## 物理代写|几何光学代写Geometrical Optics代考|OPTI502

statistics-lab™ 为您的留学生涯保驾护航 在代写几何光学Geometrical Optics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。我们的专家在代写几何光学Geometrical Optics代写方面经验极为丰富，各种代写几何光学Geometrical Optics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|几何光学代写Geometrical Optics代考|Set of Maxwell Equations for Electrostatic Field

First, we introduce the set of Maxwell equations for the electrostatic field in free space. Using Gauss’s Law (see Chap. 2), we can write the electric flux of electric field created by continuous charge distribution in a volume $V$ enclosed by the surface $A$ as
$$\oint_A \mathbf{E} \cdot d \mathbf{A}=\frac{Q_{i n}}{\epsilon_0}$$
Note that in Eq. (4.65) $\mathbf{E}$ is the electrostatic field created by all charges in space, and $Q_{i n}$ is the electric charge inside the volume $V$ enclosed by the surface $A$. The left-hand side of Eq. (4.65) can be written in the following form using Gauss formula: $$\oint_A \mathbf{E} \cdot d \mathbf{A}=\int_V \nabla \cdot \mathbf{E} d V$$
where $V$ is the volume enclosed by the surface $A$. In addition, the right-hand side of Eq. (4.65) can be written as
$$\frac{Q_{i n}}{\epsilon_0}=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$
Combining Eqs. (4.65), (4.66) and (4.67), we get
$$\int_V \nabla \cdot \mathbf{E} d V=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$
where $\nabla \cdot \mathbf{E}$ is the divergence of the vector $\mathbf{E}$, which produces a scalar.
Comparing both sides of Eq. (4.68), we obtain the first Maxwell equation in free space:
$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})}{\epsilon_0}$$
where both $\mathbf{E}$ and $\rho$ can be functions of the position $\mathbf{r}$.
Using the expression of the electrostatic potential difference in free space, Eq. (4.10) (Chap.3), we have
$$\Delta \phi=-\int_A^B \mathbf{E} \cdot d \mathbf{s}$$
where $A$ and $B$ are two points in free space, and $d \mathbf{s}$ is an infinitesimal displacement along the curve joining points $A$ and $B$. If we consider a closed path, that is, $A=B$, then $\Delta \phi=\phi_B-\phi_A=\phi_A-\phi_A=0$, and hence
$$\oint_{\mathcal{L}} \mathbf{E} \cdot d \mathbf{s}=0$$

## 物理代写|几何光学代写Geometrical Optics代考|Maxwell Equations for Dielectric Media Electrostatic Field

We mentioned that in the dielectric medium, an average over macroscopically small volumes, which are microscopically large, is necessary to obtain the Maxwell equations of the macroscopic phenomena.
The first observation is that Eq. (4.74) holds microscopically, that is
$$\nabla \times \mathbf{E}_{\text {micro }}=0$$
When averaging is made of the homogeneous Eq. (4.75), we obtain
$$\nabla \times \mathbf{E}=0$$
Equation (4.76) indicates that Eq. (4.74) holds for the averaged macroscopic electric field $\mathbf{E}$.

Using Eq. (4.57) for the effective charge density in the medium, Eq. (4.69) becomes
$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})-\nabla \cdot \mathbf{P}(\mathbf{r})}{\epsilon_0}$$
Rearranging Eq. (4.77), we get
$$\nabla \cdot\left(\epsilon_0 \mathbf{E}(\mathbf{r})+\mathbf{P}(\mathbf{r})\right)=\rho(\mathbf{r})$$
Using the definition of the electric displacement vector given by Eq. (4.58), we write Eq. (4.78) as
$$\nabla \cdot \mathbf{D}(\mathbf{r})=\rho(\mathbf{r})$$
Note that Eqs. (4.76) and (4.79) are the macroscopic Maxwell equations in the dielectric medium, which are the counterparts of Eqs. (4.69) and (4.74).

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Set of Maxwell Equations for Electrostatic Field

$$\oint_A \mathbf{E} \cdot d \mathbf{A}=\frac{Q_{i n}}{\epsilon_0}$$

$$\oint_A \mathbf{E} \cdot d \mathbf{A}=\int_V \nabla \cdot \mathbf{E} d V$$

$$\frac{Q_{i n}}{\epsilon_0}=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$

$$\int_V \nabla \cdot \mathbf{E} d V=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$

$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})}{\epsilon_0}$$

$$\Delta \phi=-\int_A^B \mathbf{E} \cdot d \mathbf{s}$$

$$\oint_{\mathcal{L}} \mathbf{E} \cdot d \mathbf{s}=0$$

## 物理代写|几何光学代写Geometrical Optics代考|Maxwell Equations for Dielectric Media Electrostatic Field

$$\nabla \times \mathbf{E}_{\text {micro }}=0$$

$$\nabla \times \mathbf{E}=0$$

$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})-\nabla \cdot \mathbf{P}(\mathbf{r})}{\epsilon_0}$$

$$\nabla \cdot\left(\epsilon_0 \mathbf{E}(\mathbf{r})+\mathbf{P}(\mathbf{r})\right)=\rho(\mathbf{r})$$

$$\nabla \cdot \mathbf{D}(\mathbf{r})=\rho(\mathbf{r})$$

## 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|几何光学代写Geometrical Optics代考|PHYSICS134A

statistics-lab™ 为您的留学生涯保驾护航 在代写几何光学Geometrical Optics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。我们的专家在代写几何光学Geometrical Optics代写方面经验极为丰富，各种代写几何光学Geometrical Optics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|几何光学代写Geometrical Optics代考|Energy Stored in Capacitor

The energy stored in the absence of the dielectric is
$$U_0=\frac{Q_0^2}{2 C_0}$$
After the battery is removed and the dielectric inserted, the charge on the capacitor remains the same. Hence, the energy stored in the presence of the dielectric is
$$U=\frac{Q_0^2}{2 C}$$
Using the relation $C=\varepsilon C_0$, then
$$U=\frac{Q_0^2}{2 \varepsilon C_0}$$
or
$$U=\frac{U_0}{\varepsilon}$$
Because $\varepsilon>1$, the final energy is less than the initial energy (see also Eq. (4.48)) $\Delta U=U-U_0<0$. We can account for the “missing” energy by noting that the dielectric, when inserted, gets pulled into the device. An external agent must do negative work to keep the dielectric from accelerating.
This work is simply the difference
$$W_a=U-U_0$$
Alternatively, the positive work done on the external agent by the system is
$$W=-W_a=U_0-U$$

## 物理代写|几何光学代写Geometrical Optics代考|Electric Polarization

Consider an electric field applied to a medium made up of a large number of particles, such as atoms or molecules. The charges bound in molecules will then respond to the external electric field, and they will follow the perturbed motion to align with the external field. Thus, the charge density within the molecules will be distorted. The dipole moments ${ }^3$ of each molecule will be different in comparison to the dipole moments in the absence of the applied electric field. That is, in the absence of the external field, the average dipole moments over all molecules of the substance are zero because the dipole vectors are oriented randomly. In contrast, in the presence of the applied electric field, the net dipole moment of the substance is different from zero. Therefore, in the medium, there is an average dipole moment per unit volume, which is called electric polarization $\mathbf{P}$, given as
$$\mathbf{P}(\mathbf{r})=\sum_i n_i\left\langle\mathbf{p}_i\right\rangle$$
In Eq. (4.51), $\mathbf{p}_i$ is the dipole moment of the molecule type $i$ in the medium, $\langle\cdots\rangle$ denotes the average over a small volume around $\mathbf{r}$, and $n_i$ is the average number per unit volume of the molecule type $i$ at the position $\mathbf{r}$.

If the net charge of the molecule $i$ is $Q_i$, and there is a macroscopic excess or free charge, the charge density at the macroscopic level is
$$\rho(\mathbf{r})=\sum_i n_i\left\langle Q_i\right\rangle+\rho_{\text {free }}$$
Note that, in general, average charge of a molecule $i$ is zero, $\left\langle Q_i\right\rangle=0$, and hence, the charge density $\rho$ is equal to the macroscopic excess or free charge, $\rho_{\text {free }}$.

In the following, we will consider the case of a continuous charge distribution, as in Fig. $3.6$ (Chap. 3), and see the medium from a macroscopic viewpoint. The potential at some point $P$ at the position $\mathbf{r}$ from a macroscopic small volume element $d V$ at the position $\mathbf{r}^{\prime}$ is the sum of the potential created by the charge of $d V, d q-\rho\left(\mathbf{r}^{\prime}\right) d V$ and the dipole moment of $d V$ is $\mathbf{P}\left(\mathbf{r}^{\prime}\right) d V$, assuming that there are no higher macroscopic multipole moment densities:
$$d \phi\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=k_e\left(\frac{\rho\left(\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}+\frac{\mathbf{P}\left(\mathbf{r}^{\prime}\right) \cdot\left(\mathbf{r}-\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^3}\right)$$

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Energy Stored in Capacitor

$$U_0=\frac{Q_0^2}{2 C_0}$$

$$U=\frac{Q_0^2}{2 C}$$

$$U=\frac{Q_0^2}{2 \varepsilon C_0}$$

$$U=\frac{U_0}{\varepsilon}$$

$$W_a=U-U_0$$

$$W=-W_a=U_0-U$$

## 物理代写|几何光学代写Geometrical Optics代考|Electric Polarization

$$\mathbf{P}(\mathbf{r})=\sum_i n_i\left\langle\mathbf{p}i\right\rangle$$ 在等式中。(4.51), $\mathbf{p}_i$ 是分子类型的偶极矩 $i$ 在媒体中， $\langle\cdots\rangle$ 表示周围小体积的平均值 $\mathbf{r}$ ，和 $n_i$ 是分子类型 每单位体积的平均数 $i$ 在那个位置r. 如果分子的净电荷 $i$ 是 $Q_i$ ，并且存在宏观过剩或自由电荷，宏观层面的电荷密度为 $$\rho(\mathbf{r})=\sum_i n_i\left\langle Q_i\right\rangle+\rho{\text {free }}$$

$$d \phi\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=k_e\left(\frac{\rho\left(\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}+\frac{\mathbf{P}\left(\mathbf{r}^{\prime}\right) \cdot\left(\mathbf{r}-\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^3}\right)$$

## 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|几何光学代写Geometrical Optics代考|PHYS201

statistics-lab™ 为您的留学生涯保驾护航 在代写几何光学Geometrical Optics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。我们的专家在代写几何光学Geometrical Optics代写方面经验极为丰富，各种代写几何光学Geometrical Optics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|几何光学代写Geometrical Optics代考|Energy Storage in the Electric Field

To transfer an amount of charge from one plate of a capacitor to the other during the process of charging the capacitor, an external work is done against the electric field. That work stores in the capacitor in the form of the potential energy. For that, let $q$ be the charge on the capacitor at some instant during the charging process when the potential difference across the capacitor is $\Delta V=q / C$. At that instant, one of the plates is carrying a charge $+q$ and the other $-q$. To transfer an increment of charge $d q$ from the plate with charge $-q$ (which is at a lower electric potential) to the plate carrying charge $+q$ (which is at a higher electric potential) an elementary work is done against the electric field:
$$d W=\Delta V d q=\frac{q}{C} d q$$
To calculate the total work required to charge the capacitor from $q=0$ to final charge $Q$, we integrate Eq. (4.27) as follows:
$$W=\int_0^Q \frac{q}{C} d q=\frac{1}{2} \frac{Q^2}{C}$$

This work done to charge the capacitor stores in the capacitor as an electric potential energy $U$. Therefore, $U=W$. Also, we can express the potential energy $U$ in the following forms:
\begin{aligned} U & =\frac{1}{2} \frac{Q^2}{C} \ & =\frac{1}{2} Q \Delta V \ & =\frac{1}{2} C(\Delta V)^2 \end{aligned}
Note that all expressions given by Eqs. (4.29)-(4.31) are equivalent; that is, they can all be used to calculate the potential energy stored in a capacitor depending on what is known. We can consider the energy stored in a capacitor as being stored in the electric field created between the plates as the capacitor is charged. This description is reasonable from the viewpoint that the electric field is proportional to the charge $Q$ stored on a capacitor. For a capacitor of two parallel plates, the potential difference is related to the electric field through a simple relationship $\Delta V=E d$. Furthermore, its capacitance is $C=\epsilon_0 \frac{A}{d}$. Then, we obtain
$$U=\frac{1}{2}\left(\epsilon_0 \frac{A}{d}\right)(E d)^2=\frac{1}{2} \epsilon_0(A d) E^2$$
Since the volume is $A d$, then the energy density is given
$$u_E=\frac{U}{A d}=\frac{1}{2} \epsilon_0 E^2$$
This expression is generally valid. That is, the energy density in any electric field is proportional to the square of the magnitude of the electric field at a given point.

## 物理代写|几何光学代写Geometrical Optics代考|Electrostatics of Macroscopic Media and Dielectrics

There exist many materials that do not allow electric charges to move freely within them, or may allow such motion to occur only very slowly. Those materials are used to block the flow of electrical current, and to form the insulators. For example, they can create insulating layers between the plates of a capacitor. Those materials are known as dielectric materials. As an application, the use of the dielectric material for a capacitor reduces its size for a given capacitance or increases its working voltage. Note that a dielectric material subject to a high enough electric field becomes a conductor; that is, the dielectric material experiences a dielectric breakdown. Thus, there exists a maximum voltage for dielectric capacitors to work. For example, there is a maximum power that a coaxial cable can adequately function in high-power applications such as radio transmitters; similarly, for microcircuits there are maximum voltages, which can be handled.

To know about the differences between dielectric and conducting materials, we can consider their behavior in electric fields. In particular, we have shown in Fig. 4.7 a conducting and dielectric sheet between the parallel plates in which a potential difference exists. That is, there are an equal amount of opposite charges on the two plates.

In the conducting sheet, the conducting electrons are free to move, and they establish a surface charge which exactly cancels the electric field within the conductor, as shown in Fig.4.7. That is, the surface charge density of the plates and conducting sheet is the same but with opposite sign. On the other hand, the electrons in the dielectric material are bound to atoms, and the external electric field causes only a displacement of the electronic configuration of atoms (see Fig. 4.7). However, it is sufficient to produce some surface charge with density $\sigma_{\text {ind }}$ (called an induced charge). We say that the dielectric is polarized. Note that the surface charge is not able to cancel the external electric field within the sheet; however, it does reduce. In the following, we will introduce a simplified molecular theory of dielectrics to understand the behavior of dielectric materials in the presence of an external electric field. ${ }^1$ A more complicated, but more precise theory, will be introduced in the following sections, accounting for electric polarization of the ponderable media. ${ }^2$

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Energy Storage in the Electric Field

$$d W=\Delta V d q=\frac{q}{C} d q$$

$$W=\int_0^Q \frac{q}{C} d q=\frac{1}{2} \frac{Q^2}{C}$$

$$U=\frac{1}{2} \frac{Q^2}{C} \quad=\frac{1}{2} Q \Delta V=\frac{1}{2} C(\Delta V)^2$$

$$U=\frac{1}{2}\left(\epsilon_0 \frac{A}{d}\right)(E d)^2=\frac{1}{2} \epsilon_0(A d) E^2$$

$$u_E=\frac{U}{A d}=\frac{1}{2} \epsilon_0 E^2$$

## 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。