How is it proven that the magnetic lines are closed lines? Because the divergence of $\vec {B}$ is zero does not sound convincing, since fields like $\vec{F} = a\vec i +b\vec j+c\vec k$ (with $a$, $b$, and $c$ being constant) do have zero divergence; and yet the field lines in these fields are not closed. Can it be proven that the magnetic line are closed using the Biot-Savart law? If yes how can this be done?
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$\begingroup$ Closely related: Must magnetic field lines close upon themselves or go to infinity? $\endgroup$– Emilio PisantyCommented May 25, 2022 at 9:49
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In general, it is not true that the magnetic field lines will form closed loops—although that it what happens in a lot of simple geometries. What is true instead is that the magnetic field lines never terminate. (Since the points of termination of the electric field lines are charges—positive or negative depending on whether the field line runs outward or inward—this is equivalent to the statement that there are no magnetic charges.) For the example in your question, a constant magnetic field $\vec{B}=B_{0}\,\hat{n}$ (for some direction $\hat{n}$) of course has vanishing divergence, $\vec{\nabla}\cdot\vec{B}=0$. In this case, the field lines run infinitely in the $\hat{n}$-direction.
In realistic problems, we typically are interested in the field of localized current distributions, which often give rise to mostly closed loops for the field lines. The best-known example is of course the field of an infinite wire carrying current $I$, $\vec{B}=\frac{\mu_{0}I}{2\pi\rho}\hat{\phi}$. However, you can see that adding a small additional field in the same direction as the current will turn these field lines into helices that extend out to infinity; this happens because the small added field is a particular example of a constant field from the previous paragraph. However, it is also possible (as discussed here) to have lines of magnetic field that spiral around locally, without ever actually closing.
Moreover, even for fields with entirely localized sources, for which the magnitude of the field $|\vec{B}|$ goes to zero at spatial infinity, there can be isolated field lines that still extend forever. For a dipolar magnetic field (such as generated by a circular current loop), there is a single field line that runs right along the axis of the ring, which extends all the way from $-\infty$ to $+\infty$, even though all the other field lines eventually bend around and close.
