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Forces Between Magnets

Two-dimensional analysis for circular bar magnet assemblies
Vortex Technologies
2007-03-05


For forces between magnets to suspend of a platform I seek specific functional relationships for circular bar magnets (or disc magnets for short low L/D ratio). The 3 identical magnets with diameter D length L are set up as “repelling pairs” as follows:



Fig. 1 Magnet set-up

Explanation

  1. The 3 magnets are fixed to external “worlds” that are non-magnetic or with negligible magnetic properties. Due to these fixings the magnet faces remain almost parallel to within angle a such that sine(a)=0,008;

  2. Other magnetic fields in the region of the 3 magnets are negligible;

  3. The L/D Ratio is such that 1 = < L/D <= 4;

  4. The non-magnetic forces due to fixings or external loading c.q. fixings etc. are not shown. Normal forces F11 & F22 and F21 & F22 are therefore not generally identical. For the external forces being zero then x1=x2 and F12=F22. For x2>x1 then F12>F22. Other assumptions arising from the setup to be made as required;

  5. The separation variables x1 and x2 can each take on values as follows: 0 <= x <= 25L;

  6. The value of y shall be limited to 0 =< y <= D/10 for all values of x1 and x2;

Remarks

  1. The normal and lateral forces between the magnets are functions of x and y. Any non-parallelism of the magnet faces (angle a) can be assumed to be accounted for by the substitution: sin(a)=tan(a)=a for the angle in radians.

  2. Typical force relationships between magnets that are published are for attractive forces without magnet offset. See for example [1] and [2]. Due to the fact that the magnetic field intensities in the xy-plane (for circular magnets radial symmetry applies) are quite different for repelling set-ups than for attractive set-ups the usual relationships for attraction are most likely not valid. It is possible that on fundamental grounds and for certain set-ups the attractive force may be assumed equal to repelling forces (analogous to the forces on current carrying wires). Anyway, the normal force-functions are usually given only for two magnets with their centres aligned (offset y=0). For the application shown in Fig.1 the offset y may be =>0, and depending on the type of lateral constraints that are used the ymax would be stated. It is therefore required to model these forces during non-zero offset;

  3. For the 3-magnet in Fig. 1 the force of the 3rd magnet on the 2nd magnet must be calculated separately and be added to the force of magnet #1 on magnet #2. One gets an addition with the variables x1 and x2. If more magnets are used in line then the expression will also have the x3 terms. If the offset y is to be taken into account one needs to derive the expression for the forces as a function of y as well from first principles. At this time the force expressions as a functions of x and y are not jet developed for the 3-magnet set-up in this analysis . . .it needs to be done for a variable diameter as well;

  4. The question rises if a formal representation of a magnet as a dipole is valid for deriving an accurate expression for the forces. The equations in notes 1 and 2 are OK for so far it rough estimates is concerned. If a calculation is to be accurate then the “poles” of a magnet are distributed in space and the magnetic flux distribution in the magnet is not a simple function. It would be in principle be required to model the magnetic poles as distributed dipoles in de magnet.

Object for two magnets

  1. To find a function Fx(repelling normal force)= f1(x,y): sine a=tan a=a with a= small angular displacement of a bar magnet rotated on its centre between the pole faces.

  2. To find a function Fy(repelling lateral force)= f1(x,y): sine a=tan a=a with a= small angular displacement of a bar magnet rotated on its centre between the pole faces.
[1] http://instruct.tri-c.edu/fgram/web/Mdipole.htm Equation (8).

[2] http://en.wikipedia.org/wiki/Magnet#Force_between_two_bar_magnets. Note that for this expression x=L is a minimum value for x is as in that case the pole faces are touching. The x is the distance form the magnet centres between the N- and Z-pole.