# Appendices

__Appendix 1.__ *Calculation of the force and force gradient on a cylindrical
ferromagnetic probe in the magnetic field of a ring with current.*

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. Let magnetization vector
be directed throughout the tip volume on the angle
to the cylinder base normal **(Fig. 1)**.

**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 acting on the elementary volume over all the ferromagnetic material. It shall be convenient to perform the integration of an arbitrary function
over the cylinder volume in cylindrical coordinates
which gives

(1)

Thus, changing variables
,
,
we find that the force and its Z-derivative in accordance with formulas (1, 2) in **chapter 2.7.3** and (1) are determined through function
as follows

(2)

where derivatives of magnetic field
in analytic form are given by formulas (6,7) in **chapter 2.7.10**.

__Appendix 2.__ *Calculation of the force and force gradient on a spherical
ferromagnetic tip in the magnetic field of a ring with current.*

Let us determine the force and derivative of the force acting on spherical ferromagnetic tip of radius
and magnetization
placed in a magnetic field of a current ring. Let the magnetization vector
be directed on the angle
to the Z-axis **(Fig. 2)**.

**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 acting on the elementary volume over all the ferromagnetic material. It shall be convenient to perform the integration of an arbitrary function
over the sphere volume in spherical coordinates
which gives

(3)

Thus, changing variables
,
,
we find that the force and its Z-derivative in accordance with formulas (1, 2) in **chapter 2.7.3** and (3) are determined as follows

(4)

where derivatives of magnetic field in analytic form are given by formulas (6,7) in **chapter 2.7.10**.

__Appendix 3.__ *Calculation of the force and force gradient on a conical tip covered
with a ferromagnetic film in the magnetic field of a current ring.*

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 ring. The tip has radius
, length
and cone angle
; the ferromagnetic film has thickness
and magnetization
directed along the Z-axis **(Fig. 3)**.

**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 acting on the 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. For the sake of mathematical convenience we will perform the integration over the sphere volume in spherical coordinates (see **Appendix 2**). Thus, changing variables
we find that the force and its Z-derivative on the spherical part of the tip in accordance with formulas (1, 2) in **chapter 2.7.3** are determined as follows

(5)

where derivatives of magnetic field in analytic form are given by formulas (6,7) in **chapter 2.7.10**.

The second step is the determination of the force and its derivative acting on the remaining part of ferromagnetic film between two cones. In this case, it is convenient to perform the integration in cylindrical coordinates (see **Appendix 1**) with a changing radius of integration that depends on the Z-coordinate. Thus, changing variables
,
,
we find that the force and its Z-derivative acting on the nonspherical part of the tip in accordance with formulas (1, 2) in **chapter 2.7.3** are determined as follows

(6)

where и .

The total force and its derivative on this tip are then given by:

(7)

__Appendix 4.__ *Calculation of the force and force gradient on a conical
ferromagnetic probe in the magnetic field of a ring with current.*

Let us determine the force and derivative of the force acting on a conical ferromagnetic tip placed in the magnetic field of a current carrying ring. The tip has radius
, length
and cone angle
; its magnetization
is directed at angle of
to the Z-axis **(Fig. 4)**.

**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 acting on the 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. It is convenient to perform the integration over the sphere volume in spherical coordinates (see **Appendix 2**). Thus, changing variables
we find that the force and its Z-derivative acting on the spherical part of the tip in accordance with formulas (1, 2) in **chapter 2.7.3** are determined as follows

(8)

where
and derivatives of magnetic field
in analytic form are given by formulas (6,7) in **chapter 2.7.10**.

The second step is the determination of the force and its derivative acting on the cone excluding its spherical part. In this case, it is convenient to perform the integration in cylindrical coordinates (see **Appendix 1**) with a changing radius of integration that depends on the Z-coordinate. Thus, changing variables
,
,
we find that the force and its Z-derivative acting on the conical part of the tip in accordance with formulas (1, 2) in **chapter 2.7.3** are determined as follows

(9)

where .

The total force and its derivative on this tip are then given by:

(10)