# Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields

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## Abstract

**:**

## 1. Introduction

## 2. Particles with Electric Charge and a Magnetic Dipole

#### 2.1. Equation of Motion

#### 2.2. Energy Efficiency

## 3. Spin of Kerr BH versus Magnetic Interactions

#### 3.1. In the Same ISCO Radius

#### 3.2. In the Same Accretion Disk Luminosity

## 4. Particle Collisions

#### 4.1. Critical Angular Momentum

#### 4.2. Collisions of Neutral and Electrically Charged Particles

#### 4.3. Collisions of Electrically Neutral and Magnetized Particles

#### 4.4. Charged-Magnetized

#### 4.4.1. Positively Charged Particle–Magnetized Particle with Positive $\beta $

#### 4.4.2. Negatively Charged Particle–Magnetized Particle with Positive $\beta $

#### 4.4.3. Positively Charged Particle–Magnetized Particle with Negative $\beta $

#### 4.5. Particles Collision Having Electric Charge and a Magnetic Dipole

## 5. Particles with Electric Charge and a Magnetic Dipole Moment in Astrophysics

#### 5.1. Neutron Stars and White Dwarfs as Test Particles with a Magnetic Dipole Moment and Electric Charge

#### 5.2. Electron and Protons as Candidates for Particles with Electric Charge and a Magnetic Dipole Moment

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

$\mathit{Gaussian}$ | $\mathit{Geometrized}$ | $\mathit{Conv}.$ | |
---|---|---|---|

$Period$ | 1 s | $2.99\times {10}^{10}$ cm | c |

Mass | 1 g | $7.42\times {10}^{-29}$ cm | $G/{c}^{2}$ |

Electric charge | 1 $statC$ | $2.87\times {10}^{-25}$ cm | $\sqrt{G}/{c}^{2}$ |

Magnetic field | 1 $Gauss$ | $8.16\times {10}^{-15}$ 1/cm | $\sqrt{G}/c$ |

$\mathit{q},\phantom{\rule{4pt}{0ex}}\mathit{e}$ | $\mathit{m},\phantom{\rule{4pt}{0ex}}{\mathit{m}}_{\mathit{e}}$ | $\mathit{\mu},\phantom{\rule{4pt}{0ex}}{\mathit{\mu}}_{\mathit{B}}$ | |
---|---|---|---|

$\mathrm{Electron},e$ | $-1$ | 1 | − |

$\mathrm{Proton},p$ | 1 | 1836 | $1.5\xb7{10}^{-3}$ |

$\mathrm{Neutron},n$ | 0 | 1839 | $-{10}^{-3}$ |

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**Figure 1.**The radial dependence of the effective potential for magnetized and charged test particles.

**Figure 2.**Radial profiles of the specific energy (

**left panel**) and angular momentum (

**right panel**) for magnetized and charged particles at the circular orbits.

**Figure 3.**The dependence of the ISCO radius on the parameters ${\omega}_{B}$ (

**left panel**) and $\beta $ (

**right panel**).

**Figure 4.**Possible values of parameters ${\omega}_{B}$ and $\beta $ for a ISCO radius between 5 and 10 M.

**Figure 5.**Dependence of the energy efficiency on the magnetic-coupling parameter $\beta $ for different values of the parameter $\omega $.

**Figure 6.**Degeneracy between parameter $\beta $ and a, i.e., for a given value of $\beta $ there exists a Kerr geometry with a given value of spin parameter $a/M$ that has the same ISCO radius for various combinations of parameter ${\omega}_{B}$. The point to be noted here is that $a/M$ represents a black hole rotation (spin) parameter.

**Figure 7.**Degeneracy between the parameter $\beta $ and the spin parameter of Kerr BH, providing the same energy efficiency for different values of parameter $\omega $.

**Figure 8.**The dependence of critical angular momentum on parameter $\beta $ with the different values of ${\omega}_{B}$.

**Figure 9.**Figure shows the radial dependence of ${\mathcal{E}}_{\mathrm{cm}}$ of the collisions of two positively and negatively charged particles with electrically neutral particles on the different values of the charge.

**Figure 10.**The same as Figure 9 but for collisions of neutral-magnetized particles with positive magnetic dipole particles.

**Figure 11.**The same as Figure 9 but for collisions of positively charged particles with no magnetic dipole-positive magnetized particles with no charge.

**Figure 12.**The same as Figure 9 but for the collisions of negatively charged particles with non-dipole-positive magnetized particles with no charge.

**Figure 13.**The same as Figure 9 but for collisions of positively charged particles with non-dipole-negative magnetized particles with no charge.

**Figure 14.**The same as Figure 9 but for the collisions of two positively charged particles with a magnetic dipole for different values of ${\omega}_{B}$ (

**left panel**) and $\beta $ (

**right panel**).

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**MDPI and ACS Style**

Rayimbaev, J.; Shaymatov, S.; Abdulxamidov, F.; Ahmedov, S.; Begmatova, D.
Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields. *Universe* **2023**, *9*, 135.
https://doi.org/10.3390/universe9030135

**AMA Style**

Rayimbaev J, Shaymatov S, Abdulxamidov F, Ahmedov S, Begmatova D.
Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields. *Universe*. 2023; 9(3):135.
https://doi.org/10.3390/universe9030135

**Chicago/Turabian Style**

Rayimbaev, Javlon, Sanjar Shaymatov, Farrux Abdulxamidov, Saidmuhammad Ahmedov, and Dilfuza Begmatova.
2023. "Magnetized Particles with Electric Charge around Schwarzschild Black Holes in External Magnetic Fields" *Universe* 9, no. 3: 135.
https://doi.org/10.3390/universe9030135