The basic law of magnetism is that like poles repel one another, and unlike poles attract each other. Even though Gauss’ law for magnetic flux density (B-field) indicates that there is no free magnetic charge, we can still define the effective bound magnetic charges locally from the magnetization of magnetic material1. The distribution of positive magnetic charge can be defined as the “north pole”, and correspondingly, the negative magnetic charge can be defined as the “south pole”. The interaction between the local magnetic charges is governed by Coulomb’s law so that like charges repel and unlike charges attract2. However, George Mizzell observed some cases in which two like poles attracted each other near the central area for a pair of permanent magnets with significantly different dimensions. Mizzell first reported it in May 20073 on YouTube, then reported it again in March 20194. We reported our preliminary investigation for this seemingly unacceptable behaviour in November 20195. Other researchers also reported such a fact in 20196. The phenomenon does not mean that the fundamental laws of magnetism are violated, but it is important to understand the physics underlying the phenomenon. Why do these “like poles” attract each other instead of repelling? To understand the mechanism and reveal the mystery of this unique interaction, a series of experiments were conducted in our labs.

The permeance coefficient (Pc) is defined as the ratio of magnetic induction Bd and magnetic field Hd inside a standalone magnet at the operating point, i.e., Pc =|Bd/Hd|7,8, which depends on the geometry of the magnet. For example, in the case of cylindrical magnets with the same diameter and same magnetization along the axis, the longer the magnet is, the higher the Pc. Combined with the B–H curves, the Pc can determine how easily a magnet will be demagnetized, especially when the B–H curve in the 2nd quadrant is nonlinear. In this work, we find that the Pc and the B–H curves are the key factors to explain the interesting phenomena.

Experiment method

N55 and N48SH samples of Nd–Fe–B, and SmCo30 samples of Sm2Co17 were tested. The dimensions and values of Pc are shown in Table 1, and their demagnetization B–H curves are shown in Fig. 1. These cylindrical magnet pairs with various Pc values range from 0.13 to 24. For the material itself, N55 has a nonlinear B–H curve in the 2nd quadrant, while N48SH and SmCo30 have linear B–H curves in the 2nd and even in part of the 3rd quadrant. SmCo30 was tested so that the surface degradation can be excluded, which may be more obvious for some small or thin Nd–Fe–B magnets9.

Table 1 The magnet samples tested in this investigation (the unit is mm, where OD and L are the diameter and length of the cylinders, respectively).
Figure 1
figure 1

The B–H curves of the three magnets in the 2nd and part of the 3rd quadrant, and the working points Bd/Hd marked for N55 and SmCo30 for four load lines with Pc = 0.13–24.

These magnet pairs were tested for repelling and attracting forces at the gaps between 0 and 50 mm at the center position, using an Instron 5944 force tester. In order to observe what happened during the test process, four forces were recorded and marked by their testing sequence, F1, F2, F3, and F4, as shown in Fig. 2. F1 and F4 were recorded for N → S unlike poles, and F2 and F3 for N →← N like poles. In addition to the force test, some of the magnets with OD = 32 mm were also tested for flux density on the surface to estimate the level of the localized demagnetization LD, using a Brockhaus XYZ Scanner. The Hall sensor was 1.2 mm above the magnet surface due to the Hall probe construction and the clearance, and this gap was maintained throughout the experiment. There were two steps in sequence to test the surface flux, as shown in Fig. 3. Step 1, the pairs were magnetized together aligning at the center, then scanned the surfaces of both sides N and S on the bottom magnet after the top magnet was removed; Step 2, turned the top magnet upside down and touched the bottom magnet at the center for 1 min, then separated, scanned the bottom magnet at the surface for both sides N and S.

Figure 2
figure 2

Four forces testing setup: F1, F2, F3, and F4 as the above sequence. F1 and F4 should be all attracting and F2 and F3 should be all repelling, and “d” is the gap in between.

Figure 3
figure 3

Surface flux testing setup: scan on the surface with 0.5 mm intervals, and gap d is 1.2 mm between the Hall sensor and the surface due to Hall probe construction and the clearance.

Results and analysis

Unusual behaviours shown in F 2 and F 4

Figure 4 plots F1, F2, and F4 forces vs. the gap d for some N55 unequally sized pairs. (F3 is not plotted here as F3 is the same as F2). Two unusual behaviours are observed in the plots: (1) an inflection point IP appears on the curve of F2 vs d for the N →← N pairs of like poles. Normally, F2 is a repelling force (F2 < 0), and |F2| increases when the gap decreases, but this IP results in nonmonotonic curves, even attracting force. When the gap reduces to d < IP, |F2| starts to decrease and eventually transforms from repulsion to attraction (F2 > 0) at gap d < δ (~1 mm). It should be noted that the IP is caused by a localized demagnetization LD, which starts playing its effective role well before the IP point. (2) For some NdFeB magnets with a low coercivity and nonlinear B–H curve in the 2nd quadrant, repulsion can occur for these unequally sized unlike poles, after previously pairing with their like poles that left an unrecoverable LD and reversed polarity area. A force difference ∆F occurs to the unlike pole pairs, testing before and after pairing with their like poles. F4 < F1 and even F4 < 0 are observed for these N55 N → S pairs. Normally, F1 and F4 have the same attracting force (F1 = F4 > 0), as they were tested on the same N → S pairs. The only difference is that F4 was tested after F2, and F2 was tested with N →← N like poles, which experienced the LD. When gap d reduces to a range (~ 15 to 8 mm), where the LD starts playing its role to cause an unusual effect, an unusual behaviour of F4 < F1 occurs. Three pairs with Pc1/Pc2 = 4.69, 10.8 and 185 have F4 < 0 when the gap d < σ (~ 2–3 mm), where the force transforms from attraction to repulsion, and only one pair (Pc1/Pc2 = 2.15) does not have F4 < 0. This unusual F4 < F1 does not occur to N48SH and SmCo30, which have linear B–H curves.

Figure 4
figure 4

Forces vs gap for some unequally sized N55 pairs, showing the following: (1) an inflection point IP appears on the curve of F2 vs d; (2) F4 < F1 or even F4 < 0 when d < σ, and F2 > 0 when d < δ. (a) F3 is not plotted here as F3F2. (b) δ is the point where F2 = 0, and σ is the point where F4 = 0.

Figure 5 shows F2 vs d curves of the N →← N pairs for N55, N48SH and SmCo30, with both unequally sized and equally sized pairs. An unusual IP is observed for all the magnet pairs, except the equally sized pairs with Pc1 = Pc2. Figure 5a,b for N55 show five out of eight pairs having F2 > 0 at d < δ (~1 mm). Figure 5c,d for N48SH display two pairs having F2 > 0 at d < δ. Figure 5e,f for SmCo30 present one pair (Pc1/Pc2 = 10.8) with F2 = 0 at d = δ (0.05 mm), which indicates that certain SmCo30 pairs can also have F2 > 0 at a small gap. It is noticeable in Fig. 5 that the higher the Pc ratio is, the higher the IP will be, and the IP ranges from 0.4 to 6 mm when Pc1/Pc2 is 2.1 to 185. When Pc1/Pc2 = 1 for the equal sized pairs, the IP disappears (or IP → 0).

Figure 5
figure 5

F2 vs gap d for N55, N48SH, and SmCo30 with both unequally sized and equally sized pairs. (a) IP is the inflection point. (b) δ is the point where F2 = 0.

Figure 6 shows the green tape views for the magnets with  a Pc ratio of 4.69, before and after the N →← N paring. The reversed polarity is clearly seen on the D32 × 2 magnet after the paring, and no change is observed for the D8 × 2 magnet.

Figure 6
figure 6

The green-tape views for N55 pair of D8 × 2 and D32 × 2 (Pc ratio = 4.69) before and after N →← N paring.

Most of the results are listed in Table 2, from which the unusual behaviours of F2 and F4 can be analysed, and the localized demagnetization LD can be estimated. The data in Table 2 are plotted in Figs. 7 and 8 to make reasonable analyses. Figure 7 shows the force difference ΔF2 (= |(F2@d<0.5 − F2@IP)/F2@IP|) vs Pc1/Pc2. In general, a higher Pc ratio results in a larger ΔF2, which is as high as 1100% for N55 with Pc1/Pc2 = 10.8 in Series #1, and 400% for N55 with Pc1/Pc2 = 185 in Series #2. The ΔF2 is 2–250% for N48SH pairs, and 4–100% for SmCo30 pairs. Figure 8 demonstrates the force difference ΔF (= |(F4 − F1)/F1|) vs Pc1/Pc2 for N55 pairs only, as N48SH and SmCo30 do not have this unusual behaviour of F4 < F1. It is clear that a higher Pc ratio results in a larger ΔF for both Series #1 with the bottom size fixed and Series #2 with the top size fixed. The ΔF is as high as 193% for Pc1/Pc2 = 185 in Series #2, and 175% for Pc1/Pc2 = 10.8 in Series #1. A larger ΔF or ΔF2 indicates a greater localized demagnetization LD.

Table 2 Unusual behaviours observed for the magnet pairs (IP appears on F2 vs d curves and F4 < F1 or F4 < 0).
Figure 7
figure 7

Unusual ΔF2 for all the magnet pairs with Pc1 ≠ Pc2: A higher Pc1/Pc2 leads to a higher ΔF2. All the pairs with ΔF2 > 100% show attraction for N →← N pairs.

Figure 8
figure 8

Unusual F4 < F1 for the N55 pairs with Pc1 ≠ Pc2: a higher Pc1/Pc2 results in a larger ΔF. Note: All the pairs with ΔF > 100% show repulsion for N → S pairs.

Localized demagnetization LD plots and maps and the determined LD levels

The LD can be visualized using a surface flux density test (the testing setup is illustrated in Fig. 3), from which the LD levels can be determined. Figures 9, 10 and 11 show the flux density at 1.2 mm above the N-side of the bottom magnets of N55 and SmCo30 with three Pc ratios of 4.69, 185, and 17, respectively. The 2D magnetic flux maps are shown on the right, and the curves of flux density vs position and the magnet size are shown on the left. The curves of flux density vs position along a diameter are curves 1 and 2 for N55, and curves 3 and 4 for SmCo30. Curves 1 and 3 are the original fluxes labelled “A”, corresponding to Step 1-Scan A in Fig. 3. Curves 2 and 4 are the fluxes after 1 minute of repulsion by the top magnet, labelled “B”, corresponding to Step 2-Scan B in Fig. 3. The flux densities at the surface of the magnets with OD = 32, as well as the LD values (LD = 100%*(BB − BA)/BA) for various pairs are summarized in Table 3. Figure 9 shows pair 2-A with Pc1/Pc2 = 4.69, in which N55 has a larger LD (− 55%) at the center of curve_2, comparing to original curve_1, while SmCo30 has a small LD (− 2.1%). Figure 10 shows the pair 4-A with Pc1/Pc2 = 185, and N55 has a huge LD with its polarity being totally reversed at the center (− 114%) of curve_2, while SmCo30 has a small LD (− 0.44%). Figure 11 shows pair 4-D with Pc1/Pc2 = 17, and N55 has LD = − 6.1% while SmCo30 has LD = 0.64%. In general, LD on the N-side is higher than that on the S-side. When the Pc1/Pc2 ratio = 1 for the A–A pairs, the LD for both N55 and SmCo30 is approximately “0”, and the tiny difference is within the measurement error. Since the flux density was tested at 1.2 mm above the surface, the actual LD should be higher than the levels described in this section.

Figure 9
figure 9

The surface field @ 1.2 mm above the N side of N55 and SmCo30 after pairing with Pc1/Pc2 = 4.69.

Figure 10
figure 10

The surface field @1.2 mm above the N side of N55 and SmCo30 after pairing with Pc1/Pc2 = 185.

Figure 11
figure 11

The surface field @ 1.2 mm above the N side of N55 and SmCo30 after pairing with Pc1/Pc2 = 17.

Table 3 The LD detected at 1.2 mm above the surface of bottom magnets (BA – original testing after magnetization, BB tested after repulsion on N →← N pairing).

The effects of the linearity of B–H curves and the load-lines with working points

As shown in the previous sections, the LD level is related to the Pc ratio, and it is also linked to the linearity of the B–H curves in the 2nd and part of the 3rd quadrant. Figure 1 shows the B–H curves of N55, N48SH, and SmCo30, and the linearity limits (knee positions) of the B–H curves are 13 kOe, 21.4 kOe, and > 21.5 kOe, respectively. Among all three magnets, SmCo30 has the best linearity of the B–H curve. The small circles at the cross points of the load-lines and the B–H curves in Fig. 1 are the working points (Bd/Hd), which are listed in Table 4 for some magnets tested in this investigation. When two unequally sized magnets with different Pc pair with the like poles N →← N, the one with a lower Pc, which has an internal self-demagnetization field Hd, will be affected by an external field Hex from the one with a higher Pc. The total demagnetizing field is the sum of Hd and Hex as shown in Fig. 12. If the sum exceeds the linearity limits, the B loss will be unrecoverable. Figure 12a shows how a N55 magnet with Pc = 0.13 loses its flux B after paring as N →← N with a magnet having a higher Pc. The stand-alone magnet has operating point “a”, and it drops to point “b” while pairing with one having a higher Pc. After the pair separates, the magnet can only return to point “c” as it needs to return to its stand-alone condition along the line parallel to its relative permeability μr = 1.045. This large B loss is due to the nonlinear B–H curve of N55, and the working point “a” of this magnet is very close to the knee of the B–H curve. On this aspect, N48SH and SmCo30 would not have such unrecoverable losses as long as the total demagnetizing field is less than 21.4 kOe and 21.5 kOe, respectively, where they are about to lose their linearity of their B–H curves (see Fig. 1 for the B–H curve details). Figure 12b shows how a N48SH with Pc = 0.13 maintains its flux B after paring N →← N with a magnet with a higher Pc. This explains why N48SH and SmCo30 do not show the unusual behaviour of F4 < F1 at a small gap, and this recoverable LD is a unique characteristic to be utilized in some novel applications in the near future.

Table 4 Working points Bd and Hd, for the following tested magnets (see  Fig. 1 for the B–H curves).
Figure 12
figure 12

The B–H curves in the 2nd and part of the 3rd quadrant and the load-line function.

See Refs.7,8,10,11 for the concept used in the analysis. It is clear that the linearity of the B–H curve and the magnet’s load-line play important roles in the LD level and its recoverability after the magnet separates from the N →← N paring.


A localized demagnetization (LD) is identified for unequally sized magnetic like poles as their Pc values are different, in which the pole with a higher Pc causes a LD to the pole with a lower Pc. If the LD is large enough, the polarity of the localized area can be reversed, resulting in an attraction between two like poles in the LD area in a small gap. Two unusual behaviours are observed in this investigation.

  1. 1.

    An inflection point, IP, appears on the curves of the force vs gap for all the unequally sized like poles. The IP results in nonmonotonic curves, even an attraction for the like poles.

  2. 2.

    For some NdFeB magnets with a low coercivity and nonlinear B–H curve in the 2nd quadrant, a repulsion can occur for these unequally sized unlike poles after previously pairing with their like poles that left an unrecoverable LD and a reversed polarity area.

The unusual behaviours are not contradictory to the basic law of magnetism, and they are caused by the localized demagnetization LD. A higher Pc ratio results in a greater LD; the linearity of the B–H curves and the load-line also play important roles in the LD level. The LD level can be visualized and determined by mapping the surface flux.


The linearity of the B–H curve and the magnet’s load-line play important roles in the LD level and its recoverability after the magnet separates from the N →← N paring. For N55 with a nonlinear B–H curve, especially with small thickness, the LD is mostly unrecoverable. Similar to N55, Alnico magnets may also show the same unusual phenomena. For N48SH and SmCo30 magnets with linear B–H curves in the 2nd and part of the 3rd quadrant, the LD is mostly recoverable after the like pair is separated. If SmCo30 or N48SH pairs have a proper Pc ratio, the like poles can also appear attracting each other; since the LD is recoverable, some novel applications may be developed for utilizing these newly discovered unique characteristics.