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Appendix 1. Calculation of the force and force gradient on a cylindrical
ferromagnetic probe in the magnetic field of a conductor.
Let us determine the force and derivative of the force acting on cylindrical ferromagnetic tip of radius  , length  and magnetization  placed in a magnetic field of a current carrying conductor. Assume that magnetization vector  is directed throughout the tip volume on the angle  to the cylinder base normal (Fig. 1).
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Fig. 1. Schematic representation of a cylindrical probe.
Vector  has coordinates  . |
As shown in chapter 2.7.3, force  acting on magnetic cantilever and its derivative 
can be calculated integrating the force on elementary volume over all
the ferromagnetic material. Taking into consideration the tip shape and
supposing that magnetic field along the Y-axis is equal to zero, the
force and its Z-derivative can be expressed through function  in accordance with formulas (1, 2) in chapter 2.7.3 as follows:
where derivatives of magnetic field  in the analytic form are given by formulas (9,10) in chapter 2.7.9.
Appendix 2. Calculation of the force and force gradient on a spherical
ferromagnetic tip in the magnetic field of a conductor.
Let us determine the force and derivative of the force acting on a spherical ferromagnetic tip of radius and magnetization placed in a magnetic field of a current carrying conductor. Let the magnetization vector be directed throughout the tip volume on the angle to the Z-axis (Fig. 2).
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Fig. 2. Schematic representation of a spherical probe.
Vector has coordinates . |
As shown in chapter 2.7.3, force acting on magnetic cantilever and its derivative
can be calculated integrating the force on elementary volume over all
the ferromagnetic material. Taking into consideration the tip shape and
supposing that magnetic field along the Y-axis is equal to zero, the
force and its Z-derivative can be expressed in accordance with formulas
(1, 2) in chapter 2.7.3 as follows:
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(2) |
where  and derivatives of magnetic field in analytic form are given by formulas (9,10) in chapter 2.7.9.
Appendix 3. Calculation of the force and force gradient on a conical tip covered
with a ferromagnetic film in the magnetic field of a conductor.
Let us determine
the force and derivative of the force acting on a conical tip covered
with a ferromagnetic film placed in a magnetic field of a
current-carrying conductor. The tip has radius , length and cone angle  ; the ferromagnetic film has thickness  and magnetization directed along the Z-axis (Fig. 3).
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Fig. 3. Section of a conical probe covered with a ferromagnetic film.
Vector has coordinates . |
As shown in chapter 2.7.3, force acting on magnetic cantilever and its derivative
can be calculated integrating the force on elementary volume over all
the ferromagnetic material. It shall be convenient to perform the
volume integration in three steps.
The first step is
the determination of the force and its derivative acting on the
spherical part of the tip. Supposing the magnetic field along the
Y-axis for this part equals to zero, the force and its Z-derivative can
be expressed in accordance with formulas (1, 2) in chapter 2.7.3 as follows:
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(3) |
 |
where  and derivatives of magnetic field in analytic form are given by formulas (9,10) in chapter 2.7.9.
The second step is
the determination of the force and its derivative on the nonmagnetic
internal cone supposing that the magnetization vector throughout the
cone volume is nonzero and coincides with . This operation is also performed in accordance with formulas (1), (2) in chapter 2.7.3 to give:
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(4) |
 |
where  and  .
The third step is
the determination of the force and its derivative on the outer part of
the cone excluding the tip spherical part supposing that the
magnetization vector throughout the cone volume is nonzero and
coincides with . In this case
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(5) |
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The total force and its derivative on this tip are then given by:
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(6) |
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Appendix 4. Calculation of the force and force gradient on a conical
ferromagnetic probe in the magnetic field of a conductor.
Let us determine
the force and derivative of the force acting on a conical tip covered
by a ferromagnetic film placed in a magnetic field of a
current-carrying conductor. The tip has radius , length and cone angle ; its magnetization is directed on the angle to the Z-axis (Fig. 4).
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Fig. 4. Section of a conical ferromagnetic probe.
Vector has coordinates . |
As shown in chapter 2.7.3, force acting on magnetic cantilever and its derivative
can be calculated integrating the force on elementary volume over all
the ferromagnetic material. It shall be convenient to perform the
volume integration in two steps.
The first step is
the determination of the force and its derivative acting on the
spherical part of the tip. Supposing the magnetic field along the
Y-axis for this part equal to zero, the force and its Z-derivative can
be expressed in accordance with formulas (1, 2) in chapter 2.7.3 as follows:
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(7) |
where , and derivatives of magnetic field in analytic form are given by formulas (9,10) in chapter 2.7.9.
The second step is
the determination of the force and its derivative on the cone excluding
its spherical part. This operation is also performed in accordance with
formulas (1), (2) in chapter 2.7.3 to give:
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(8) |
where  .
The total force and its derivative on this tip are then given by:
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(9) |
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