# Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam

## Abstract

**:**

## 1. Introduction

## 2. Basic Equations

_{e}, charge -e, and immobile ions with density n

_{i}(x ≥ 0), forming a sharp boundary, to which a beam of plane-polarized electromagnetic radiation with a frequency ω and a wave number k propagates along itsnormal on the axis 0X (Figure 1). In the region (x ≤ 0) surrounding the plasma, the following expression can be written for the intensity of

**E**

_{0}= {0, E

_{0},0} of the electric field having auniform spatial distribution in azimuth in the front plane 0YZ.

_{e}(

**r**) is the electrostatic potential of the surface charge formed at the plasma boundary [9,10,11], and E

_{0r}is the amplitude of the reflected electromagnetic signal.

**v**is described by the equation:

**B**is the strength of the magnetic field of external radiation, v

_{T}

^{2}= T

_{e}/m, T

_{e}is the temperature of electrons, thermal pressure is taken into account in (2) only to estimate the parameters of the equilibrium state, and n

_{0}is the equilibrium density of plasma particles in the stationary state (n

_{e}= n

_{i}≡ n

_{0}).

_{0}is the dielectric density of a vacuum, and μ

_{0}is its magnetic permeability.

_{0}(y, z) woulddepend only on the coordinate ρ, and it waspossible to consider different intensity distributions in the plane of the wave front, for example, an exponential one.

**v**should be divided into a fast-variable

**v**

_{E}component and a static δ

**v**(

**r**) part (

**v**

_{E}= e

**E**

_{0}

**/**mω). As a result, the following expression can be derived from Equation (2)

**v**is determined by the potential ψ (δ

**v =**∇ψ). With its help, it waspossible to estimate the change in individual physical quantities, as compared to their values at selected points.

## 3. Analytical and Numerical Results

_{0}changedsmoothly, and when the density of n

_{e}reachedthe critical value of n

_{c}, that is ω = ω

_{p}(ω

_{p}

^{2}= e

^{2}n

_{e}/(m·ε

_{0})), the plasma becameopaque to this wave field, as a result of which it droppedexponentially rapidly into the dense plasma, the density of which, in turn, increasedas rapidly, according to (9). At this increase in density on the surface (x = x

_{b}, ρ = ρ

_{b}), thermal pressure v

_{T}

^{2}dominated one side, and on the other, electromagnetic, characterized by v

_{E}

^{2}, which in a stationary state wouldbalance each other. Therefore, an approximate condition

_{0}(x

_{b},ρ

_{b}) for the formation of the cavern.

#### 3.1. Conditions for the Formation of Globe-Shaped Resonators of the Electromagnetic Field

_{e}, which attemptedto return the displaced electrons back to their positions. The shape of the cavern varied depending on the ratio of the characteristics of the task. However, for example, in the case of a spherical cavity, its radius R in the equilibrium state was determined bycondition (8), in which the potential φ

_{e}that dependedonly on the radial coordinate rhad tobe substituted from the solution of Equation (5) for the sphere, within which n

_{e}≈ 0. In this case, one couldobtain from (5):

_{r}of the electric field

_{m}(m = 1,2,3,…), as defined from the following dispersion equation:

_{p}= ω

_{p}R/c, the approximate value of the frequency of natural oscillations was in the form ω

_{m}= a

_{m}c/R, where the constant a

_{m}is determined from the solution of the following transcendental equation:

#### 3.2. Generation of Electrostatic Fields of Surface Charge near Plasma Space by a Beam of Electromagnetic Radiation

_{e}(

**r**), which hada large localization region near the plasma boundary and couldaccelerate charged particles [12,13,14]. The description of this surface charge, performed in [9,10,11], was based on the theory of the potential [8,15], which could express the value of the potential φ

_{e}(

**r**) throughout space viaits value on the surface of the plasma:

_{ρ}= −∂

_{ρ}φ

_{e}decreased with increasing distance |x| proportionally 1/|x| (component E

_{x}= −∂

_{x}φ

_{e}fell 1/|x|

^{2}). As compared tothe amplitude of the electromagnetic beam, the magnitude of the electrostatic strength of the field decreasedat a distance from its center not according to the exponential but according to the power law, that is, the area of its localization wasmuch larger.

_{0}and a mass M from a point (x,ρ) and calculate the final velocity of its movement at infinity. From the equation of motion for the velocity

**v**

_{p}of the particle:

^{1/2}times wasless than the amplitude of electron oscillation at the center of the electromagnetic beam.

## 4. Summary and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Gradov, O.M.
Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam. *Technologies* **2022**, *10*, 78.
https://doi.org/10.3390/technologies10040078

**AMA Style**

Gradov OM.
Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam. *Technologies*. 2022; 10(4):78.
https://doi.org/10.3390/technologies10040078

**Chicago/Turabian Style**

Gradov, O. M.
2022. "Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam" *Technologies* 10, no. 4: 78.
https://doi.org/10.3390/technologies10040078