## 物理代写|电磁学代写electromagnetism代考|ELEC3104

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电磁学代写electromagnetism代考|Quasi-Static Models

More general approximate models can be obtained by discriminating the time variations, respectively, of the electric field and the magnetic induction. Hence, after the scaling step in Maxwell’s equations in vacuum, that is, in Eqs. (1.107-1.110), if we suppose that
$$\bar{v} \frac{\bar{B}}{\bar{E}} \ll 1 \quad \text { and } \quad \frac{\bar{v}}{c} \frac{\bar{E}}{c \bar{B}} \approx 1$$

we easily obtain that we may neglect the time derivative $\partial_t \boldsymbol{B}$ in Faraday’s law, whereas the coefficient of the time derivative $\partial_t \boldsymbol{E}$ in Ampère’s law is comparable to one. We then obtain the electric quasi-static model, which can be written in the physical variables $\boldsymbol{E}, \boldsymbol{B}$ as
\begin{aligned} &\operatorname{curl} \boldsymbol{E}=0 \ &\operatorname{div} \boldsymbol{E}=\frac{1}{\varepsilon_0} \varrho \ &\operatorname{curl} \boldsymbol{B}=\mu_0 \boldsymbol{J}+\frac{1}{c^2} \frac{\partial \boldsymbol{E}}{\partial t} \ &\operatorname{div} \boldsymbol{B}=0 \end{aligned}
It can be proven (see Sect. 6.4) that this model is a first-order approximation of Maxwell’s equations. As mentioned, it is formally built by assuming that the time variations of the magnetic induction are negligible.
In a similar way, let us suppose, contrastingly, that
$$\frac{\bar{v}}{c} \frac{\bar{E}}{c \bar{B}} \ll 1 \quad \text { and } \quad \bar{v} \frac{\bar{B}}{\bar{E}} \approx 1$$
thus we may neglect the time derivative $\partial_t \boldsymbol{E}$ in Ampère’s law, whereas the coefficient of the time derivative $\partial_t \boldsymbol{B}$ in Faraday’s law is comparable to one.

## 物理代写|电磁学代写electromagnetism代考|Darwin Model

Let us introduce another approximate model, also known as the Darwin model [90]. It consists in introducing a Helmholtz decomposition of the electric field as
$$\boldsymbol{E}=\boldsymbol{E}^L+\boldsymbol{E}^T$$
where $\boldsymbol{E}^L$, called the longitudinal part, is characterized by curl $\boldsymbol{E}^L=0$, and $\boldsymbol{E}^T$, the transverse part, is characterized by div $\boldsymbol{E}^T=0$. Starting from Maxwell’s equations in vacuum, one then assumes that $\varepsilon_0 \partial_t \boldsymbol{E}^T$ can be neglected in Ampère’s law: one neglects only the transverse part of the displacement current, whereas, in the quasi-static model, the total displacement current $\varepsilon_0 \partial_t \boldsymbol{E}$ is neglected. In this sense, it is a more sophisticated model than the quasi-static one. Moreover, it can be proven (see Sect. 6.4), by using the low frequency approximation (1.111) and the resulting dimensionless form of Maxwell’s equations, that this model yields a second-order approximation of the electric field and a first-order approximation of the magnetic induction.
The Darwin model in vacuum is written in the physical variables $\boldsymbol{E}, \boldsymbol{B}$ as
\begin{aligned} &\operatorname{curl} \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t}, \quad \operatorname{div} \boldsymbol{E}=\frac{\varrho}{\varepsilon_0}, \ &\operatorname{curl} \text { curl } \boldsymbol{B}=\mu_0 \text { curl } \boldsymbol{J}, \quad \operatorname{div} \boldsymbol{B}=0 . \end{aligned}
Then, if one uses the Helmholtz decomposition (1.120) with div $\boldsymbol{E}^T=0$ and $\boldsymbol{E}^L=-\operatorname{grad} \phi$, we see that the three fields $\boldsymbol{B}, \boldsymbol{E}^T$ and $\phi$ solve three elliptic PDEs, namely (1.121) and
\begin{aligned} &-\Delta \phi=\frac{\varrho}{\varepsilon_0} \ &\operatorname{curl} \boldsymbol{E}^T=-\frac{\partial \boldsymbol{B}}{\partial t}, \quad \operatorname{div} \boldsymbol{E}^T=0 . \end{aligned}

## 物理代写|电磁学代写电磁学代考|准静态模型

$$\bar{v} \frac{\bar{B}}{\bar{E}} \ll 1 \quad \text { and } \quad \frac{\bar{v}}{c} \frac{\bar{E}}{c \bar{B}} \approx 1$$

\begin{aligned} &\operatorname{curl} \boldsymbol{E}=0 \ &\operatorname{div} \boldsymbol{E}=\frac{1}{\varepsilon_0} \varrho \ &\operatorname{curl} \boldsymbol{B}=\mu_0 \boldsymbol{J}+\frac{1}{c^2} \frac{\partial \boldsymbol{E}}{\partial t} \ &\operatorname{div} \boldsymbol{B}=0 \end{aligned}
。可以证明(见第6.4节)，该模型是麦克斯韦方程的一阶近似。如前所述，它是通过假设磁感应的时间变化可以忽略的形式建立的。以类似的方式，让我们假设，相比之下，
$$\frac{\bar{v}}{c} \frac{\bar{E}}{c \bar{B}} \ll 1 \quad \text { and } \quad \bar{v} \frac{\bar{B}}{\bar{E}} \approx 1$$
，因此我们可以忽略Ampère定律中的时间导数$\partial_t \boldsymbol{E}$，而法拉第定律中时间导数$\partial_t \boldsymbol{B}$的系数相当于1

## 物理代写|电磁学代写电磁代考|达尔文模型

$$\boldsymbol{E}=\boldsymbol{E}^L+\boldsymbol{E}^T$$
，其中$\boldsymbol{E}^L$，称为纵向部分，以旋度$\boldsymbol{E}^L=0$为特征，$\boldsymbol{E}^T$，横向部分，以div $\boldsymbol{E}^T=0$为特征。从真空中的麦克斯韦方程出发，假设Ampère定律中可以忽略$\varepsilon_0 \partial_t \boldsymbol{E}^T$:只忽略位移电流的横向部分，而在准静态模型中，总位移电流$\varepsilon_0 \partial_t \boldsymbol{E}$被忽略。从这个意义上说，它是一个比准静态模型更复杂的模型。此外，可以证明(见第6.4节)，通过使用低频近似(1.111)和得到的麦克斯韦方程组的无量纲形式，该模型产生电场的二阶近似和磁感应的一阶近似。真空中的达尔文模型在物理变量$\boldsymbol{E}, \boldsymbol{B}$中被写为
\begin{aligned} &\operatorname{curl} \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t}, \quad \operatorname{div} \boldsymbol{E}=\frac{\varrho}{\varepsilon_0}, \ &\operatorname{curl} \text { curl } \boldsymbol{B}=\mu_0 \text { curl } \boldsymbol{J}, \quad \operatorname{div} \boldsymbol{B}=0 . \end{aligned}

\begin{aligned} &-\Delta \phi=\frac{\varrho}{\varepsilon_0} \ &\operatorname{curl} \boldsymbol{E}^T=-\frac{\partial \boldsymbol{B}}{\partial t}, \quad \operatorname{div} \boldsymbol{E}^T=0 . \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电磁学代写electromagnetism代考|PHYC20014

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电磁学代写electromagnetism代考|Magnetostatics

In a similar manner, a static formulation can be written for the magnetic induction $\boldsymbol{B}^{\text {stat }}$. By applying the curl operator to equation $\operatorname{curl}\left(\mu^{-1} \boldsymbol{B}^{\text {stat }}\right)=\boldsymbol{J}$, we obtain
$$\operatorname{curl} \operatorname{curl}\left(\mu^{-1} \boldsymbol{B}^{s t a t}\right)=\operatorname{curl} J$$
In a homogeneous medium (for instance, in vacuum $\mu=\mu_0 \mathbb{I}_3$ ), and using the identity (1.36) again, we obtain the magnetostatic problem
$$-\Delta \boldsymbol{B}^{s t a t}=\mu_0 \operatorname{curl} \boldsymbol{J}, \quad \operatorname{div} \boldsymbol{B}^{s t a t}=0,$$
whose solution, $\boldsymbol{B}^{\text {stat }}$, is called the magnetostatic field. This is a vector Poisson equation, i.e.s, an elliptic PDE (left Eq.), with a constraint (right Eq.). Again, this formulation leads to problems that are easier to solve than the complete set of Maxwell’s equations.

Note also that one has $\boldsymbol{B}^{\text {stat }}=\operatorname{curl} \boldsymbol{A}^{\text {stat }}$ (see (1.35)). If, moreover, the Coulomb gauge is chosen to remove the indetermination on the vector potential $\boldsymbol{A}^{\text {stat }}$, one finds the alternate magnetostatic problem
$$-\Delta A^{s t a t}=\mu_0 J, \quad \operatorname{div} A^{s t a t}=0,$$
with $A^{\text {stat }}$ as the unknown. Then, one sets $B^{\text {stat }}=\operatorname{curl} A^{\text {stat }}$ to recover the magnetostatic field.

## 物理代写|电磁学代写electromagnetism代考|A Scaling of Maxwell’s Equations

In order to define an approximate model, one has to neglect one or several terms in Maxwell’s equations. The underlying idea is to identify parameters, whose value can be small (and thus, possibly negligible). To derive a hierarchy of approximate models, one can perform an asymptotic analysis of those equations with respect to the parameters. This series of models is called a hierarchy, since considering a supplementary term in the asymptotic expansion leads to a new approximate model. An analogous principle is used, for instance, to build approximate (paraxial) models when simulating data migration in geophysics modelling (cf. among others [41, 85]). From a numerical point of view, the approximate models are useful, first and foremost, if they coincide with a physical framework, and second, because in general, they efficiently solve the problem at a lower computational cost.

In the sequel, let us show how to build such approximate models formally (i.e., without mathematical justifications), recovering, in the process, static models, but also other intermediate ones.

Let us consider Maxwell’s equations in vacuum (1.26-1.29). As a first step, we introduce a scaling of these equations based on the following characteristic values:
$\bar{l}$ : characteristic length,
$\bar{t}$ : characteristic time,
$\bar{v}:$ characteristic velocity, with $\bar{v}=\bar{l} / \bar{t}$,
$\bar{E}, \bar{B}$ : scaling for $\boldsymbol{E}$ and $\boldsymbol{B}$,
$\bar{\varrho}, \bar{J}$ : scaling for $\varrho$ and $\boldsymbol{J}$.
In order to build dimensionless Maxwell equations, we set
\begin{aligned} \boldsymbol{x} &=\bar{l} \boldsymbol{x}^{\prime} \quad \Rightarrow \frac{\partial}{\partial x_i}=\frac{1}{\bar{l}} \frac{\partial}{\partial x_i^{\prime}} \ t &=\bar{t} t^{\prime} \quad \Rightarrow \frac{\partial}{\partial t}=\frac{1}{\bar{t}} \frac{\partial}{\partial t^{\prime}} \ \boldsymbol{E} &=\bar{E} \boldsymbol{E}^{\prime}, \text { etc. } \end{aligned}

## 物理代写|电磁学代写电磁学代考|静磁学

$$\operatorname{curl} \operatorname{curl}\left(\mu^{-1} \boldsymbol{B}^{s t a t}\right)=\operatorname{curl} J$$

$$-\Delta \boldsymbol{B}^{s t a t}=\mu_0 \operatorname{curl} \boldsymbol{J}, \quad \operatorname{div} \boldsymbol{B}^{s t a t}=0,$$
，其解$\boldsymbol{B}^{\text {stat }}$称为静磁场。这是一个矢量泊松方程，即椭圆型PDE(左Eq.)，有一个约束(右Eq.)。同样，这个公式导致的问题比完整的麦克斯韦方程组更容易解决

$$-\Delta A^{s t a t}=\mu_0 J, \quad \operatorname{div} A^{s t a t}=0,$$
，其中$A^{\text {stat }}$为未知数。然后设置$B^{\text {stat }}=\operatorname{curl} A^{\text {stat }}$来恢复静磁场。

## 物理代写|电磁学代写电磁学代考|麦克斯韦方程组的缩放

$\bar{l}$ :特征长度，
$\bar{t}$ :特征时间，
$\bar{v}:$ 特征速度， $\bar{v}=\bar{l} / \bar{t}$，
$\bar{E}, \bar{B}$ :缩放。 $\boldsymbol{E}$ 和 $\boldsymbol{B}$，
$\bar{\varrho}, \bar{J}$ :缩放。 $\varrho$ 和 $\boldsymbol{J}$.

\begin{aligned} \boldsymbol{x} &=\bar{l} \boldsymbol{x}^{\prime} \quad \Rightarrow \frac{\partial}{\partial x_i}=\frac{1}{\bar{l}} \frac{\partial}{\partial x_i^{\prime}} \ t &=\bar{t} t^{\prime} \quad \Rightarrow \frac{\partial}{\partial t}=\frac{1}{\bar{t}} \frac{\partial}{\partial t^{\prime}} \ \boldsymbol{E} &=\bar{E} \boldsymbol{E}^{\prime}, \text { etc. } \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电磁学代写electromagnetism代考|PHYS3040

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电磁学代写electromagnetism代考|The Static Models

Let us consider problems (and solutions) that are time-independent, namely static equations, in a perfect medium. In other words, we assume that $\partial_t \cdot=0$ in Maxwell’s equations (1.22-1.25). This assumption leads to (with non-vanishing charge and current densities)
$$\left{\begin{array}{l} \operatorname{curl} \boldsymbol{E}^{\text {stat }}=0, \quad \operatorname{curl}\left(\mu^{-1} \boldsymbol{B}^{\text {stat }}\right)=\boldsymbol{J}, \ \operatorname{div}\left(\mathbb{E} \boldsymbol{E}^{s t a t}\right)=\varrho, \operatorname{div} \boldsymbol{B}^{\text {stat }}=0, \end{array}\right.$$
where the superscript ${ }^{\text {stat }}$ indicates that we are dealing with static unknowns. In the following two subsubsections, we will consider the electric and the magnetic cases separately. Again, they are set in all space, $\mathbb{R}^3$.

Remark 1.4.1 Within the framework of the time-harmonic Maxwell equations (see Sect. 1.2), we looked for solutions to Maxwell’s equations with an explicit timedependence. In this setting, the static equations can be viewed as time-harmonic Maxwell equations with a pulsation $\omega$ “equal to zero”. This interpretation can be useful, for instance, for performing an asymptotic analysis.

## 物理代写|电磁学代写electromagnetism代考|Electrostatics

Equation curl $\boldsymbol{E}^{\text {stat }}=0$ yields $\boldsymbol{E}^{\text {stat }}=-\operatorname{grad} \phi^{\text {stat }}$, where $\phi^{\text {stat }}$ denotes the electrostatic potential; see the connection to (1.33) when $\partial_{t^*}=0$. As $\operatorname{div}\left(\mathbb{C} \boldsymbol{E}^{\text {stat }}\right)=$ $\varrho$, the potential $\phi^{s t a t}$ solves the elliptic ${ }^{15}$ problem
$$-\operatorname{div}\left(\mathbb{C} \operatorname{grad} \phi^{s t a t}\right)=\varrho .$$
Moreover, in a homogeneous medium (for instance, in vacuum $\Subset=\varepsilon_0 \mathbb{』}_3$ ), we obtain the electrostatic problem with unknown $\phi^{\text {stat }}$
$$-\Delta \phi^{\text {stat }}=\frac{\varrho}{\varepsilon_0} .$$
This is the Poisson equation in variable $\phi^{\text {stat }}$ (see, for instance, Chapter 3 of [103, Volume II]), which is an elliptic partial differential equation (PDE), and by definition, a static problem, much cheaper to solve computationally than the complete set of Maxwell’s equations. Then, one sets $\boldsymbol{E}^{\text {stat }}=-\operatorname{grad} \phi^{\text {stat }}$ to recover the electrostatic field.

## 物理代写|电磁学代写电磁学代考|静态模型

$$\left{\begin{array}{l} \operatorname{curl} \boldsymbol{E}^{\text {stat }}=0, \quad \operatorname{curl}\left(\mu^{-1} \boldsymbol{B}^{\text {stat }}\right)=\boldsymbol{J}, \ \operatorname{div}\left(\mathbb{E} \boldsymbol{E}^{s t a t}\right)=\varrho, \operatorname{div} \boldsymbol{B}^{\text {stat }}=0, \end{array}\right.$$
，其中上标${ }^{\text {stat }}$表示我们正在处理静态未知数。在接下来的两个子小节中，我们将分别考虑电的和磁的情况。同样，它们设置在所有空间$\mathbb{R}^3$ .

## 物理代写|电磁学代写电磁学代考|静电学

$$-\operatorname{div}\left(\mathbb{C} \operatorname{grad} \phi^{s t a t}\right)=\varrho .$$此外，在均匀介质中(例如，在真空中) $\Subset=\varepsilon_0 \mathbb{』}_3$ )，得到未知数的静电问题 $\phi^{\text {stat }}$
$$-\Delta \phi^{\text {stat }}=\frac{\varrho}{\varepsilon_0} .$$这是变量中的泊松方程 $\phi^{\text {stat }}$ (例如，参见[103，卷II]的第3章)，这是一个椭圆型偏微分方程(PDE)，从定义上讲，它是一个静态问题，比完整的麦克斯韦方程组要便宜得多。然后，一组 $\boldsymbol{E}^{\text {stat }}=-\operatorname{grad} \phi^{\text {stat }}$ 恢复静电场。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电动力学代写electromagnetism代考|ELEC3104

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电动力学代写electromagnetism代考|ELECTRIC POTENTIAL

We often come across the term potential when applied to the potential energy of a body or the potential difference between two points in a circuit. In the former case, the potential energy of a body is related to its height above a certain reference level. Thus, a body gains potential energy when we raise it to a higher level. This gain in energy is equal to the work done against an attractive force, gravity in this example. Figure 2.6a shows this situation.

As Figure 2.6a shows, the body is placed in an attractive, gravitational force field. So, if we raise the body through a certain distance, we have to do work against the gravitational field. The difference in potential energy between positions 1 and 2 is equal to the work done in moving the body from 1 to 2 , a distance of $l$ metres. This work done is given by
$$F \times l=m \times 9.81 \times l$$

where $m$ is the mass of the body $(\mathrm{kg})$ and $9.81$ is the acceleration due to gravity $\left(\mathrm{m} \mathrm{s}^{-2}\right.$ ). (Although the effects of gravity vary according to the inverse square law, the difference in gravitational force between positions 1 and 2 is small. This is because the Earth is so large. Thus, we can take the gravitational field to be linear in form, and so this equation holds true.)

In an electrostatic field, we have an electrostatic force field instead of a gravitational force field. However, the idea of potential energy is the same. Let us consider the situation in Figure 2.6b. We have a positive test charge of $1 \mathrm{C}$ at a distance $\mathrm{d}_1$ from the fixed negative charge, $-q_1$. This test charge will experience an attractive force whose magnitude we can find from Coulomb’s law. Now, if we move the test charge from position 1 to position 2, we have to do work against the field. If the distance between positions 1 and 2 is reasonably large, the strength of the force field decreases as we move away from the fixed charge. Thus, we say that we have a non-linear field.
As the field decreases when we move away from the fixed charge, let us move the test charge a very small distance, $\mathrm{d} r$. The electric field strength will hardly alter as we move along this small distance. So, the work done against the field in moving the test charge a small distance $\mathrm{d} r$ will be given by
\begin{aligned} \text { work done } &=\text { force } \times \text { distance } \ &=-F \times \mathrm{d} r \ &=-1 \times E \times \mathrm{d} r \end{aligned}

## 物理代写|电动力学代写electromagnetism代考|EQUIPOTENTIAL LINES

Let us consider the three paths $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ shown in Figure $2.7 \mathrm{a}$. All of these paths link points 1 and 2, but only path A does so directly. Now, let us take the circular lines in Figure 2.7a as the contours on a hill. In moving from position 1 to position 2 by way of path $\mathrm{A}$, we clearly do work against gravity. The work done is equal to the gain in potential energy which, in turn, is equal to the gravitational force times the change in vertical height. (This is shown in Figure 2.7b.)

Now let us take path B. We initially walk left from position 1, around the contour line, to a point directly below position 2. As we have moved around a contour line, we have not gained any height, and so the potential energy remains the same, i.e., we have not done any work against gravity. We now have to walk uphill to position 2 . In doing so we do work against gravity equal to the gain in potential energy. This gain in potential energy is clearly the same as with path $\mathrm{A}$. (Although we have to do more physical work in travelling along path $\mathrm{B}$, the change in potential energy is the same.) If we use path $\mathrm{C}$, the same argument holds true. So, we can say that the work done against gravity is independent of the path we take.

Let us now turn our attention to the electrostatic field in Figure 2.8. As with the contour map, we have three different paths. As we have just seen, we do no work against the field when we move in a circular direction. We only do work when we move in a radial direction. Thus, the potential difference between points 1 and 2 is independent of the exact path we take. This implies that we do no work against the field when we move around the plot in a circular direction. Thus, the circular ‘contours’ in Figure $2.8$ are lines of equal potential or equipotential lines.

We should be careful when using the term equipotential lines. This is because we are considering a point charge, and so the equipotential surfaces are actually spheres with the charge at their centre. As we are not yet able to draw in a three-dimensional holographic world, we have to make do with two-dimensional diagrams drawn on pieces of paper!

## 物理代写|电动力学代写electromagnetism代考|ELECTRIC POTENTIAL

$$F \times l=m \times 9.81 \times l$$

$$\text { work done }=\text { force } \times \text { distance } \quad=-F \times \mathrm{d} r=-1 \times E \times \mathrm{d} r$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电动力学代写electromagnetism代考|PHYS3040

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电动力学代写electromagnetism代考|COULOMB’S LAW

As we have seen in Chapter 1, electronic charge comes in two forms: negative charge from an electron and positive charge from a proton. In both cases, a single isolated charge has a charge of $1.6 \times 10^{-19}$ Coulomb. If there are two charges close to each other, they tend to repel each other if the charges are alike or attract each other if they are dissimilar. Thus, we can say that these charges exert a force on each other.
Charles Augustin de Coulomb (1736-1806) determined by direct experimental observation that the force between two charges is proportional to the product of the two charges and inversely proportional to the square of the distance between them. In terms of the SI units, the force between two charges, a vector quantity, is given by
$$\boldsymbol{F}=\frac{q_1 q_2}{4 \pi \varepsilon r^2} \boldsymbol{r}$$
where
$\boldsymbol{F}$ is the force between the charges (N)
$q_1$ and $q_2$ are the magnitudes of the two charges (C)
$\varepsilon$ is a material constant $\left(\mathrm{F} \mathrm{m}^{-1}\right)$
$r$ is the distance between the charges (m)
and $r$ is a unit vector acting in the direction of the line joining the two charges

This is Coulomb’s law. The force, as given by Equation (2.1), is positive (i.e. repulsive) if the charges are alike, and negative (i.e. attractive) if the charges are dissimilar (see Figure 2.1). As Equation (2.1) shows, the force between the charges is inversely dependent on a material constant, $\varepsilon$, the permittivity. Good insulators have very high values of permittivity, typically ten times that of air for glass and so the electrostatic force is correspondingly smaller.

If no material separates the charges, i.e., if they are in a vacuum, the permittivity has the lowest possible value of $8.854 \times 10^{-12}$ or $1 / 36 \pi \times 10^{-9} \mathrm{~F} \mathrm{~m}^{-1}$. (These rather obscure values result from the adoption of the SI units.) As permittivity has such a low value, it is more usual to normalize the permittivity of a material to that of free-space. This normalized permittivity is commonly known as the relative permittivity, $\varepsilon_{\mathrm{r}}$, given by
$$\varepsilon_{\mathrm{r}}=\frac{\varepsilon}{\varepsilon_{\mathrm{o}}}$$

## 物理代写|电动力学代写electromagnetism代考|ELECTRIC FLUX AND ELECTRIC FLUX DENSITY

One definition of flux is that it is the flow of material from one place to another. Some familiar examples of flow are water flowing out of a tap or spring, air flowing from areas of high pressure to low pressure and audio waves flowing outward from a source of disturbance. In general, we can say that flux flows away from a source and towards a sink.

If we adapt this to electrostatics, we can say that a positive charge is a source of electric flux, and a negative charge acts as a sink. We must exercise extreme caution here. Nothing physically flows out of positive charges – a charge does not run out of electric flux! What we are doing is adapting the general definition of flux, so that we can visualize what is happening. If we consider isolated point charges, we can draw a diagram as in Figure 2.2. (A point charge is simply a physically small charge or collection of charges. This raises the question of how small is small? The answer lies with relative sizes. Relative to the distance between the Earth and the Sun, the height of Mount Everest is insignificant. Similarly, we can regard a collection of individual charges, arranged in a 10-nm diameter sphere, as a point charge when viewed from $10 \mathrm{~m}$ away.)

Now, what happens to the distribution of electric flux if we bring two positive charges together? As the charges are both sources of electric flux, the fluxes repel each other to produce the distribution shown in Figure 2.3. One of the main things to note from this diagram is the distortion of the lines of flux in the space between the charges. This causes the force of repulsion between the two charges, in agreement with Coulomb’s law.
If we now return to Coulomb’s law, we can rewrite it as
$$\boldsymbol{F}=\frac{q_1}{4 \pi r^2} \frac{1}{\varepsilon} q_2 \boldsymbol{r}$$

The first term in Equation (2.3) consists of the electronic charge, $q_1$, divided by the surface area of a sphere, $4 \pi r^2$. Thus, $q_1 / 4 \pi r^2$ has units of $\mathrm{C} \mathrm{m}^{-2}$ and would appear to be a surface density of some sort – the flux density. To explain this, we must use Gauss’ law (Karl Friedrich Gauss, 1777-1855) which states that the flux through any closed surface is equal to the charge enclosed by that surface.

Figure $2.4$ shows an imaginary spherical surface surrounding an isolated point charge. Application of Gauss’ law shows that the flux, $\psi$, radiating outwards in all directions has a value of $q_1$ – the amount of charge enclosed by the sphere. The area of the Gaussian surface is simply that of a sphere, i.e., a surface area of $4 \pi r^2$. Thus, we get a flux density, $\boldsymbol{D}$, of
$$\boldsymbol{D}=\frac{q_1}{4 \pi r^2} \boldsymbol{r}$$

## 物理代写|电动力学代写electromagnetism代考|COULOMB’S LAW

Charles Augustin de Coulomb (1736-1806) 通过直接实验观察确定，两个电荷之间的力与两个电荷的 乘积成正比，与它们之间距离的平方成反比。就 SI 单位而言，两个电荷之间的力，一个向量，由下式 给出
$$\boldsymbol{F}=\frac{q_1 q_2}{4 \pi \varepsilon r^2} \boldsymbol{r}$$

$\boldsymbol{F}$ 是电荷之间的力 (N)
$q_1$ 和 $q_2$ 是两个电荷的大小 (C)
$\varepsilon$ 是材料常数 $\left(\mathrm{Fm}^{-1}\right)$
$r$ 是电荷之间的距离 $(\mathrm{m})$

• 径向单位向量
这是库仑定律。如公式 (2.1) 所给出的，如果电荷相同，则力为正（即排斥），如果电荷不同， 则力为负（即吸引）（见图 2.1）。如等式 (2.1) 所示，电荷之间的力与材料常数成反比， $\varepsilon$ ， 介电常数。好的绝缘体具有非常高的介电常数值，通常是玻璃的空气的十倍，因此静电力相应较 小。
如果没有材料分离电荷，即，如果它们处于真空中，则介电常数的最低值可能为 $8.854 \times 10^{-12}$ 或者 $1 / 36 \pi \times 10^{-9} \mathrm{~F} \mathrm{~m}^{-1}$. (这些相当模糊的值是采用 SI 单位造成的。) 由于介电常数的值如此之低， 因此更通常将材料的介电常数归一化为自由空间的介电常数。这种归一化的介电常数通常称为相对介电 常数， $\varepsilon_{\mathrm{r}}$ ，由
$$\varepsilon_{\mathrm{r}}=\frac{\varepsilon}{\varepsilon_{\mathrm{o}}}$$

## 物理代写|电动力学代写electromagnetism代考|ELECTRIC FLUX AND ELECTRIC FLUX DENSITY

$$\boldsymbol{F}=\frac{q_1}{4 \pi r^2} \frac{1}{\varepsilon} q_2 \boldsymbol{r}$$

$$\boldsymbol{D}=\frac{q_1}{4 \pi r^2} \boldsymbol{r}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电动力学代写electromagnetism代考|PHYC20014

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电动力学代写electromagnetism代考|HISTORICAL BACKGROUND

Flectromagnetic field theory is really the result of the union of three distinct sciences. The oldest of these is electrostatics, which was first studied by the Greeks. They discovered that if they rubbed certain substances, they were able to attract lighter bodies to them. One of these substances was amber, whose Greek name is electron – this is where we get the name ‘electricity’. It was in 1785 that French physicist, Charles Augustin de Coulomb (1736-1806), showed that electrically charged materials sometimes attract and sometimes repel each other. This was the first indication that there were two types of charge – positive and negative.

In the late $1700 \mathrm{~s}$, two Italians were working on the new science of current electricity. One, Luigi Galvani (1737-1798), was a physiologist and physician who thought that animal tissues generate electricity. Although he was later proved wrong, his experiments stimulated Count Alessandro Volta (1745-1827) to invent the first electric battery in 1800 . Most of the early experiments in current electricity were performed on frog’s legs – this was a result of Galvani’s work.

Later, a favourite party trick was to get a group of people to hold hands and then connect them to a voltaic cell (a battery). The cell produced quite a large voltage, which then caused current to flow through the guests. This made them jump uncontrollably! It wasn’t until 1833 that the British experimenter Michael Faraday (17911867) showed that the current electricity of Volta and Galvani was the same as the electrostatic electricity of Coulomb. Rather than linking these two phenomena, it was shown that the current and electrostatic electricity were one and the same thing.

(Faraday’s contribution is all the more remarkable when it is realized that his theories were formulated by direct experimentation and not by manipulating mathematics!)
Although the ancient Greeks also knew about magnetism in the form of lodestone, the Chinese invented the magnetic compass, and in 1600, William Gilbert of Gloucester laid down some fundamentals. However, it was not until 1785 that Coulomb formulated his law relating the strengths of two magnetic poles to the force between them. Magnetism may have been laid to rest here if it wasn’t for the Danish physicist Hans Christian Oersted (1777-1851). It was Oersted who demonstrated to a group of students that a current-carrying wire produces a magnetic field. This was the first sign that electricity and magnetism could he interlinked. This link was strengthened in 1831 by the work of Faraday who showed that a changing magnetic field could induce a current into a wire. It was a French physicist André Marie Ampèree who first formulated the idea that the field of a permannent magnent could be due to currents in the material. (We now accept that electrons orbiting the nucleus constitute a current, and this produces the magnetic field.)

## 物理代写|电动力学代写electromagnetism代考|VECTORS AND COORDINATE SYSTEMS

When we use a thermometer, we read the temperature off a graduated scale. The temperature of a body is independent of direction (it is simply measured at a certain point), and so it is known as a scalar quantity. Scalar quantities are those that have no direction associated with them.

If we push an object, we have to exert a force on it. This force has direction associated with it – we could push the object to the left, to the right or in any direction we choose. The force is a vector quantity because it has magnitude and direction.

At this point, we could launch into a discussion of vector theory – addition, multiplication, etc. Unfortunately this would complicate matters, and mask the underlying ideas. Instead, we will avoid vector algebra in favour of discussion and reasoning. In spite of this, Figure $1.3$ shows the standard Cartesian, spherical and cylindrical systems that we will use as we progress with our studies. (We will use unit vectors in most of the text, however. This is to help readers get used to vector notation, which will aid future studies.)

## 物理代写|电动力学代写electromagnetism代考|HISTORICAL BACKGROUND

（当意识到他的理论是通过直接实验而不是通过操纵数学来制定时，法拉第的贡献就更加显着了！）

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电磁学代写electromagnetism代考|ELEC3104

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电磁学代写electromagnetism代考|Solvability of Maxwell’s Equations

What about the proof of the existence of electromagnetic fields on $\mathbb{R}^3$ ?
To begin with, there exist many “experimental proofs” of the existence of electromagnetic fields! These experiments actually led to the definition of the equations that govern electromagnetic phenomena, and of the related electromagnetic fields, by Maxwell and many others during the nineteenth and twentieth centuries. So, it is safe to assume that these fields exist, the challenge being mathematical and computational nowadays…

Where does the theory originate? Let us give a brief account of one of the more elementary (mathematically speaking!) results on charged particles at rest (results have also been obtained for circuits, involving currents).

The fundamental experimental results we report here were obtained by Charles Augustin de Coulomb in 1785, when he studied repulsive or attractive forces between charged bodies, small elder balls. In the air-a homogeneous medium respective positions are $x_1$ and $\boldsymbol{x}$, whereas their respective electric charges are $q_1$ and $q$. In short, Coulomb’s results (now known as Coulomb’s law) state that the two particles interact electrically ${ }^7$ with one another, in the following way. The force $\boldsymbol{F}$ acting on particle part and originating from particle part $_1$ is such that:

• it is repulsive if $q_1 q>0$, and attractive if $q_1 q<0$;
• its direction is parallel to the line joining the two particles;
• its modulus is proportional to $\left|x-x_1\right|^{-2}$;
• its modulus is also proportional to $q_1$ and $q$.

## 物理代写|电磁学代写electromagnetism代考|Potential Formulation of Maxwell’s Equations

Let us introduce another formulation of Maxwell’s equations. For the sake of simplicity, we assume that we are in vacuum (in all space, $\mathbb{R}^3$ ), with Maxwell’s equations written in differential form as Eqs. (1.26-1.29). According to the divergencefree property of the magnetic induction $\boldsymbol{B}$, there exists a vector potential $\boldsymbol{A}$ such that
$$B=\operatorname{curl} A$$

Plugging this into Faraday’s law (1.27), we obtain
$$\operatorname{curl}\left(\frac{\partial \boldsymbol{A}}{\partial t}+\boldsymbol{E}\right)=0$$
Then, there exists a scalar potential $\phi$ such that
$$\frac{\partial A}{\partial t}+\boldsymbol{E}=-\operatorname{grad} \phi .$$
This allows us to introduce a formulation in the variables $(\boldsymbol{A}, \boldsymbol{\phi})$ – the vector potential and the scalar potential, respectively – since it holds there that
\begin{aligned} &\boldsymbol{E}=-\operatorname{grad} \phi-\frac{\partial \boldsymbol{A}}{\partial t} \ &\boldsymbol{B}=\operatorname{curl} \boldsymbol{A} \end{aligned}
This formulation requires only the four unknowns $\boldsymbol{A}$ and $\phi$. instead of the six unknowns for the $\boldsymbol{E}$ and $\boldsymbol{B}$-field formulation. Moreover, any couple $(\boldsymbol{E}, \boldsymbol{B})$ defined by Eqs. (1.34-1.35) automatically satisfies Faraday’s law and the absence of free magnetic monopoles. From this (restrictive) point of view, the potentials $\boldsymbol{A}$ and $\phi$ are independent of one another. Now, if one takes into account Ampère’s and Gauss’s laws, constraints appear in the choice of $\boldsymbol{A}$ and $\phi$ (see Eqs (1.37-1.38) below). Also, the vector potential $\boldsymbol{A}$ governed by Eq. (1.35) is determined up to a gradient of a scalar function: there lies an indetermination that has to be removed. On the other hand, for the scalar potential, the indetermination is up to a constant: it can be removed simply by imposing a vanishing limit at infinity.

## 物理代写|电磁学代写electromagnetism代考|Solvability of Maxwell’s Equations

• 如果 $q_1 q>0$, 并且如果 $q_1 q<0$;
• 它的方向平行于连接两个粒子的线;
• 它的模量与 $\left|x-x_1\right|^{-2}$;
• 它的模量也与 $q_1$ 和 $q$.

## 物理代写|电磁学代写electromagnetism代考|Potential Formulation of Maxwell’s Equations

$$B=\operatorname{curl} A$$

$$\operatorname{curl}\left(\frac{\partial \boldsymbol{A}}{\partial t}+\boldsymbol{E}\right)=0$$

$$\frac{\partial A}{\partial t}+\boldsymbol{E}=-\operatorname{grad} \phi$$

$$\boldsymbol{E}=-\operatorname{grad} \phi-\frac{\partial \boldsymbol{A}}{\partial t} \quad \boldsymbol{B}=\operatorname{curl} \boldsymbol{A}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电磁学代写electromagnetism代考|PHYC20014

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电磁学代写electromagnetism代考|Equivalent Reformulation of Maxwell’s Equations

Starting from the integral form of Maxwell’s equations (1.1-1.4), one can reformulate them in a differential form, ${ }^3$ with the help of Stokes and Ostrogradsky formulas
$$\int_S \operatorname{curl} \boldsymbol{F} \cdot \boldsymbol{d} S=\int_{\partial S} \boldsymbol{F} \cdot \boldsymbol{d} \boldsymbol{l} \text { and } \int_V \operatorname{div} \boldsymbol{F} d V=\int_{\partial V} \boldsymbol{F} \cdot \boldsymbol{d} S .$$
One easily derives the differential Maxwell equations (system of units SI):
\begin{aligned} \frac{\partial \boldsymbol{D}}{\partial t}-\operatorname{curl} \boldsymbol{H} &=-\boldsymbol{J}, \ \frac{\partial \boldsymbol{B}}{\partial t}+\operatorname{curl} \boldsymbol{E} &=0, \ \operatorname{div} \boldsymbol{D} &=\varrho, \ \operatorname{div} \boldsymbol{B} &=0 . \end{aligned}
The differential charge conservation equation can be expressed as
$$\frac{\partial \varrho}{\partial t}+\operatorname{div} \boldsymbol{J}=0 .$$
However, the above set of equations is not equivalent to the integral set of equations. As a matter of fact, two notions are missing.

The first one is related to the behavior of the fields across an interface between two different media. Let $\Sigma$ be such an interface.

Starting from the volumic integral equations (1.3)-(1.4), we consider thin volumes $V_\epsilon$ crossing the interface. As $\epsilon$ goes to zero, their height goes to zero, and so does the area of their top and bottom faces (parallel to the interface), with proper scaling. The top and bottom faces are disks whose radius is proportional to $\epsilon$, while the height is proportional to $\epsilon^2$. As a consequence, the area of the lateral surface is proportional to $\epsilon^3$ and its contribution is negligible as $\epsilon$ goes to zero.

## 物理代写|电磁学代写electromagnetism代考|Constitutive Relations

Maxwell’s equations are insufficient to characterize the electromagnetic fields completely. The system has to be closed by adding relations that describe the properties of the medium in which the electromagnetic fields propagate. These are the so-called constitutive relations, relating, for instance, $\boldsymbol{D}$ and $\boldsymbol{B}$ to $\boldsymbol{E}$ and $\boldsymbol{H}$, namely
$$\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{H}) \quad \text { and } \quad \boldsymbol{B}=\boldsymbol{B}(\boldsymbol{E}, \boldsymbol{H})$$
(We could also choose $a$ priori to use such a relation as $\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{B})$, etc.)

These constitutive relations can be very complex. For this reason, we will make a number of assumptions on the medium (listed below), which lead to generic expressions of the constitutive relations. This will yield three main categories of medium, which are, from the more general to the more specific:

1. the chiral medium, a linear and bi-anisotropic medium;
2. the perfect medium, a chiral, non-dispersive and anisotropic medium;
3. the inhomogeneous medium, a perfect and isotropic medium, and its subcategory, the homogeneous medium, which is, in addition, spatially homogeneous.

In what follows, $\boldsymbol{E}(t)$ (or $\boldsymbol{B}(t)$, etc.) denotes the value of the electric field on $\mathbb{R}^3$ at time $t: \boldsymbol{x} \mapsto \boldsymbol{E}(t, \boldsymbol{x})$. Let us now list the assumptions about the medium.

• The medium is linear. This means that its response is linear with respect to electromagnetic inputs (also called excitations later on). In addition, it is expected that when the inputs are small, the response of the medium is also small.
• The medium satisfies a causality principle. In other words, the value of $(\boldsymbol{D}(t), \boldsymbol{B}(t))$ depends only on the values of $(\boldsymbol{E}(s), \boldsymbol{H}(s))$ for $s \leq t$.
• The medium satisfies a time-invariance principle. Let $\tau>0$ be given. II the response to $t \mapsto(\boldsymbol{E}(t), \boldsymbol{H}(t))$ is $t \mapsto(\boldsymbol{D}(t), \boldsymbol{B}(t))$, then the response to $t \mapsto$ $(\boldsymbol{E}(t-\tau), \boldsymbol{H}(t-\tau))$ is $t \mapsto(\boldsymbol{D}(t-\tau), \boldsymbol{B}(t-\tau))$.

## 物理代写|电磁学代写electromagnetism代考|Equivalent Reformulation of Maxwell’s Equations

$$\int_S \operatorname{curl} \boldsymbol{F} \cdot \boldsymbol{d} S=\int_{\partial S} \boldsymbol{F} \cdot \boldsymbol{d} \boldsymbol{l} \text { and } \int_V \operatorname{div} \boldsymbol{F} d V=\int_{\partial V} \boldsymbol{F} \cdot \boldsymbol{d} S .$$

$$\frac{\partial \boldsymbol{D}}{\partial t}-\operatorname{curl} \boldsymbol{H}=-\boldsymbol{J}, \frac{\partial \boldsymbol{B}}{\partial t}+\operatorname{curl} \boldsymbol{E}=0, \operatorname{div} \boldsymbol{D}=\varrho, \operatorname{div} \boldsymbol{B}=0 .$$

$$\frac{\partial \varrho}{\partial t}+\operatorname{div} \boldsymbol{J}=0$$

## 物理代写|电磁学代写electromagnetism代考|Constitutive Relations

$$\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{H}) \quad \text { and } \quad \boldsymbol{B}=\boldsymbol{B}(\boldsymbol{E}, \boldsymbol{H})$$
(我们也可以选择 $a$ 先验地使用这样的关系 $\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{B})$ ， ETC。)

1. 手性介质，一种线性和双各向异性介质；
2. 完美介质，手性、非色散和各向异性介质；
3. 非均质介质，一种完美的各向同性介质，及其子类别，均质介质，此外，它在空间上是均质的。
在接下来的内容中， $\boldsymbol{E}(t)$ (或者 $\boldsymbol{B}(t)$ 等) 表示电场的值 $\mathbb{R}^3$ 有时 $t: \boldsymbol{x} \mapsto \boldsymbol{E}(t, \boldsymbol{x})$. 现在让我们列出关于介质的 假设。
• 介质是线性的。这意味着它的响应相对于电磁输入 (后面也称为激励) 是线性的。此外，预计当输入较小 时，介质的响应也较小。
• 媒介满足因果关系原则。换句话说，价值 $(\boldsymbol{D}(t), \boldsymbol{B}(t))$ 仅取决于的值 $(\boldsymbol{E}(s), \boldsymbol{H}(s))$ 为了 $s \leq t$.
• 该介质满足时不变原理。让 $\tau>0$ 被给予。二、回应 $t \mapsto(\boldsymbol{E}(t), \boldsymbol{H}(t))$ 是 $t \mapsto(\boldsymbol{D}(t), \boldsymbol{B}(t))$ ，然后响应 $t \mapsto(\boldsymbol{E}(t-\tau), \boldsymbol{H}(t-\tau))$ 是 $t \mapsto(\boldsymbol{D}(t-\tau), \boldsymbol{B}(t-\tau)) .$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|电磁学代写electromagnetism代考|PHYS3040

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电磁学代写electromagnetism代考|Physical Framework and Models

The aim of this first chapter is to present the physics framework of electromagnetism, in relation to the main sets of equations, that is, Maxwell’s equations and some related approximations. In that sense, it is neither a purely physical nor a purely mathematical point of view. The term model might be more appropriate: sometimes, it will be necessary to refer to specific applications in order to clarify our purpose, presented in a selective and biased way, as it leans on the authors’ personal view. This being stated, this chapter remains a fairly general introduction, including the foremost models in electromagnetics. Although the choice of such applications is guided by our own experience, the presentation follows a natural structure.
Consequently, in the first section, we introduce the electromagnetic fields and the set of equations that governs them, namely Maxwell’s equations. Among others, we present their integral and differential forms. Next, we define a class of constitutive relations, which provide additional relations between electromagnetic fields and are needed to close Maxwell’s equations. Then, we briefly review the solvability of Maxwell’s equations, that is, the existence of electromagnetic fields, in the presence of source terms. We then investigate how they can be reformulated as potential problems. Finally, we relate some notions on conducting media.

In Sect. 1.2, we address the special case of stationary equations, which have timeperiodic solutions, the so-called time-harmonic fields. The useful notion of plane waves is also introduced, as a particular case of the time-harmonic solutions.

Maxwell’s equations are related to electrically charged particles. Hence, there exists a strong correlation between Maxwell’s equations and models that describe the motion of particles. This correlation is at the core of most models in which Maxwell’s equations are coupled with other sets of equations: two of them-the Vlasov-Maxwell model and an example of a magnetohydrodynamics model (or MHD)—will be detailed in Sect. 1.3.

## 物理代写|电磁学代写electromagnetism代考|Integral Maxwell Equations

The propagation of the electromagnetic fields in continuum media is described using four space- and time-dependent functions. If we respectively denote by $\boldsymbol{x}=\left(x_1, x_2, x_3\right)$ and $t$ the space and time variables, these four $\mathbb{R}^3$-valued, or vectorvalued, functions defined in time-space $\mathbb{R} \times \mathbb{R}^3$ are

1. the electric field $\boldsymbol{E}$,
2. the magnetic induction $\boldsymbol{B}$,
3. the magnetic field ${ }^2 \boldsymbol{H}$,
4. the electric displacement $\boldsymbol{D}$.
These vector functions are governed by the integral Maxwell equations below. These four equations are respectively called Ampère’s law, Faraday’s law, Gauss’s law and the absence of magnetic monopoles. They read as (system of units SI)
\begin{aligned} \frac{d}{d t}\left(\int_S \boldsymbol{D} \cdot \boldsymbol{d} \boldsymbol{S}\right)-\int_{\partial S} \boldsymbol{H} \cdot d \boldsymbol{l} &=-\int_S \boldsymbol{J} \cdot \boldsymbol{d} \boldsymbol{S} \ \frac{d}{d t}\left(\int_{S^{\prime}} \boldsymbol{B} \cdot \boldsymbol{d} \boldsymbol{S}\right)+& \int_{\partial S^{\prime}} \boldsymbol{E} \cdot d \boldsymbol{l}=0 \ & \int_{\partial V} \boldsymbol{D} \cdot \boldsymbol{d} \boldsymbol{S}=\int_V \varrho d V \ \int_{\partial V^{\prime}} \boldsymbol{B} \cdot \boldsymbol{d} \boldsymbol{S} &=0 \end{aligned}
Above, $S, S^{\prime}$ are any surface of $\mathbb{R}^3$, and $V, V^{\prime}$ are any volume of $\mathbb{R}^3$. One can write elements $d S$ and $d l$ as $d S=n d S$ and $d l=\tau d l$, where $n$ and $\tau$ are, respectively, the unit outward normal vector to $S$ and the unit tangent vector to the curve $\partial S$. When $S$ is the closed surface bounding a volume, then $\boldsymbol{n}$ is pointing outward from the enclosed volume. Similarly, the unit tangent vector to $\partial S$ is pointing in the direction given by the right-hand rule.

## 物理代写|电磁学代写electromagnetism代考|Integral Maxwell Equations

1. 电场 $\boldsymbol{E}$,
2. 磁感应 $\boldsymbol{B}$,
3. 磁场 ${ }^2 \boldsymbol{H}$,
4. 电位移 $\boldsymbol{D}$.
这些向量函数由下面的积分麦克斯韦方程控制。这四个方程分别称为安培定律、法拉第定律、高斯定律和不 存在磁单极子。它们读作（单位制 SI)
$$\frac{d}{d t}\left(\int_S \boldsymbol{D} \cdot \boldsymbol{d} \boldsymbol{S}\right)-\int_{\partial S} \boldsymbol{H} \cdot d \boldsymbol{l}=-\int_S \boldsymbol{J} \cdot \boldsymbol{d} \boldsymbol{S} \frac{d}{d t}\left(\int_{S^{\prime}} \boldsymbol{B} \cdot \boldsymbol{d} \boldsymbol{S}\right)+\int_{\partial S^{\prime}} \boldsymbol{E} \cdot d \boldsymbol{l}=0 \int_{\partial V} \boldsymbol{D}$$
以上，S, $S^{\prime}$ 是任何表面 $\mathbb{R}^3$ ，和 $V, V^{\prime}$ 是任何体积 $\mathbb{R}^3$. 一个可以写元素 $d S$ 和 $d l$ 作为 $d S=n d S$ 和 $d l=\tau d l$ ，在哪里 $n$ 和 $\tau$ 分别是单位外向法向量 $S$ 和曲线的单位切向量 $\partial S$. 什么时候 $S$ 是包围一个体积的封闭曲面，那 么 $\boldsymbol{n}$ 从封闭的体积向外指向。类似地，单位切向量为 $\partial S$ 指向右手定则给出的方向。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|ECE310

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号处理Digital Signal Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号处理Digital Signal Processing方面经验极为丰富，各种代写数字信号处理Digital Signal Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|The Basic Network

Consider the simplest configuration for impedance matching, namely, the L-network shown in Fig. 3.1. $Z_{\mathrm{s}}$ is an $L C$ impedance and $Y_{p}$ is an $L C$ admittance such that when the network is terminated in $R_{L}$, the input impedance is $R_{S}$ at the frequencies of interest. Let, at $s=j \omega, Z_{s}=j X_{s}$ and $Y_{P}=j B_{P}$, where $X$ denotes reactance and $B$ denotes susceptance. Then, at the frequencies of interest, we should have
$$R_{S}=j X_{s}+1 /\left(j B_{p}+G_{L}\right)$$
where $G_{L}=1 / R_{L}$. Cross multiplying, simplifying, and equating the real and imaginary parts on both sides give the two equations
$$X_{S} B_{p}=1-R_{S} G_{L} \text { and } R_{S} B_{p}=X_{S} G_{L}$$
The second equation shows that $X_{s}$ and $B_{p}$ must be of the same sign, both positive or both negative. Combining this fact with the first Equation in (3.2), we note that $R_{S} G_{L}$ must be less than unity, i.e. $R_{S}$ must be less than $R_{L}$. However, this is no restriction because the other situation, i.e. $R_{S}>R_{L}$, can be taken care of by simply interchanging the positions of $R_{S}$ and $R_{L}$ in Fig. 3.1. Eliminating $B_{p}$ from the two Equations in (3.2), we get
$$X_{s}^{2}=R_{S}\left(R_{L}-R_{S}\right)=R_{1}^{2}, \text { say }$$

Or,
$$X_{\mathrm{s}}=\pm R_{1}$$
Combining this with the second Equation in (3.2) gives
$$B_{p}=\pm R_{1} /\left(R_{L} R_{S}\right)=\pm G_{2} \text {, say }$$
As already stated, the signs in Eqs. (3.4) and (3.5) should be either both positive or both negative.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Impedance Matching at a Single Frequency

For matching $R_{L}$ to $R_{S}$ at a single frequency $\omega_{0}$, we can choose either an inductance $L_{s}$ for $X_{s}$ and a capacitance $C_{p}$ for $B_{p}$, or a capacitance $C_{s}$ for $X_{s}$ and an inductance $L_{p}$ for $B_{p}$. In the first case, to be referred to as Design 1 (D1)
$$L_{s}=R_{1} / \omega_{0} \text { and } C_{p}=G_{2} / \omega_{0}$$
while for the alternative design, to be called Design 2 (D2),
$$C_{s}=1 /\left(R_{1} \omega_{0}\right) \text { and } L_{p}=1 /\left(G_{2} \omega_{0}\right)$$
We shall mostly use D1 in our further discussions, it is being implied that D2 is equally applicable, giving another set of solutions.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|The Basic Network

$$R_{S}=j X_{s}+1 /\left(j B_{p}+G_{L}\right)$$

$$X_{S} B_{p}=1-R_{S} G_{L} \text { and } R_{S} B_{p}=X_{S} G_{L}$$

$$X_{s}^{2}=R_{S}\left(R_{L}-R_{S}\right)=R_{1}^{2} \text {, say }$$

$$X_{\mathrm{s}}=\pm R_{1}$$

$$B_{p}=\pm R_{1} /\left(R_{L} R_{S}\right)=\pm G_{2}, \text { say }$$

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Impedance Matching at a Single Frequency

$$L_{s}=R_{1} / \omega_{0} \text { and } C_{p}=G_{2} / \omega_{0}$$

$$C_{s}=1 /\left(R_{1} \omega_{0}\right) \text { and } L_{p}=1 /\left(G_{2} \omega_{0}\right)$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。