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Magnetic Bonding of Round Current Loops

Abstract. Powerful bonds may be possible with current loops.

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 xy-plane 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 Biot-Savart:

. (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 z-direction 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 Bz(x,0,0) where x = nDx, n=0, 1, , 59; Dx = R/60. Note that Bz is practically constant over most of the range of x. But as x approaches R, Bz rapidly approaches positive infinity.

Figure 1.2

 

Bz(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. 1.4a-d give the components of dl and r. Fig. 1.4 depicts Bz(x,0,0) for this range of x. Note that Bz approaches negative infinity as x approaches R, but not far from the loop Bz is practically zero.

Figure 1.4

 

Bz(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 x-axis 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 As 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 left-most negative charge (moving in the negative y-direction) encounters an increasingly more powerful magnetic field pointing in the negative z-direction (see Fig. 1.4). Thus Loop B experiences a magnetic force toward Loop A.

If the left-most part of Loop B overshoots into x<R, then the negative charge of Loop B (still traveling in the negative y-direction) encounters a powerful magnetic field pointing in the positive z-direction (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 left-most part of Loop B gets close enough to the right-most part of Loop A, the two loops will magnetically bond.