There are various ways of determining the force acting on an object in a magnetic field in air or in vacuum. A very straightforward and general method is one in which Maxwell's equations in vector notation are used to calculate a fictitious force density at a surface that completely encloses the object:

(2) |

where the
vector
is the magnetic flux
density,
the outward-directed unit vector perpendicular to the enclosing surface
and the
magnetic permeability of free space; see *fig. 2a*. The total vector
force
acting on the object is then found from

(3) |

where *S'* is the total area of the enclosing surface with surface element
d*s*. The enclosing surface *S'* may coincide with the surface
*S* of the object.

After some manipulation, expressions for the
component
of the force density perpendicular to the surface ('tensile or compressive
stress') and the
component
along the surface ('shear stress') can be derived from (2):

(4) | |

(5) |

where
is the normal component
and the
tangential component
of . The
relation between the two components
of and
the angle
that
makes with the surface is shown in fig. 2*b*:
for =
0 and for
the
component
and is
at a maximum, and
for the
component
and is
at a maximum.

In a magnetic bearing we have a cylindrical object that is levitated in a
magnetic field. For the cylindrical surface of the shaft it is then desirable
that *p _{n}* should be large and

The question is now whether the situation shown in fig. 2*c* is in conflict
with Earnshaw's theorern as stated above. We shall find that this is not
the case.

A situation as shown in fig. 2*c* can be obtained by making use of the
effect of superconductivity. If niobium or lead ('superconductors of the
first kind') are in the superconducting state, any change in the magnetic
flux density causes permanent eddy currents at the surface of the material,
their magnitude and direction being such as to cancel the effect of this
change. No magnetic flux can therefore be generated in these superconductors.
This situation corresponds to that in a hypothetical material whose magnetic
permeability is equal to zero. The lines of force representing the magnetic
field cannot penetrate into the material; see fig. 2*c*. In *fig.
3a* this situation is shown again for the case where the centre-line of
the shaft coincides with the plane of the diagram and for a different direction
of the flux density. The maximum force density that can be achieved
is , from
(4). The flux density at the boundary surface of superconductors of the first
kind must remain limited to between 0.1 and 0.3 T (tesla or
Wb/m^{2}), since otherwise the superconductivity would vanish. The
corresponding force density is at the most 3 10^{4} N/m^{2}
or 0.3 bar. The load-carrying capacity that can be achieved is thus fairly
small. Also, there is the great disadvantage that part of the rotor has to
be cooled to a temperature of 10 K or lower. As explained above, the magnetic
bearing with superconducting shaft ends does yield a stable equilibrium of
forces. This is not in conflict with Earnshaw's theorem, since this theorem
only applies to cases
where
> 1.

Fig. 3*b* shows the situation for the case where the shaft ends are
made of diamagnetic material. The angle that the lines of force make with
the surface now has a value between 0
and .
In the material the lines of force diverge, because the flux density here
is less than in air.
Since
differs very little from unity, the force density is limited. The load-carrying
capacity is even less than in the case of fig. 3*a*. It should be noted
that the situation shown in fig. 3*b* can also be obtained by means
of an alternating electromagnetic field, because eddy currents opposing the
alternating magnetic field are then induced into the shaft ends. The eddy
currents cause losses, however. A rough calculation shows that the loss on
levitating a copper plate, irrespective of its thickness, is about 2.8 kW
per m^{2} of bearing surface. The loss with an aluminium plate is
about 1.3 kW/m^{2}

Fig. 3*c* shows a situation where the direction of the flux is perpendicular
to the bearing surface. As in fig. 3*a*, the force density is equal
to . We
can work here with materials of high magnetic permeability and high saturation
flux density. In cobalt iron, for example, the flux density can amount to
as much as 2.4 T. The maximum force density is then 23 10^{5}
N/m^{2} or 23 bars. The equilibrium of forces is not stable, however,
since Earnshaw's theorem now applies. To obtain an effective bearing the
size of the bearing gap must be controlled. The method of control will be
discussed later on in this article.

The mean force
density
for the three cases of fig. 3 can be expressed in mathematical terms. We
assume that the magnetic field along the surface of a medium of magnetic
permeability
is generated by means of a sinusoidal surface-current density of peak
value
(in A/m); see *fig. 4a*. Using equation (4) we then find
for

(6) |

where is the wavelength of the surface-current density.

Equation (6) can be derived as follows. We assume that the surface-current
density
is located in a rectangular co-ordinate system
(*x*,*y*,__z__) in the plane *z* = 0, with the currents
perpendicular to the plane *x* = 0 and the surface-current density (current
sheet) extending to infinity in the positive and negative
*y*-directions.
For the
permeability
is not equal to unity and is constant, and for *z* < *a* we
have
= 1. Let the space be divided into three parts. In part *1*,
for ,
the magnetic potential function has the
value ;
in part *2*,
for ,
it has the
value ;
in part 3, for *z* < 0, it has the
value .
In each of these parts Laplace's equation
à
= 0 applies, with *i* = 1, 2 and 3. We are considering a two-dimensional
problem, so that the derivatives with respect to *x* are equal to 0
and .
For the magnetic field-strengths in the three parts we can have:

With the aid of the above differential equations we can calculate the
field-strengths along the plane *z* = *a*. The following boundary
conditions then apply:

*H*_{1y} = *H*_{2y}
and *H*_{1z}
= *H*_{2z}, for *z* = *a*;

*H*_{1y} = *H*_{1z} = 0, for *z*
= ;

*H*_{3y} - *H*_{2y} = *A*, and
*H*_{2z} = *H*_{3z}, for *z*
= 0;

*H*_{3y} = *H*_{3z} = 0, for *z*
=

The solutions for the potential functions must then satisfy:

in which the boundary conditions for *z*
= and
*z* =
have already been included. The remaining boundary conditions give four
equations, which we can solve for *A*_{1},
*A*_{2}, *A*_{3} and *B*_{2}. We thus
obtain equations for the field-strengths at the boundary plane z = *a*:

from this and (4) we obtain an equation for the force density
*p _{n}* perpendicular to the boundary plane:

Integrating *p _{n}* with respect to

Since
is very small for diamagnetic materials, the force density obtained with
these materials is not very large, as indicated earlier. For bismuth, the
diamagnetic material with the lowest permeability, we
have .
The force density is thus
about
times smaller than for materials in the superconducting state
with = 0 or
for ferromagnetic materials
with
>> 1; see fig. 4*b*.

The mean force density at *z* = *a* in the *z*-direction
is . After
differentiating , with respect to *a* we then find the mean vertical stiffness
per unit area in the *z*-direction:

(7) |

We see from this that for diamagnetic and superconducting materials the mean
vertical stiffness is negative. Thus, with these materials, as we have already
seen, a stable equilibrium of forces is possible for levitation. With other,
'ordinary' materials this stiffness is positive; the equilibrium of forces
is then unstable, as would be expected from Earnshaw's theorem. It also follows
from (6) that the conductors of the surface-current density must lie as close
as possible to the boundary plane *z* = *a* to produce the maximum
force density. For *a* = 0 the mean absolute force
density
for ferromagnetic and superconducting materials is given by:

(8) |

In practice the force densities for superconducting materials and ferromagnetic materials are unequal, since, as we noted earlier, in superconducting materials the maximum flux density at the boundary plane must remain limited.