The force of electrodynamic origin which is exerted on the reel in the positive direction of the  axis can be calculated through two methods. The first one is based on  law because the reel forming conductors are situated in the  field from the air-gap. the second method is based on the calculation of the co-energy variation depending on the position.
The first method
To calculate the value of induction in any point from the air-gap, we use Ampere's theorem applied on a countour that travels radialy the air-gap and axialy the permanent magnet (figure 3). Results:
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(1) |
as we assume that:
- the permeability of iron is infinite and thus the
 field in iron is null;
- the
 field in the magnet is uniform and perfectly axial.
Figure 3
The  field having the same  value in any point from the air-gap, from (1) results
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(2) |
The transition from field within the air-gap to  induction is made using  relation.
The y flux which traverses the air-gap is
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(3) |
As induction is perfectly radial and is the same in any point within the air-gap, it results:
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(4) |
The y flux closes through the permanent magnet and the core. Through the permanent magnet the flux will be  , where  =  is the perpendicular section of the magnet; from here results:
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(5) |
The following relation is obtained:
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(6) |
Combining (6) and (2) ecuations and taking into consideration that  results:
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(7) |
Ecuation (7) expresses the connection between  and  .
There is, yet, another relation between the two, respectively the  characteristic of the permanent magnet.
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(8) |
The operating point is situated at the straight lines intersection given by (7) and (8) (figure 4):
Figure 4
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(9) |
and then
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(10) |
Assuming that the positive sign/direction of the i current through the reel traverses the sectioning plan as indicated in figure 3 ( 'enters' the superior side of the symmetry axis and 'exits' the inferior side), the  electrodynamic force which will be exerted upon a whirl, in the positive direction of the  axis, will be:
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(11) |
where  is the lenght of a whirl .
As the reel has  whirls, the total  force will be:
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(12) |
The second method
The calculation of the  induction in the air-gap is made similarly to the first method. In any point within the air-gap the  induction is radial and its value is (figure 1) :
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(13) |
The flux caused by the magnet which traverses a perpendicular section situated in  , position is equal to the flux which traverses the air-gap from to  (figure 5):
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(14) |
Figure 5
The direction of the flux is the positive one of the axis. This flux has the direction of the permanent magnet, which intersects a whirl of the reel situated in the same position, if we choose as positive direction of the i current, the one indicated by figure 5.
The y0 flux through the reel, when its median point is situated at  , will be (figure 5):
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(15) |
We obtain
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(16) |
The total flux induced by the reel is:
 , |
(17) |
in which  is the own inductance of the reel. By analogiy to (2.45) (Chapter 2, § 2.6 of the book), the magnetic co-energy is:
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(18) |
The term, which is the correspondent co-energy to the null current (or the opposite of the magnetic energy stored at null current), is a term which does not depend on the moving-reel's position, because the magnetic circuit, as seen from the permanent magnet, is invariant, respectively, does not depend on the reel's position.
The force which is exerted upon the reel is written as being the partial derivative of the co-energy in ratio to the reel's position, is then:
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(19) |
The term corresponds to the electrodynamic force. The  term corresponds to the force caused by the modification of the reluctance.
The electrodynamic force will be:
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(20) |
We notice, as expected, that (20) relation expresses a force with the same value to that obtained by applying  (12) rule.
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