# Integral Equations of the First Kind for Calculating Electro- and Magnetostatic Fields Perturbed by Conductors and Ferro-Magnets

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## Abstract

**:**

## 1. Introduction

_{0}or higher, then the field perturbed by such a material will practically not differ in any way from the field perturbed by a ferromagnet with an infinite magnetic permeability [16,17,18,19].

- -
- a brief overview of the comparative characteristics of various types of magnetometers;
- -
- a schematic representation of the development of methods, equipment, and data processing of magnetic prospecting;
- -
- the goal of this research as the development of a method for magnetic prospecting of, predominantly, fossils containing materials with a high magnetic permeability.

- (A)
- Electrostatic fields. Three-dimensional case, where a conducting body bounded by a closed surface S carrying a charge q is introduced into the field of charges located in volume V
_{0}. - (B)
- The field of a permanent magnet. Three-dimensional case, where a magnetic system consists of a permanent magnet with a given distribution of the magnetization vector (V
_{0}is the volume occupied by the magnet) and a homogeneous ferromagnet bounded by a closed surface S. - (C)
- Electrostatic field. The plane-parallel case, where the system is extended along the z axis.
- (D)
- The field of a permanent magnet. Plane-parallel case, where the magnetic system is extended along the z axis.

_{0}(Figure 3) [24,25,26].

_{0}, and $\sigma \left(P\right)$ is an unknown distribution of the surface-electric-charge density on a given surface S.

## 2. Materials and Methods

_{k}approximately in the center of the site $\Delta {S}_{k}$; if $i\ne k$, then approximately replace the integrals in (8) with the value:

_{k}is in the center of the rectangle; this integral is equal to (calculations are omitted):

## 3. Results and Discussion

_{1}. We find the solution of the equation:

#### 3.1. The Field of a Permanent Magnet—Three-Dimensional Case

_{0:}

_{n}is the normal component of the vector $\overrightarrow{M}$ on the surface of the magnet S

_{0}(the direction of the normal is chosen on the exterior of the magnet).

_{0}is the point of the zero value of the potential.

#### 3.2. The Electrostatic Field—The Plane-Parallel Case

#### 3.3. The Field of a Permanent Magnet—The Plane-Parallel Case

#### 3.4. Examples of Calculating Magnetic Fields Using Integral Equations of the First Kind

_{1}= 1.749 Tl (the first method), B

_{1}= 1.749 Tl (the second method) and at a point on the y axis at a distance of 0.0025 m from the ferromagnet, i.e., at the point y = 0.0275 m. It was equal to B

_{2}= 0.644 Tl (the first method) and B

_{2}= 0.644 Tl (the second method). At the same point, the induction was calculated in the absence of a ferromagnet. It was equal to ${{B}^{\prime}}_{2}$ = 0.365 Tl.

_{1}= 1.817 Tl (first method), B

_{1}= 1.817 Tl (second method), B

_{2}= 0.44 Tl (first method), and B

_{2}= 0.444 Tl (second method). ${{B}^{\prime}}_{2}$ = 0.268 Tl.

## 4. Conclusions

- -
- bring geophysical services to the service market on a new scientific and technical production level;
- -
- reduce the environmental burden on nature by replacing magnetometric measurements with energy-saving, environmentally safe technology;
- -
- ensure the export potential of magnetometric equipment.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the development of methods, equipment, and data processing of magnetic prospecting.

**Figure 3.**A conducting body bounded by a closed surface S carrying a charge q is introduced into the field of charges located in volume V

_{0}.

**Figure 4.**A magnetic system consisting of a permanent magnet with a given distribution of the magnetization vector $\overrightarrow{M}\left(P\right)$ ($P\in {V}_{0}$, ${V}_{0}$ is the volume occupied by the magnet) and a homogeneous ferromagnet bounded by a closed surface S.

**Figure 6.**Field of a permanent magnet, plane-parallel case, under the condition of a magnetic system extended along the z axis.

**Figure 7.**The magnetic field of the system for which the calculation was made using integral equations of the first kind.

Magnetometer Type | Magnetic Sensitive Element | Measured Components |
---|---|---|

Optical-mechanical | Permanent magnet | Z, ΔZ |

Proton | Hydrogen liquid | T, ΔT, ∂T ∂x, ∂T ∂x |

Overhauser | Hydrogen-containing liquid with the addition of free radicals with unpaired electrons | |

Quantum | Alkali metal vapors | |

Ferroprobe | Ferrosonde | X, Y, Z, ΔX, ΔY, ΔZ |

Cryogenic | Superconducting quantum interferometer | T, ΔT |

Magnetometer Type | Advantages | Disadvantages |
---|---|---|

Optical-mechanical | Able to measure Z, X, Y and H components. | Zero point creep, presence of azimuth correction, temperature drift, low measurement speed, low accuracy. |

Proton | This magnetometer type is impervious to shaking and vibrations, measurements are practically independent of changes in external conditions (temperature, humidity, pressure), there is no need for precise orientation of the sensor, there is no need to stake out reference networks, zero-point shift is negligible. | Instability and signal loss at high magnetic field gradients. |

Overhauser | All the benefits of proton magnetometers, plus reduced measurement time, lower uncertainty due to increased signal-to-noise ratio, small sensor size. | Short lifetime of the working substance, the appearance of a systematic error, due to the influence of the microwave unit. |

Quantum | High measurement speed, high resolution. | The need for orientation of the sensor is present, but with small values: orientation and azimuth errors, temperature drift. Sensitivity to mechanical influences (shock, vibration). |

Ferroprobe | Able to measure Z, X, Y and H components with high accuracy. | The bulkiness of the equipment, the need to orient the sensor. |

Cryogenic | High accuracy. | The need to maintain very low temperatures for a superconductor. There are no mass-produced devices. |

**Table 3.**Calculation of the potential on the segment L of the upper face of the ferromagnet after calculating the distribution $\sigma $.

The First Way | The Second Way |
---|---|

−2410.29 | −2410.29 |

−2368.50 | −2368.50 |

−2354.42 | −2354.42 |

−2353.95 | −2353.95 |

−2354.71 | −2354.71 |

−2354.71 | −2354.71 |

−2353.95 | −2353.95 |

−2354.42 | −2354.42 |

−2368.50 | −2368.50 |

−2410.29 | −2410.29 |

**Table 4.**Calculation of the potential on the line L in the case of a plane-parallel version of this system, when $c=\infty $ and ${c}_{\mathcal{M}}=\infty $. N was taken to be 256.

The First Way | The Second Way |
---|---|

−6020.20 | −6020.20 |

−6008.95 | −6008.95 |

−6004.63 | −6004.63 |

−6004.21 | −6004.21 |

−6004.25 | −6004.25 |

−6004.25 | −6004.25 |

−6004.21 | −6004.21 |

−6004.63 | −6004.63 |

−6008.95 | −6008.95 |

−6020.20 | −6020.20 |

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**MDPI and ACS Style**

Plugatar, Y.; Filippov, D.; Chabanov, V.; Kazak, A.; Korzin, V.; Oleinikov, N.; Mayorova, A.; Nekhaychuk, D. Integral Equations of the First Kind for Calculating Electro- and Magnetostatic Fields Perturbed by Conductors and Ferro-Magnets. *Inventions* **2023**, *8*, 55.
https://doi.org/10.3390/inventions8020055

**AMA Style**

Plugatar Y, Filippov D, Chabanov V, Kazak A, Korzin V, Oleinikov N, Mayorova A, Nekhaychuk D. Integral Equations of the First Kind for Calculating Electro- and Magnetostatic Fields Perturbed by Conductors and Ferro-Magnets. *Inventions*. 2023; 8(2):55.
https://doi.org/10.3390/inventions8020055

**Chicago/Turabian Style**

Plugatar, Yurij, Dmitriy Filippov, Vladimir Chabanov, Anatoliy Kazak, Vadim Korzin, Nikolay Oleinikov, Angela Mayorova, and Dmitry Nekhaychuk. 2023. "Integral Equations of the First Kind for Calculating Electro- and Magnetostatic Fields Perturbed by Conductors and Ferro-Magnets" *Inventions* 8, no. 2: 55.
https://doi.org/10.3390/inventions8020055