Magnetic Bonding of Round Current Loops G.R.Dixon, 10/01/03
1. The Magnetic Field of a Circular Current. Given: A circular current loop, I = 1 amp, centered on the origin and with a radius of R=1 meter. The loop lies in the xyplane and the current circulates counterclockwise, looking down from positive z. Problem: Find B(x,0,0) over the ranges 0<x<R and R<x<2R. Solution: Use BiotSavart: . (1.1) 0<x<R The directions of dl and r are as indicated in Fig. 1.1, with . (1.2) Figure 1.1 Directions of dl and r
We shall assume that the loop is a circular line charge (infinitesimal cross section). Since dl x r points in the positive zdirection for x<R, (1.3) where , (1.4a) , (1.4b) , (1.4c) . (1.4d) And . (1.5) Thus . (1.6) The integral can be numerically evaluated, say using Dq = 2p/1000. Fig. 1.2 depicts B_{z}(x,0,0) where x = nDx, n=0, 1, …, 59; Dx = R/60. Note that B_{z} is practically constant over most of the range of x. But as x approaches R, B_{z} rapidly approaches positive infinity. Figure 1.2
B_{z}(x,0,0), Circular Current Loop, 0<x<R
R<x<2R For this range dl x r points toward negative z for some values of q, and toward positive z for others. Fig. 1.3 illustrates the two cases. Figure 1.3 Cases where dl x r points toward negative and positive z
Here again Eqs. 1.4ad give the components of dl and r. Fig. 1.4 depicts B_{z}(x,0,0) for this range of x. Note that B_{z} approaches negative infinity as x approaches R, but not far from the loop B_{z} is practically zero. Figure 1.4
B_{z}(x,0,0), Circular Current Loop, R<x<R
2. Magnetic Bonding. Let us now consider two current loops: (1) Loop A, centered on the origin (see Sect. 1), and (2) Loop B, with its center on the xaxis and free to approach Loop A from the right. We shall assume that each loop is electrically neutral so that there are no electric forces to consider. (That is, a given loop is the superposition of a resting, positive, circular line charge, and a circulating negative charge.) Let the circulation of Loop A’s negative charge be clockwise, so that the loop's magnetic field is as indicated in Figs. 1.2 and 1.4. And let the negative charge of Loop B circulate counterclockwise. As Loop B (slowly) approaches Loop A, its leftmost negative charge (moving in the negative ydirection) encounters an increasingly more powerful magnetic field pointing in the negative zdirection (see Fig. 1.4). Thus Loop B experiences a magnetic force toward Loop A. If the leftmost part of Loop B overshoots into x<R, then the negative charge of Loop B (still traveling in the negative ydirection) encounters a powerful magnetic field pointing in the positive zdirection (see Fig. 1.2). This will cause Loop B to experience a magnetic force away from Loop A. Equilibrium occurs when the overlap results in zero force. Evidently if the leftmost part of Loop B gets close enough to the rightmost part of Loop A, the two loops will magnetically bond. 3. Physical Implications. Experiment indicates that the nuclear binding force between atomic nucleons (protons and neutrons) is described by a single function of nucleon separation. Of course protons are positively charged and repel one another electrically. They will get sufficiently close to one another for the nuclear force (magnetic bonding force?) to engage only when (a) they are on a collision course, and (b) their initial kinetic energies are great enough to overcome the electric repulsive force. Neutrons, however, can closely approach one another at relatively low kinetic energies. Thus the most practical way to reap the attendant release of energy, whenever a magnetic bond forms, would appear to be with neutrons. Such magnetic bonding might be expected to occur at relatively low temperatures (or mean kinetic energies) using these particles (and assuming the magnetic bonding mechanism described above applies). Such lowtemperature bonding has aptly been dubbed "cold fusion." Recent claims that such bonding has actually been witnessed have been challenged by many on theoretical grounds. However, the magnetic bonding mechanism described herein may provide the needed theoretical basis for such "cold fusion" processes among neutrons.
