110. Closed B field lines - Abteilung für Didaktik der Physik

Historical burdens on physics 110 Closed B field lines

Subject: It is often said that magnetic field lines are closed: “The […] difference is that electric field lines always begin on positive charges and end on negative charges, whereas for magnetic field lines there are no points in space, where they begin or end, since magnetic monopoles do not exist. Instead magnetic field lines form closed loops.” “The magnetic field of a current has always closed force lines, in contrast to the electrostatic field lines, which start on positive charges (sources) and end on negative charges (sinks).” Deficiencies: 1. Normally, when referring to magnetic field lines, the field lines of the magnetic flux density B are meant. The fact that field lines can have a beginning and an end is not a peculiarity of the electric field. Just as the E field lines of an electrostatic arrangement have a beginning and an end, the H field lines of a magnetostatic arrangement have a beginning and an end: They begin on a north pole of a magnetized body and end on a south pole. 2. The fact that the B field is divergence-free does not allow for the conclusion that the B field lines are closed, and indeed, in general they are not. The equation div B = 0 only tells us, that the field lines do not have a beginning and or an end. What do we mean in the first place when we say, field lines are closed? Probably anybody who hears the statement will imagine something like the following: We have an electric current that is flowing on a well-defined path, typically within a wire. An arbitrarily chosen field line runs around this current. When beginning at one point of the line and following the line we come back to our starting point after one single turn around the current. However, there is no physical reason why the line should close after one turn. And practically there is only a vanishing small chance that this would happen. If exceptionally the line does so, the reason is not so much physical but rather geometrical. It is indeed so in the case of a straight wire of infinite length, or in the case of an electric circuit that is completely confined to a plane. The smallest deviation from this restriction makes that the line, after executing one turn misses its point of departure. One might believe that the field of a cylindrical or toroidal solenoid has closed field lines, but they don’t [1]. The unavoidable helicity of the coils is the reason why a field line does not meet its starting point after one turn.

Fig. 1. Straight current and circular current. The field lines in the plane of the ring form a lefthand helix inside and a right-hand helix outside of the ring.

B vectors

Electric current

Fig. 1. Straight current and circular current. The field lines in the plane of the ring form a lefthand helix inside and a right-hand helix outside of the ring.

From Fig. 1, which shows a straight current-carrying wire and a circular current, it can be seen that field lines are in general not closed. We consider the field vectors in the plane of the ring. At the inner side of the ring the superposition of the fields of the circular current and of the straight conductor result in a left-hand helix; outside of the ring it is a right-hand helix. Obviously, the field lines cannot be closed. Fig. 2 shows another example of field lines that are not closed. A cylindrical homogeneously magnetized flexible permanent magnet is twisted around the axis of the cylinder, and then bended and closed to form a torus. The lines of magnetization, and thus the B lines now spiral round the torus axis (which previously was the axis of the cylinder) and never close.

Fig. 2. A flexible permanent magnet which initially was cylindrical has been twisted and bent.

Magnetic fields in nature, for instance those of the Earth or cosmical magnetic fields are so intricate that nobody would suspect that field lines might ever be closed. Another technical example where a field line after running around a current misses its point of departure by far is the magnetic field of a fusion reactor. It was shown in a beautiful article in the American Journal of Physics in 1951 by Joseph Slepian [2] that B field lines are in general not closed. The article does not contain a single equation or figure. In the following decades several other articles were written about the subject, see [1, 3] and the literature which is cited there. Origin: 1. At school and in the University lecture we usually only treat simple magnetic fields: the field of the bar magnet and of the horse shoe magnet, the field of electric currents in a straight conductor or in a conducting ring, and the field in a solenoid. The B field lines of an ideal bar magnet, i.e. a bar magnet with a perfectly homogeneous magnetization, or the field lines of a perfect horseshoe magnet are indeed closed; the same is true for the field

of a perfectly straight wire or a perfect circular conductor. In the case of a solenoid it is true only approximately. The fact that mainly these sources of magnetic fields are considered may explain why it seems plausible that the field lines are always closed. 2. Field lines are a graphical tool for the representation of field strength distributions. However, students often perceive them as something to which a physical reality can be attributed. If one imagines the field lines as physical objects, there is an argument in favor of the idea that field lines are closed, even if it is not after one single turn around a current carrying conductor. Instead of a field line consider a thread. Someone has made of the thread a mazy clew and assures us that the thread has no beginning and no end. In this case the conclusion that the thread forms a closed loop or several closed loops is correct. Why does this argument not hold for magnetic field lines? Field lines are no physical objects but mathematical objects, i.e. lines. All one could say in the best case is that, when following a field line, one may come as near as one wants to the point of departure. Disposal: Avoid saying that field lines are closed. It is enough to say that they have no starting point and no end. However thereby one would only try to eliminate a symptom; the roots of the evil are deeper. The proper cause of the error is the misconception of the field lines as physical objects. Therefore, it is more important that when introducing the field concept one does not begin with representing the field by field lines. The picture to show first could rather be a representation of the energy distribution of the field by means of a gray shading, Fig. 3a.

Fig. 3. (a) Representation of the field of a solenoid by means of gray shading; (b) Representation of the field of a solenoid by means of gray shading, field lines and field surfaces

Only thereafter one shows that the field has in every point a preferential direction, i.e. it is not isotropic. To graphically represent this fact one begins by drawing vector arrows. Next one comes to a representation that is more convenient for practical purposes: one draws the field lines, but also the field surfaces, fig. 3b. By field surfaces we mean the orthogonal surfaces to the field lines. For conservative fields they are identical with the equipotential surfaces. For Maxwell it was a matter of course to represent in all of his figures field lines and field surfaces [4]. Both elements have a simple intuitive meaning: In the direction of the field lines the field is under tensional stress, in all the orthogonal directions, i.e. the directions within the field surfaces there is compressional stress. When knowing that, one will no longer interpret the field lines as filaments that run through the field, but as a means to represent graphically the mechanical stress within the field.

[1] M. Lieberherr: The magnetic field lines of a helical coil are not simple loops, Am. J. Phys. 78 (2010), S. 1117-1119 [2] J. Slepian: Lines of Force in Electric and Magnetic Fields, Am. J. Phys. 19 (1951), S. 87-90 [3] M. Schirber: Magnetic Fields in Chaos, Phys. Rev. Focus, http://focus.aps.org/story/v24/st24 [4] J. C. Maxwell: Lehrbuch der Electricität und des Magnetismus, Verlag von Julius Springer, Berlin 1838, Tafeln XII bis XXI

Friedrich Herrmann and Ralph von Baltz, Karlsruhe Institute of Technology