A taxonomy of magnetostatic field lines

Magnetic field lines were introduced by Michael Faraday, who was inspired by the pattern of iron filings in the vicinity of a magnet. They are sometimes called “magnetic flux lines” or even “magnetic lines of force” (which is grossly misleading).

R. P.

Feynman

,

R. L.

Leighton

, and

M.

Sands

, ( , , ), Vol. , Secs. 1–5.

Arguably, even number 3 is more a choice than a requirement. But in the magnetic case, as we shall see, the density of field lines cannot (in general) be taken to represent the strength of the field—unless we are prepared to countenance the abomination of field lines that are discontinuous at arbitrary points.

Zilberti traces the belief that magnetic field lines always form closed loops back to Faraday (1855) and Maxwell (1855) and suggests that Liénard (1921) and Tamm (1929) were the first to challenge it:

L.

Zilberti

, “ ,” , ().

The modern literature on non-closed magnetostatic field lines begins with

J.

Slepian

, “ ,” , – ()

and

K. L.

McDonald

, “ ,” , – ().

Much of the material in our paper is well known to those who study the sun's corona. See, for instance,

D. W.

Longcope

, “ ,” , ().

In highly conductive plasmas (such as the sun's corona), the special environment afforded by magnetohydrodynamics endows magnetic field lines with a kind of physical “reality” that they do not enjoy more generally. They can be spectacularly visible in photographs of solar flares for instance (but of course we are not really seeing the field lines themselves—only radiation from the charged particles that track them).

These are known as “null points” in the literature. In non-static situations, magnetic field lines can disconnect and reconnect (to a different line) as a null point passes by, a phenomenon observed in the solar corona. See, for instance,

A.

Zangwill

, ( , , ), Sec. 10.6.1.

For the general theory of null points, see

C. E.

Parnell

et al, “ ,” , – ().

One advantage of using $λ = | F ( r ) |$ is that it finesses the ambiguity in $F / | F |$ at null points. Of course $λ ≯ 0$ at such a point, but since the field line never actually gets there, it remains the case that $λ > 0$ for all points on the field line.

Note that there is no violation here of $∇ · B = 0$—this is a saddle point. Incidentally, a point where $B = 0$ is (obviously) a minimum of $| B |$ (and so also of $| B | 2$⁠), but there's no law against that: $| B |$ can have local minima but not local maxima. See

M. V.

Berry

and

A. K.

Geim

, “ ,” , – ().

The formula for the field, in terms of elliptic integrals, is given in

W. R.

Smythe

, , 3rd ed. ( , , ), Sec. 7.10.

This striking example was apparently first proposed by I. E. Tamm, in his 1929 textbook. For an English translation of the 9th edition, see

I. E.

Tamm

, ( , , ); the relevant sections are 412 and 413.

For recent discussions, and further references, see

M.

Lieberherr

, “ ,” , – (); L. Zilberti, Ref. 4.

I. S.

Veselovsky

and

A. T.

Lukashenko

, “ ,” , –, https://doi.org/10.1134/S0016793216070161 ().

It might be a spinning sphere with surface charge $σ ( θ ) = σ 0 cos θ$ and angular velocity ω₀ (about the z axis). In that case, $K 0 = σ 0 ω 0 R / 2$⁠.

P. J.

Morrison

, “ ,” , – ();

J.

Aguirre

and

D.

Peralta-Salas

, “ ,” , ();

M.

Hosoda

et al, “ ,” , (); Lieberherr, Ref. 10 and Veselovsky and Lukashenko, Ref. 11.

It is not always clear from simulations whether field lines that start out chaotic remain so indefinitely, as we follow them out; Hosoda et al, Ref. 13, refers to this as “transient chaos.”

S. M.

Ulam

, ( , , ), Chap. VIII, Sec. 6;

S. M.

Ulam

, ( , , ), Chap. 4, Sec. 5.

Lieberherr, Ref. 10, recommends adding a small constant magnetic field, to prevent (or at least delay) the field line's tendency to run off to infinity. M. Hosoda et al, Ref. 13, do the same thing for a different reason, noting that in practice such a perturbation is typically present in the form of the earth's magnetic field.

For further discussion, see

F. G.

Gascon

and

D.

Peralta-Salas

, “ ,” , – ().

The calculation is essentially the same for each symmetry, so we will do it just once, in  Appendix A.

D. J.

Griffiths

, , 5th ed. ( , , ), Problem 5.19.

See Griffiths, Ref. 19, Example 5.10.

Others have reached a somewhat different conclusion: Ştefănesçu says “[The assertion that all magnetic field lines are closed] is correct in the case of co-planar electric circuits [i.e. filamentary currents].” Lieberherr writes “[the field lines of] planar polygons of [arbitrary] shape … close after one loop or they do not return [i.e. they run off to infinity].” But already in 1954, McDonald provided a striking counter-example, his Fig. 2(b) (our Eq. (24) is another).

S. S.

Ştefănesçu

, “ ,” , – (). Lieberherr, Ref. 10 and McDonald, Ref. 4.

The examples discussed here (spheres, circular cylinders, symmetrically placed circles) also exhibit mirror symmetry, but that does not appear to be relevant. Other spinning figures of revolution—cones, for example—also give rise to closed field lines.

We take $κ = 1 / R$ (where R is the radius of the osculating circle) to be non-negative; then $n ̂$ points toward the center of that circle.

See, for example,

K. F.

Riley

,

M. P.

Hobson

, and

S. J.

Bence

, ( ., , ), Sec. 8.3.