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## 物理代写|电磁学代写Electromagnetism代考|Energy and Momentum of Electromagnetic Waves

Using the relation in Eq. (12.24), the contributions of electric and magnetic field to the total energy density are equal; that is,
$$\epsilon_m \frac{E^2}{2}=\frac{B^2}{2 \mu_m} .$$
Therefore, the total energy density in the electromagnetic wave is
$$u=\left(\epsilon_m \frac{E^2}{2}+\frac{B^2}{2 \mu_m}\right)=\epsilon_m E^2=\frac{B^2}{\mu_m} .$$

## 物理代写|电磁学代写Electromagnetism代考|Coherence of Electromagnetic Waves

The monochromatic plane electromagnetic waves are an idealization picture because they have infinite extent in all directions. Furthermore, they are just sinusoidal waves, and hence their angular frequency spread is zero, $\Delta \omega=0$. The same would have been true for the monochromatic spherical waves. These waves (that is, having $\Delta \omega=0$ ) are called completely coherent. If two identical of such waves superimpose, then the interference can be seen. For instance, consider two electromagnetic waves with magnitudes of electric field vectors as
\begin{aligned} & E_1=E_0 \cos \left(\mathbf{k} \cdot \mathbf{r}1-\omega t\right), \ & E_2=E_0 \cos \left(\mathbf{k} \cdot \mathbf{r}_2-\omega t+\delta\right), \end{aligned} where $\delta$ is a phase angle shift; moreover, we have assumed they have the same amplitude $E_0$. The resultant wave is given as follows: \begin{aligned} E_R & =E_1+E_2 \ & =E_0 \cos \left(\mathbf{k} \cdot \mathbf{r}_1-\omega t\right)+E_0 \cos \left(\mathbf{k} \cdot \mathbf{r}_2-\omega t+\delta\right) \ & =2 E_0 \cos \left(\frac{\mathbf{k} \cdot \mathbf{r}_2-\omega t+\delta-\mathbf{k} \cdot \mathbf{r}_1+\omega t}{2}\right) \ & \times \sin \left(\frac{\mathbf{k} \cdot \mathbf{r}_2-\omega t+\delta+\mathbf{k} \cdot \mathbf{r}_1-\omega t}{2}\right) \ & =2 E_0 \cos \left(\frac{\mathbf{k} \cdot\left(\mathbf{r}_2-\mathbf{r}_1\right)+\delta}{2}\right) \sin \left(\frac{\mathbf{k} \cdot\left(\mathbf{r}_2+\mathbf{r}_1\right)}{2}-\omega t+\frac{\delta}{2}\right), \end{aligned} which is the equation of a traveling wave, and its amplitude is $$E{R 0}=2 E_0 \cos \left(\frac{\mathbf{k} \cdot\left(\mathbf{r}_2-\mathbf{r}_1\right)+\delta}{2}\right)$$

## 物理代写|电磁学代写Electromagnetism代考|Energy and Momentum of Electromagnetic Waves

$$\epsilon_m \frac{E^2}{2}=\frac{B^2}{2 \mu_m} \text {. }$$

$$u=\left(\epsilon_m \frac{E^2}{2}+\frac{B^2}{2 \mu_m}\right)=\epsilon_m E^2=\frac{B^2}{\mu_m} .$$

## 物理代写|电磁学代写Electromagnetism代考|Coherence of Electromagnetic Waves

$$E_1=E_0 \cos (\mathbf{k} \cdot \mathbf{r} 1-\omega t), \quad E_2=E_0 \cos \left(\mathbf{k} \cdot \mathbf{r}_2-\omega t+\delta\right),$$

$$E_R=E_1+E_2 \quad=E_0 \cos \left(\mathbf{k} \cdot \mathbf{r}_1-\omega t\right)+E_0 \cos \left(\mathbf{k} \cdot \mathbf{r}_2-\omega t+\delta\right)=2 E_0 \cos \left(\frac{\mathbf{k} \cdot \mathbf{r}_2-\omega t+\delta-\mathbf{k} \cdot \mathbf{r}_1+\omega t}{2}\right)$$

$$E R 0=2 E_0 \cos \left(\frac{\mathbf{k} \cdot\left(\mathbf{r}_2-\mathbf{r}_1\right)+\delta}{2}\right)$$

## 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:Electromagnetism, 物理代写, 电磁学

## avatest™帮您通过考试

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## 物理代写|电磁学代写Electromagnetism代考|Mixed Boundary Value Problem

To demonstrate the separation of variable method, we choose for simplicity reasons a plane and, mixed boundary value problem for Laplace’s equation $\nabla^2 \varphi=0$. Consider the rectangular piece of a uniform conductor with the sides $a$ and $b$ where a voltage is applied as shown in Fig. 4.19. The two sides $y=0$ and $y=b$ are coated with a very good conductor (like silver) and are grounded. The side $x=a$

shall also be coated with silver but the potential there shall be $\varphi=\varphi_0$. There is no conducting material at the side $x=0$ and therefore, the current density lines have to be parallel to that boundary, which means that $\partial \varphi / d n=0$. We summarize the boundary conditions:
In order to meet the boundary conditions for $\mathrm{y}$, the $y$-dependent part is written as
$$A \cos (k y)+B \sin (k y) \text {. }$$
This determines the form of the $x$-dependency, since the problem is independent of $z$.
$$C \cosh (k x)+D \sinh (k x)$$

## 物理代写|电磁学代写Electromagnetism代考|Basic Equations

Maxwell’s equations were introduced in Chapter 1 eq. (1.72). In the limit of just time-independent problems, the system of Maxwell’s equations nicely splits into two electrostatic and two magnetostatic equations. The latter consists of Ampere’s law, and the fact that the magnetic field is always source free (i.e. solenoidal).
\begin{tabular}{c}
$\nabla \times \mathbf{H}=\mathbf{g}$ \
\hline$\nabla \bullet \mathbf{B}=0$ \
\hline
\end{tabular}
Beyond that, we have to establish a relation between $\mathbf{B}$ and $\mathbf{H}$
$$\mathbf{B}=\mathbf{B}(\mathbf{H}) \text {. }$$
The relation for vacuum is
$$\mathbf{B}=\mu_0 \mathbf{H} .$$
The B-field is perceivable because it exhibits a velocity dependent force on charged particles (Lorentz force). If an electric field exists simultaneously, the force becomes
$$\mathbf{F}=Q(\mathbf{E}+\mathbf{v} \times \mathbf{B}) .$$
Integrating eq. (5.1) over an arbitrary area gives its integral representation
$$\int_A(\nabla \times \mathbf{H}) d \mathbf{A}=\int_A \mathbf{g} d \mathbf{A} .$$

## 物理代写|电磁学代写|电磁学代考|混合边界值问题

$$A cos (k y)+B sin (k y) \text {. }$$

$$C\cosh (k x)+D\sinh (k x)$$

## 物理代写|电磁学代写电磁学代考|基本方程

$$\mathbf{B}=\mathbf{B}(\mathbf{H}) .$$

$$\mathbf{B}=mu_0\mathbf{H}。$$
B场是可感知的，因为它对带电粒子表现出一种与速度有关的力（洛伦兹力）。如果同时存在一个电场，这个力就变成了
$$\σmathbf{F}=Q(\mathbf{E}+\mathbf{v}\times σmathbf{B})$$

$$\int_A(\nabla \times \mathbf{H}) d \mathbf{A}=int_A \mathbf{g} d \mathbf{A}。$$

## 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:Electromagnetism, 物理代写, 电磁学

## avatest™帮您通过考试

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## 物理代写|电磁学代写Electromagnetism代考|Self-inductance

There is a difference between the emfs and the current produced by a battery or other sources and those induced by changing the magnetic field flux.

In general, the source emf and source current describe parameters associated with a physical source. In contrast, the induced emf and induced current describe parameters associated with changing magnetic field flux.

Consider the simple electric circuit shown in Fig. 10.5. It consists of a source emf, $\epsilon$, a resistance, $R$, and a switch, $S$. When the switch is closed the source current does not instantly increase to its maximum value $I_{\max }$ :
$$I_{\max }=\frac{\epsilon}{R}$$
At some instance of time $t$ the current in the circuit is $I(t)$. Besides, the current passing through straight wire line produces a magnetic field $B=\mu_0 I / 2 \pi r$ (where $r$ is the distance from the wire), and thus $B \sim I$. Since the current increases to reach its maximum value $I_{\max }$, so does the magnetic field. Furthermore, the magnetic flux passing through the surface area enclosed by the circuit is
$$\Phi_B=\int_{\mathcal{S}} \mathbf{B} \cdot d \mathbf{S}$$
where $\mathcal{S}$ is the surface area enclosed by the circuit. Therefore, since $B$ increases, $\Phi_B$ increases with time, that is, $d \Phi_B / d t \neq 0$, which in turn creates an induced emf in the circuit:
$$\epsilon_L=-\frac{d \Phi_B}{d t}$$

## 物理代写|电磁学代写Electromagnetism代考|Mutual Inductance

The magnetic flux passing through the surface area of a loop is (see also Fig. 10.8):
$$\Phi_B=\int_{\mathcal{S}} \mathbf{B} \cdot d \mathbf{S}$$
where $|\mathbf{B}| \sim I$. Therefore, $\Phi_B$ varies with time because $I$ varies with time. Thus, an induced emf occurs through the process of mutual inductance. This is related with the fact that it depends on the interaction between two circuits.

Consider two parallel coils of $N_1$ and $N_2$ turns, respectively, as shown in Fig. $10.9$. Through the coil I is passing the current $I_1$ and coil II the current $I_2$. Suppose the current $I_1$ is creating a magnetic field with magnetic field lines as depicted in Fig. 10.9. Some of these lines pass through the coil II. We denote by $\Phi_{12}$ the magnetic flux of the magnetic field created by coil I through the coil II. The mutual inductance, namely, $M_{12}$, of coil II with respect to coil I is

$$M_{12}=N_2 \frac{\Phi_{12}}{I_1}$$
Assuming that the current $I_1$ is varying with time, then an induced emf is created at coil II, given as
$$\epsilon_{\text {ind }, 2}=-N_2 \frac{d \Phi_{12}}{d t}$$
where $\Phi_{12}$ is calculated from Eq. (10.34) as
$$\Phi_{12}=I_1 \frac{M_{12}}{N_2}$$

## 物理代写|电磁学代写Electromagnetism代考|Self-inductance

$$I_{\max }=\frac{\epsilon}{R}$$

$$\Phi_B=\int_{\mathcal{S}} \mathbf{B} \cdot d \mathbf{S}$$

$$\epsilon_L=-\frac{d \Phi_B}{d t}$$

## 物理代写|电磁学代写Electromagnetism代考|Mutual Inductance

$$\Phi_B=\int_{\mathcal{S}} \mathbf{B} \cdot d \mathbf{S}$$

$$M_{12}=N_2 \frac{\Phi_{12}}{I_1}$$

$$\epsilon_{\text {ind }, 2}=-N_2 \frac{d \Phi_{12}}{d t}$$

$$\Phi_{12}=I_1 \frac{M_{12}}{N_2}$$

## 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。