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Problem: A flat irregular shaped surface (A) need be affixed securely to a flat magnetic surface (B). The available means is small disc magnets of uniform dimensions and flux (the flat sides are N and S) to be glued onto (A). For a given thickness of the magnetic surface (B), what pattern of distributing the magnets will produce the strongest adhesive force? (Imagine you want to stick a clothes iron on the fridge door.)

Some example patterns:

1    NNNSSS
     NNNSSS
     NNNSSS

2    NSNSNS
     SNSNSN
     NSNSNS

3    NNNNNN
     SSSSSS
     NNNNNN
  1. The surface is covered in uniform halves.
  2. Every magnet is surrounded by opposite poles.
  3. Rows of alternating poles.

Keeping in mind that the shape my prohibit effectively accomplishing a particular pattern (especially ex. 2), will certain patterns be more effective in certain shapes?

For example:

1   N
   SS
  NNN

as opposed to:

2    N
    SN
   SNS
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    $\begingroup$ @ChiralAnomaly It can be. Question is about the best tiling and how boundary shape can affect it. $\endgroup$
    – christo183
    Commented Oct 30, 2019 at 14:13
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    $\begingroup$ I think you probably want a planar Halbach array, with the array's weak side glued onto surface A. $\endgroup$
    – PM 2Ring
    Commented Nov 1, 2019 at 9:31
  • $\begingroup$ @PM2Ring Maybe with cubic magnets? But how would a planar implementation look when all magnets are uniformly magnetized? $\endgroup$
    – christo183
    Commented Nov 1, 2019 at 11:25
  • $\begingroup$ Right, you can't make a proper Halbach array using those disc magnets. I suggest you pack your discs in a hex pattern to minimize the area. The polarities should be in some kind of alternating pattern, but I'm not sure what would be the best arrangement, so I won't write an answer. Maybe have all magnets in a row with the same polarity, alternating polarity from row to row. $\endgroup$
    – PM 2Ring
    Commented Nov 1, 2019 at 12:19
  • $\begingroup$ @PM2Ring A hex pattern would probably work well when the overall shape has a high area to circumference ratio. As it is I am favoring the alternating rows approach since it has good shape filling capability (getting more units into narrow angles) - Still, I'm hoping there is a more rigorous solution... $\endgroup$
    – christo183
    Commented Nov 1, 2019 at 12:44

1 Answer 1

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Definitely number '2'. The flux from multiple magnets is not a constant. For example, if I place two magnets against each other N to N, much of the flux is cancelled. But if I place them N to S, they both add to the external flux. You can then move one of those magnets along the flux lines of the other magnet, keeping the flux aligned, such that the two magnets end up side by side and in opposite directions (the magnet we moved also rotated 180 degrees). So when magnets are placed side by side, you get more external flux when they are pointing opposite directions than you do when they are pointing the same direction, and this is what we see empirically. In fact, the effect is pretty dramatic. Alternating magnets can produce upwards of ten times the near field force than same-direction magnets. Of course, the far field is weaker, but you don't care about that.

Another way to look at this is: When you bring magnets together, if you must add energy to do that then you are reducing the flux, but if bringing them together yields energy, then you are increasing the flux.

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