils vanishes. The magnetic field lines approaching this symmetry point along the axis from either of the coils therefore have to terminate at the symmetry point. On the other hand, there are magnetic field lines starting at the symmetry point and running perpendicular to the axis. The overall effect is that the flux through a small sphere around the symmetry point vanishes, consistent with a divergenceless magnetic field. A second more generic situation are magnetic field lines that continue forever without beginning or end, but also without being closed. Imagine the closed field lines around a single Helmholtz coil, which you then perturb by a current running through some wire nearby. For a weak perturbation the field lines still wrap around the Helmholtz coil, but in general they will do so in an incommensurate manner without being closed. As a matter of fact, closed magnetic field lines are a set of measure zero in the space of current geometries. However, many textbook examples consider specific simple geometries leading to closed lines, which one then believes to always be the case. Putting it otherwise: Simply solvable cases can really give you the wrong idea about what is generic.
All of this is perfectly well known since at least 1929 (textbook by I. Y. Tamm), but the wrong picture is prevalent. Just go around and ask your fellow students/colleagues, the reactions can be quite entertaining. For more information including lots of references and the two examples above (plus many other examples) I recommend the following two articles: The Misconception of Closed Magnetic Flux Lines and A taxonomy of magnetostatic field lines (from 2024!).