Posted on Categories:Electrodynamics, 物理代写, 电动力学

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## 物理代写|电动力学代考Electrodynamics代写|Interaction of Langmuir and ion waves

In correspondence with the announced in the title aim we should include ions into consideration, see the basic system (14.1). Under the action of coherent sources of a prescribed frequency and polarization in a plasma the wave trains of different types can be excited. These are, for example, the ionosound and Langmuir longitudinal waves discussed in section 13.2 .2 and waves with the electric field $\vec{E}$ orthogonal to $\vec{k}$. Wave splitting in a simple way is impossible if the plasma is in an external magnetic field, that is, magnetized plasma [6]. In such magnetized plasma new important dispersion relation branches appear $[4,6,13]$. All possible plasma waves of sufficient amplitude interact [18, 19]. Let us consider the occurrence of an interaction in the case of longitudinal waves in an inhomogeneous plasma, for which, including an acoustic mode we derive a nonlinear system of equations. The weak nonlinearity dependence of the susceptibility on the electromagnetic field can be found by iterating equation (13.11) over the next order of amplitude parameter. To do this, one must modify the Vlasov term $f \rightarrow f_0+\chi$. As $\chi$ is a linear function of the amplitude the expression for the distribution function will be quadratic in the field $E^{\prime}=\sigma E, \sigma$ being the amplitude parameter. So if
$$f=f^0+\sigma \chi+\sigma^2 \Phi+\cdots$$
then
$$\frac{d \Phi}{d t}=-e E_x \int_0^t F_p\left(x-\frac{p}{m}(t-\tau), \tau\right) d \tau$$
where $F$ is determined via the rhs of equation (13.14). Since effects of the dispersion and inhomogeneity of the field are assumed to be small $(\beta \ll 1)$ within the calculation of the nonlinear terms hidden in $\Phi$ we retain only contributions of the $\beta^0$ order. Therefore
$$\Phi=i e^2 \int_0^t \frac{A e^{i \phi}+c . c}{\omega-p k / m}\left(f_{p p}^0-\frac{k f_p^0}{m \omega-p k}\right)\left(A e^{i \phi}+c . c\right) d \tau .$$

## 物理代写|电动力学代考Electrodynamics代写|Basic equations

In this section we study x-rays, as a very important example of electromagnetic waves in matter, that is described as wave equation (6.109) in isotropic matter from chapter 6, section 6.5. It arises as direct corollary of the Maxwell system for electromagnetic field in matter (6.1)-(6.4). Such phenomenon is characterized by the parameters $\varepsilon$ and $\mu$, that are either taken from experiments (phenomenology) or from a theory originated from the Drude-Lorentz model, see chapter 6 . The propagation of a monochromatic electromagnetic wave with frequency $\omega_0$ in a medium with complex refractive index $n=1-\delta+i \beta$ is described by the Helmholtz equation $[14,15]$
$$\Delta E+k_0^2 n^2 E=0$$
where $\mathrm{E}$ is an electric field component; $\vec{r}=(x ; y ; z), k_0=\omega_0 / c$ is wavenumber, $c$ is the speed of light, $\beta$ is an absorption coefficient and $\delta$ is a refractive decrement $(\delta ; \beta$ are non-negative). More convenient for practical applications, a simplified wave equation can be derived from the Helmholtz equation (15.1). Let us consider the case when the wave propagates along the $x$-axis, and the characteristic scales $l_y, l_z$ of the wave along the axes $y$ and $z$ are much larger than the characteristic scale $l_x$ of the wave along the $x$-axis: $l_x \ll l_y, l_x \ll l_z$. In this case, equation (15.1) can be significantly simplified. If we substitute
$$E(\vec{r})=A(\vec{r}) \exp \left[i k_0 x\right]$$
into equation (15.1) we get the equation for function $A(\vec{r})$. We suppose that $A(\vec{r})$ varies slowly with variables $\vec{r}$. Neglecting the small term of a second derivative of the function $A(\vec{r})$ with respect to the x-variable, we arrive at the paraxial equation:
$$\frac{\partial A}{\partial x}+\frac{1}{2 i k_0} \Delta_{\perp} A+\frac{k_0\left(n^2-1\right)}{2 i} A=0,$$
where
$$\Delta_{\perp}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Interaction of Langmuir and ion waves

$$f=f^0+\sigma \chi+\sigma^2 \Phi+\cdots$$

$$\frac{d \Phi}{d t}=-e E_x \int_0^t F_p\left(x-\frac{p}{m}(t-\tau), \tau\right) d \tau$$

$$\Phi=i e^2 \int_0^t \frac{A e^{i \phi}+c . c}{\omega-p k / m}\left(f_{p p}^0-\frac{k f_p^0}{m \omega-p k}\right)\left(A e^{i \phi}+c . c\right) d \tau .$$

## 物理代写|电动力学代考Electrodynamics代写|Basic equations

$$\Delta E+k_0^2 n^2 E=0$$

$$E(\vec{r})=A(\vec{r}) \exp \left[i k_0 x\right]$$

$$\frac{\partial A}{\partial x}+\frac{1}{2 i k_0} \Delta_{\perp} A+\frac{k_0\left(n^2-1\right)}{2 i} A=0,$$

$$\Delta_{\perp}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Electrodynamics, 物理代写, 电动力学

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## 物理代写|电动力学代考Electrodynamics代写|Hydrodynamic equations approach: flute instability

Let us return to the hydrodynamic system, the continuity and momentum ones
$$\frac{\partial n_a}{\partial t}+\nabla \cdot\left(n_a \vec{u}_a\right)=0$$

$$\frac{\partial \vec{u}a}{\partial t}+\left(\vec{u}_a \cdot \nabla\right) \vec{u}_a=\frac{e_a}{m_a}\left{\vec{E}+\frac{1}{c}\left[\vec{u}_a \times \vec{B}\right]\right}+\vec{g} .$$ Here $\vec{u}_a$ is hydrodynamic velocity and $n_a$ the density of plasma component ‘ $a$ ‘, $\vec{g}$ is the gravity field. The constants $e_a, m_a$ are corresponding charge and mass. It is used in the next chapter together with the complete electrodynamic system (14.1). The equilibrium state we will mark by the index ‘ 0 ‘. For density it will be $$n_a^0(x)$$ with neutrality condition $$e n_e^0+e_i n_i^0=0$$ The velocity at equilibrium is $$u{a y}^0=-g / \Omega_a(x)$$
where
$$\Omega_a(x)=\frac{e_a B}{m_a c}$$
is gyrofrequency of a ion rotation in the magnetic field. The field equations give
$$\frac{d}{d x} \frac{B^2}{8 \pi}=g \sum m_a n_a=\rho_m g \approx g \sum m_i n_i$$

## 物理代写|电动力学代考Electrodynamics代写|Basic equations

We write the basic plasma equations that join electrodynamics and hydrodynamics, we repeat partially the introduction of equations of section 7.2 for reader convenience. The first group of equation is the modified Maxwell system (compare with those of section 13.1.1):
$$\begin{gathered} \nabla \cdot \vec{E}=4 \pi \sum_a e_a n_a \ \nabla \cdot \vec{B}=0 \ \frac{1}{c} \frac{\partial \vec{B}}{\partial t}=-\nabla \times \vec{E} \ \frac{1}{c} \frac{\partial \vec{E}}{\partial t}=\nabla \times \vec{B}-\frac{4 \pi}{c} \sum_a e_a n_a \overrightarrow{v_a} \end{gathered}$$
while the second, hydrodynamical group consists of the continuity and the momentum balance equations, as in section 13.4 .3
$$\begin{gathered} \frac{\partial n_a}{\partial t}+\nabla \cdot\left(n_a \vec{v}_a\right)=0, \ \frac{\partial \vec{v}_a}{\partial t}+\left(\overrightarrow{v_a} \cdot \nabla\right) \overrightarrow{v_a}=\frac{e_a}{m_a}\left{\vec{E}+\frac{1}{c}\left[\overrightarrow{v_a} \times \vec{B}\right]\right}, \end{gathered}$$
where $n_a$ stands for particles ‘ $a$ ‘ concentration, $\vec{v}_a$ is velocity of such particles, $m_a$ and $e_a$ are mass and charge of the particle $a$. Index $a$ describes the type of ion, for unique ion type, $a=i, e$, mark ions and electrons.

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Hydrodynamic equations approach: flute instability

$$\frac{\partial n_a}{\partial t}+\nabla \cdot\left(n_a \vec{u}_a\right)=0$$

$$n_a^0(x)$$

$$e n_e^0+e_i n_i^0=0$$

$$u a y^0=-g / \Omega_a(x)$$

$$\Omega_a(x)=\frac{e_a B}{m_a c}$$

$$\frac{d}{d x} \frac{B^2}{8 \pi}=g \sum m_a n_a=\rho_m g \approx g \sum m_i n_i$$

## 物理代写|电动力学代考Electrodynamics代写|Basic equations

$$\nabla \cdot \vec{E}=4 \pi \sum_a e_a n_a \nabla \cdot \vec{B}=0 \frac{1}{c} \frac{\partial \vec{B}}{\partial t}=-\nabla \times \vec{E} \frac{1}{c} \frac{\partial \vec{E}}{\partial t}=\nabla \times \vec{B}-\frac{4 \pi}{c} \sum_a e_a n_a \overrightarrow{v_a}$$

|begin ${$ gathered $} \backslash$ frac $\left{\backslash\right.$ partial $\left.n_{-} a\right}{\backslash$ partial $t}+\backslash$ nabla $\backslash$ coot $\backslash$ left $\left(n_{-} a \mid\right.$ vec ${v} _a \backslash$ right $)=0, \backslash \backslash$ frac $\left{\backslash\right.$ partial $\mid$ vec $\left.{v} _a\right}{\backslash$ partial $t}+\backslash$ left $(\backslash$ ove

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Electrodynamics, 物理代写, 电动力学

## avatest™帮您通过考试

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## 物理代写|电动力学代考Electrodynamics代写|Cold plasma: general dispersion equation

We first consider the classical example of a wave in a cold plasma neglecting the ion motion. The dispersion relations conventionally used for metamaterials have common features with ones for plasma, hence we include the wave theory for such materials in this chapter, as well in section 18.3.3.

To explain the idea of the method consider the 1D perturbation, when the electromagnetic field propagates along the $x$-axis. Let the longitudinal component of electric field be formed as a wave-train, localized in a range $t>0$, the dimensionless parameter $\beta \ll 1$ is introduced to characterize the scale of slow-varied amplitude $A$ to be convenient in a perturbation theory development

$$E_x=A(\beta x, \beta t) \exp i(k x-\omega t)+c . c .$$
Equation (13.14) simplifies as
$$\chi_t+\frac{p}{m} \chi_x=F(x, t)$$
it is directly integrated by the method of characteristics
$$\chi=\int_0^t F\left(x-\frac{p}{m}(t-\tau), \tau\right) d \tau .$$
where $m$ is the electron mass and $p$ is the $x$-component of its momentum. Plugging equation (13.17) into expression for $F$ from equation (13.14) and, next, to equation (13.19), we get in the zeroth order (in the small parameter $\beta$, tearing the slow varying amplitude $A$ out from the integrand) approximation
$$\begin{gathered} \chi=e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \int_0^t \exp i\left[k\left(x-\frac{p}{m} t\right)+\left(\frac{k p}{m}-\omega\right) \tau\right] d \tau+c . c . \ =e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p}{m}\right)}+c . c . \end{gathered}$$

## 物理代写|电动力学代考Electrodynamics代写|Maxwell distribution background: Langmuir waves

For the Maxwell distribution $f_0$ it has the form
$$\omega^2=\omega_{L e}^2+3 \kappa_B T_e \frac{k^2}{m}$$
where
$$\omega_{L e}=2 e \sqrt{\frac{\pi n}{m}}$$
is the Langmuir frequency of the electrons, $\kappa_B$-Boltzmann constant, and $T_e$ is the electron temperature.
If the electron density
$$\rho=\rho_0 \exp (-i \omega t)+c . c .$$
and $\frac{\left\langle p^2\right\rangle^{1 / 2} k}{m \omega} \ll 1$ and the same condition is valid for ions, then equation (13.20) with ion terms account gives
$$\chi=\sum_a e_a \frac{\partial f_{0 a}}{\partial p_a} A \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p_a}{m_a}\right)}+c . c .,$$
where the $x$-component of ion ‘ $\mathrm{a}$ ‘ momentum is denoted as $p_a$ so as the ion mass is $m_a$

Approximate equality occurs if there is no great difference between the concentrations $n_a$. For example, this is true for a single-ion quasineutral plasma. Obviously the Langmuir frequency is basic for longitudinal plasma oscillations in the long wavelength range.

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Cold plasma: general dispersion equation

$$E_x=A(\beta x, \beta t) \exp i(k x-\omega t)+c . c .$$

$$\chi_t+\frac{p}{m} \chi_x=F(x, t)$$

$$\chi=\int_0^t F\left(x-\frac{p}{m}(t-\tau), \tau\right) d \tau$$

$$\chi=e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \int_0^t \exp i\left[k\left(x-\frac{p}{m} t\right)+\left(\frac{k p}{m}-\omega\right) \tau\right] d \tau+c . c .=e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p}{m}\right)}+c . c .$$

## 物理代写|电动力学代考Electrodynamics代写|Maxwell distribution background: Langmuir waves

$$\omega^2=\omega_{L e}^2+3 \kappa_B T_e \frac{k^2}{m}$$

$$\omega_{L e}=2 e \sqrt{\frac{\pi n}{m}}$$

$$\rho=\rho_0 \exp (-i \omega t)+c . c .$$

$$\chi=\sum_a e_a \frac{\partial f_{0 a}}{\partial p_a} A \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p_a}{m_a}\right)}+c . c .,$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。