ARE MAGNETIC FIELD LINES CLOSED?

(7/98) Y. Kantor: I have attempted to summarize the subject, and our readers are welcome to judge!

(June 8 and 10, 1998) Michael Trott wrote (slightly edited):

It does not follow from the equation div B=0, that field lines are closed. Actually if you write down the explicit equation for the field lines together with the maxwell equation, then they can be cast into the form of Hamilton's equations and then we are on save ground to decide if the trajectories are closed. Only for pure 2D current arrangements the system is integrable. Details can be found in:
E. Petrisor. Physica D 112, 319 (1998)
J. R. Cary, R. G. Littlewood. Ann. Phys. 151, 1 (1983)
E. Pina. J. Phys. A 21, 1293 (1988)
J. Slepian. Am. J. Phys. 19, 87 (1951)
G. F. T. del Castillo. J. math. Phys. 36, 3413 (1995)
V. Arnold. C.R. Acad. Sci. Paris 261, 17 (1965)
A. F. Ranada. Phys. Lett. A 232, 25 (1997)

Actually it takes a few minutes of playing around on a computer to discover this. Just take any closed 3d loop (not planar), calculate B via Biot-Savart and integrate the field line equations. They do not close. Doing a numerical integration for knotted coils then gives you quite some surprises in the shape of the field lines too. They are never restricted to the inside of the coil, they always 'escape' somewhere.



(June 20, 1998) The response of Christian von Ferber was as follows:

We here give account of some numerical investigation concerning the existence and creation of knotted magnetic field lines by a current and finally give an argument of how to create a magnetic field line along a predescribed curve.

A numerical study

A trefoil knot may be parameterized in the following way:

x(t)= -10*cos(t)-2*cos(5*t)+15*sin(2*t)
y(t)= -15*cos(2*t)+10*sin(t)-2*sin(5*t)
z(t)= 10*cos(3*t)
t=0..2*pi

Along this line we center current loops of radius a=1 with face perpendicular to the line.
We calculate the magnetic field as a superposition of the known solution for a single current loop [e.g. Landau,Lifshitz Vol.8].
We increase the number N of current loops. The field strength H at the center of the loops normalized by N approaches a limit H/N -> h.
To achieve sufficient accuracy (10-6) we use 500 loops placed equidistantly along the parameter values t.
We integrate the resulting field in real space steps of eps=10-3. After 230,000  steps the field line we followed came back to its origin within 0.04 . This is to be compared with the radius a=1.0 of the 'solenoid'. We have at this point not checked whether the accuracy may be increased further. Obviously decreasing eps may at the same time increase rounding errors.
Find here a plot of the trefoil (center of the solenoid in green) and the integrated solution (in blue) from two different viewpoints:

To check if the field lines might escape the solenoid we have also performed a less time consuming integration with the step length eps=0.1. We followed some  100 revolutions along the solenoid without finding signs of serious divergence of the curve. We plot here every 100th point of this integration in blue along with the trefoil line in green:


 

The Zero - Knot: A toroidal solenoid

To get some idea about the influence of the numerical errors we have performed  the same procedure as above also on the toroidal solenoid, for which the exact  solution is known. To receive similar dimensions we have used a torus with  centerline along a circle of radius R=36. This implies an arc length of this line of about L=226 comparable to the arc length of the trefoil knot above. Along this line we have again placed N=500 current loops each of radius a=1. The result of the integration with step length eps=10-1  was similar to the integration along the trefoil line. (see below for 10 revolutions). Integrating with high precision (eps=10-3) we find that the field line along the centerline returns to its origin within a distance of 0.004 e.g. some 1/10th of the error found in the knotted configuration. This might be an indication, that indeed the field line in the knotted configuration does not close, but is still far from a proof, which may not be provided numerically.

 

How to generate a closed knotted magnetic field line

We have seen above that the N current loops aligned along the trefoil curve do not suffice to guarantee that there is a field line along the predescribed curve. But also we know from the numerical experiments that we need only some small correction. We may thus place at each of the N points two additional current loops centered at the same point but each with some individual small angle to the curve. Choosing the current strengths in these loops while keeping the angles fixed gives us 2N degrees of freedom. This is enough to align the magnetic field at each of the N points with the curve (also 2N degrees of freedom). As the magnetic field is a linear superposition of the now  3N current loops it is even a linear problem which will be nondegenerate if the angles of the additional loops are properly chosen (the 3 loops at each point should have non-coplanar directions). Including even the choice of the angles it is clear that more that sufficient degrees of freedom exist to produce a magnetic field line along any predescribed smooth curve.



(June 22, 1998) The reply of Michael Trott was as follows:

I disagree with the conclusion. -As long as you take a finite number of turns you must connect the loops and this gives a very thiny contribution to the magnetic field which is not in the direction parallel to the knots local tangent. -Also if you write out the equations for the corresponding Hamiltonian system for the single loops along the knot, I highly doubt that it will be integrable for any field line other then the one in the center (if at all).

(June 22, 1998) Christian von Ferber's reply:

Of course, we are here coming close to splitting hairs, but let me add that taking full control over the complete configuration of the wire, rather than constraining it to just connecting the loops, gives me even more degrees of freedom which can be used to refine the contour of the magnetic field. The argument of the 3N loops was more thought of being directly convincing, as it leads to a linear equation. More generally one may use the dense winding limit (which is standard when considering solenoids). Then one may think of the solenoid as a tube around the given (knotted) line. Along this tube we now have full control of the shape of its surface plus the density of winding along the tube. obviously this gives us per point on the line an infinity of degrees of freedom, while we only have to fix two degrees of freedom to align the magnetic field there. It seems that this should even suffice to align a bunch of lines of non-zero measure around this line. But even if we had only this one closed magnetic field line, this would correspond to a `bound state' solution in an otherwise non-integrable Hamiltonian system. In fact these solutions are of some importance when one considers the quantized version of such systems -- I know people who like to count them, but this may be a matter of attitude.

(7/98) Y. Kantor: I think that the argument reached a point where only a clear analytical argument can change the opinions of people. We welcome such arguments...