# 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)