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Figure from:Magnetic forces between arrays of cylindrical permanent magnets

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Fig. 1. A scheme of two sets of cylindrical magnets. One set is formed by the four upper magnets whereas the other set is formed by the bottom magnets and may tend to be infinite (see the dashed arrows). We first assume that one set comprises four magnets whereas the other set consists of 2N x 2N magnets (N > 0). In either set, first- neighbor magnets have their magnetization anti-parallel (Fig. 1). The advantage of the anti-parallel magnetization arrangement is its stability—the magnets do not repel each other. Furthermore, such arrangement features an interesting property: the attraction force between the two sets can be larger than four times the attraction force between just two of the magnets in the set. As a complement to the anti-parallel arrangement we also intend to mention the case of parallel magnetization arrangement.

Figure 1 A scheme of two sets of cylindrical magnets. One set is formed by the four upper magnets whereas the other set is formed by the bottom magnets and may tend to be infinite (see the dashed arrows). We first assume that one set comprises four magnets whereas the other set consists of 2N x 2N magnets (N > 0). In either set, first- neighbor magnets have their magnetization anti-parallel (Fig. 1). The advantage of the anti-parallel magnetization arrangement is its stability—the magnets do not repel each other. Furthermore, such arrangement features an interesting property: the attraction force between the two sets can be larger than four times the attraction force between just two of the magnets in the set. As a complement to the anti-parallel arrangement we also intend to mention the case of parallel magnetization arrangement.

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Fig. 2. The ratio f(t) versus magnet aspect ratio Tt. magnets. In general, let us assume two arrays, 2L x 2L and 2N x 2N where N- co. We define a force ratio f(t) as follows:
Fig. 2. The ratio f(t) versus magnet aspect ratio Tt. magnets. In general, let us assume two arrays, 2L x 2L and 2N x 2N where N- co. We define a force ratio f(t) as follows:
Fig. 3. Experimental and calculated (according to Eq. (4)) data for attraction force between two magnet sets (2 x 2) and (6 x 6). To investigate further what effects the magnet coatings may have on our results, we peeled off the Ni layer covering our NdFeB magnets Fig. 3 shows measured and calculated (according to Eq. (4) for N=3) data for the attraction force between two magnet sets (2 x 2) and (6 x 6) for various vertical displacements. Due to some small misalignment and the final rigidity of the clamps we are not able to measure the contact force accurately. The offset of Fig. 3 shows the large difference between the measured and the calculated contact force. We see that the measured force values are systematically smaller than those calculated. Excluding the data point for the contact force, the difference between the measured and calculated attraction forces is always below 10% (see Table 1), as their ratio ranges between 0.96 (gap 0.1 mm) and 0.91 (gap 2mm). Since the ratio between measured and calculated values is not constant, we cannot explain the discrepancy with a simple rescaling of some quantity, either the magnets’ field or its volume. The effect is, most likely, related to the actual shape of the magnets, which differs from an ideal cylinder in the rounded edges. The effective aspect ratio of the cylinders may also be a little different from the value we set (t=2) since there may be a layer of non-magnetic material (or magnetic but with different material parameters) coating the lateral surfaces. Finally, non-uniformity effects are also very likely to play a role. In particular, it is reasonable to imagine that when the two sets are in proximity, interaction fields are effectively enhancing the alignment of the magnetic moments, resulting in a better uniformity, while when magnets are far apart, the degree of uniformity decreases.
Fig. 3. Experimental and calculated (according to Eq. (4)) data for attraction force between two magnet sets (2 x 2) and (6 x 6). To investigate further what effects the magnet coatings may have on our results, we peeled off the Ni layer covering our NdFeB magnets Fig. 3 shows measured and calculated (according to Eq. (4) for N=3) data for the attraction force between two magnet sets (2 x 2) and (6 x 6) for various vertical displacements. Due to some small misalignment and the final rigidity of the clamps we are not able to measure the contact force accurately. The offset of Fig. 3 shows the large difference between the measured and the calculated contact force. We see that the measured force values are systematically smaller than those calculated. Excluding the data point for the contact force, the difference between the measured and calculated attraction forces is always below 10% (see Table 1), as their ratio ranges between 0.96 (gap 0.1 mm) and 0.91 (gap 2mm). Since the ratio between measured and calculated values is not constant, we cannot explain the discrepancy with a simple rescaling of some quantity, either the magnets’ field or its volume. The effect is, most likely, related to the actual shape of the magnets, which differs from an ideal cylinder in the rounded edges. The effective aspect ratio of the cylinders may also be a little different from the value we set (t=2) since there may be a layer of non-magnetic material (or magnetic but with different material parameters) coating the lateral surfaces. Finally, non-uniformity effects are also very likely to play a role. In particular, it is reasonable to imagine that when the two sets are in proximity, interaction fields are effectively enhancing the alignment of the magnetic moments, resulting in a better uniformity, while when magnets are far apart, the degree of uniformity decreases.
A comparison of measured and calculated attraction force values according to various formulas and for various distances between the two sets of magnets and measured its thickness. As the mean measured thickness was 27 + 14 um, the average Ni content of our magnets was about 3% in volume. According to Ref [14], the saturation magnetization of Ni is about 0.6 T, which is smaller than the saturation magnetization of NdFeB as measured and described in part 4. In order to ascertain whether or not the presence of a thin Ni layer might influence our measurement of the NdFeB magnetization, we repeated the measure- ments on magnets with and without Ni coating, finding only a negligible influence within statistical errors. Taking also into account the non-uniformity of the magnetization distribution in the Ni layer, we conclude that the presence of Ni coating most likely results in a slight decrease of the measured attraction force. Consequently, the calculations, carried out neglecting coatings, predict higher values for the attraction forces, which might explain the systematic theoretical overestimates visible in the comparative Table 1. Table 1
A comparison of measured and calculated attraction force values according to various formulas and for various distances between the two sets of magnets and measured its thickness. As the mean measured thickness was 27 + 14 um, the average Ni content of our magnets was about 3% in volume. According to Ref [14], the saturation magnetization of Ni is about 0.6 T, which is smaller than the saturation magnetization of NdFeB as measured and described in part 4. In order to ascertain whether or not the presence of a thin Ni layer might influence our measurement of the NdFeB magnetization, we repeated the measure- ments on magnets with and without Ni coating, finding only a negligible influence within statistical errors. Taking also into account the non-uniformity of the magnetization distribution in the Ni layer, we conclude that the presence of Ni coating most likely results in a slight decrease of the measured attraction force. Consequently, the calculations, carried out neglecting coatings, predict higher values for the attraction forces, which might explain the systematic theoretical overestimates visible in the comparative Table 1. Table 1
Another interesting point is that if we use Eq. (6) for the 4 x 4 case we obtain values close to those obtained from Eq. (4) for 6 x 6 and 2 x 2 (see Table 1). This means that in the case of our magnets (t=2) the increasing size of the larger set does not matter too much for the value of the attraction force. This is most likely a consequence of the rapid decay of the field generated by the outer regions of the larger set, decay established by the anti- parallel alignment of magnets. Furthermore, for t=2, the sum in Eq. (6) converges fast since the first addend approximates the attraction force relatively well (Table 1).
Another interesting point is that if we use Eq. (6) for the 4 x 4 case we obtain values close to those obtained from Eq. (4) for 6 x 6 and 2 x 2 (see Table 1). This means that in the case of our magnets (t=2) the increasing size of the larger set does not matter too much for the value of the attraction force. This is most likely a consequence of the rapid decay of the field generated by the outer regions of the larger set, decay established by the anti- parallel alignment of magnets. Furthermore, for t=2, the sum in Eq. (6) converges fast since the first addend approximates the attraction force relatively well (Table 1).
Fig. 4. Force ratio f,(t) or its estimate, versus aspect ratio T.
Fig. 4. Force ratio f,(t) or its estimate, versus aspect ratio T.