## 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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|INDUCTANCE

So far we have only considered coils that have a steady d.c. current passing through them. This introduced us to the idea of magnetic flux, the magnetic flux density and magnetic field strength. Although d.c. circuits sometimes use coils, we more usually find them in a.c. circuits. In such circuits, we tend to characterize coils by a term called inductance.

When a d.c. voltage energizes a coil, a current flows which sets up a magnetic field around the coil. This field will not appear instantaneously as it takes a certain amount of time to produce the field. After the initial transient has passed, the resistance of the wire that makes up the coil will limit its current.

Let us now consider a very low-resistance coil connected to a source of alternating voltage. As the coil resistance is very low, the coil should appear to be a shortcircuit. This should result in a lot of current flowing! However, what we find is that the current taken by the coil depends on the frequency of the source – high frequencies result in low currents. Thus, some unknown property of the coil restricts the current.

In 1831, a British physicist, chemist and great experimenter called Michael Faraday (1791-1867) was investigating electromagnetism. As a result of his experiments, Faraday proposed that a changing magnetic field induces an emf into a coil. This was one of the most significant discoveries in electrical engineering, and it is the basic principle behind transformers and electrical machines. (Faraday’s achievement is even more remarkable in that all of his work resulted from experimentation, and not mathematical derivation.)
Faraday’s law formalizes this result as
$$e \propto \frac{\mathrm{d}}{\mathrm{d} t}(N \phi)$$
where $N$ is the number of turns in the coil and $N \phi$ is known as the flux linkage. So, the induced emf depends on the rate of change of flux linkages, i.e., the higher the frequency, the higher the rate of change, the larger the induced emf. As this emf serves to oppose the voltage that produces it, Equation (3.42) is often modified to
$$e=-\frac{\mathrm{d}}{\mathrm{d} t}(N \phi)$$

## 物理代写|电动力学代写electromagnetism代考|Simple COIL

A simple coil consists of several turns on wire wound around a former. As we have just seen, the inductance is defined as the flux linkage per unit current, i.e.,
$$L=N \frac{\mathrm{d} \phi}{\mathrm{d} i}$$
where $N$ is the number of turns in the coil. When we considered solenoids, we saw that the flux density varies along the axis of the coil. However, if the coil is very long, the field at the centre of the coil is
$$\boldsymbol{H}=\frac{N I}{l}$$
and so,
$$\boldsymbol{B}=\mu \frac{N I}{l}$$
As $B$ is the flux density, i.e., $B=\phi / A$, we can write
\begin{aligned} L &=N \frac{\mathrm{d} B}{\mathrm{~d} i} A \ &=N A \mu \frac{N}{l} \frac{\mathrm{d} i}{\mathrm{~d} i} \ &=\frac{N^2 \mu A}{l} \end{aligned}
where $A$ is the cross-sectional area of the coil. Although Equation (3.46) gives the inductance of a long coil, this equation is an approximation. This is because it assumes that the field is constant throughout the coil, and it neglects the effects of flux leakage.

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

1831 年，一位名叫迈克尔法拉第 (1791-1867) 的英国物理学家、化学家和伟大的实验家正在研究电磁学。作为 他的实验的结果，法拉第提出变化的磁场会在线圊中感应出一个电动势。这是电气工程中最重要的发现之一，也是 变压器和电机背后的基本原理。(法拉第的成就更加显着，因为他的所有工作都是实验的结果，而不是数学推 导。)

$$e \propto \frac{\mathrm{d}}{\mathrm{d} t}(N \phi)$$

$$e=-\frac{\mathrm{d}}{\mathrm{d} t}(N \phi)$$

## 物理代写|电动力学代写electromagnetism代考|Simple COIL

$$L=N \frac{\mathrm{d} \phi}{\mathrm{d} i}$$

$$\boldsymbol{H}=\frac{N I}{l}$$

$$\boldsymbol{B}=\mu \frac{N I}{l}$$

$$L=N \frac{\mathrm{d} B}{\mathrm{~d} i} A \quad=N A \mu \frac{N}{l} \frac{\mathrm{d} i}{\mathrm{~d} i}=\frac{N^2 \mu A}{l}$$

## 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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|THE SOLENOID

Figure $3.11$ shows a long coil of wire, or solenoid. Such devices are often used as actuators with a bar magnet placed along the axis of the coil. Any current passing through the coil generates a magnetic field which forces the magnet in a particular direction. The magnet can then force a pair of contacts to close or push a lever to move something and so it is an actuator.

To determine the field at any point along the axis of the solenoid, we will consider an elemental section of the coil, of thickness $\mathrm{d} x$, and calculate the field produced. We will then integrate along the length of the coil to find the total field produced.

Let us assume that the solenoid has $N$ turns and a length of $l$ metre. With these figures, the number of turns per unit length will be N/l. Now, if we take a small section of the coil, of length $\mathrm{d} l$, the number of turns in this section will be $\mathrm{d} l \times N / l$. By using the result from the last section, the magnetic field strength generated by this section of the solenoid is
$$\mathrm{d} H_p=\mathrm{d} l \frac{N}{l} \frac{I \sin ^3 \beta}{2 r}$$
acting along the axis of the coil.
We now need to integrate along the length of the solenoid. However, as we move along the axis, the angle $\beta$ changes between the limits $\beta_{\max }$ and $\beta_{\min }$. So, we have to substitute for $\mathrm{d} l$ in terms of $\beta$. As Figure $31 \mathrm{lb}$ shows,
$$\mathrm{d} l \sin \beta=\mathrm{d} \beta \sqrt{r^2+R^2}$$ and $\mathrm{so}$,
$$\mathrm{d} l=\frac{\mathrm{d} \beta}{\sin \beta} \sqrt{r^2+R^2}$$
Thus, Equation (3.30) becomes
\begin{aligned} \mathrm{d} H_p &=\frac{\mathrm{d} \beta}{\sin \beta} \sqrt{r^2+R^2} \frac{N}{l} \frac{I \sin ^3 \beta}{2 r} \ &=\frac{\sqrt{r^2+R^2}}{2 r} \frac{N}{l} I \sin ^2 \beta \mathrm{d} \beta \end{aligned}
Now, $\sin \beta=\frac{r}{\sqrt{r^2+R^2}}$ and so we can write
$$\mathrm{d} H_p=\frac{N I}{2 l} \sin \beta \mathrm{d} \beta$$

## 物理代写|电动力学代写electromagnetism代考|THE TOROIDAL COIL, RELUCTANCE AND MAGNETIC POTENTIAL

Figure $3.13$ shows the general form of a toroidal former which has a coil wound on it. This is basically a long solenoid which is bent so that the coil has no beginning or end. In practice, the formers used in toroidal coils are made of powdered ferrite which acts to concentrate the magnetic flux. Thus, leakage effects are minimal, so making the coil very efficient. This is put to good use in transformers, which we will encounter in Chapter 7. Here, we want to develop our model of electromagnetism further.

As we saw in the last section, the field at the centre of a long solenoid is given by (Equation (3.34))
$$\boldsymbol{H}=\frac{N I}{l} \mathrm{~A} \mathrm{~m}^{-1}$$
where $l$ is the length of the solenoid. As the coil is wound on a toroid, this field will be constant along the length of the coil. As the coil is circular, the length of the solenoid

will be the average circumference of the former. Now, as $B=\mu H$, the flux density in the former will be
$$\boldsymbol{B}=\mu \frac{N I}{l} \mathrm{~Wb} \mathrm{~m}^{-1}$$
with $\mu$ being the permeability of the former. As $B$ is the flux density, Equation (3.36) becomes
$$\frac{\phi}{\text { area }}=\mu \frac{N I}{l}$$
and so,
$$N I=\phi \times \frac{1}{\mu \times \text { area }}$$
Let us examine this equation closely. The first term on the right-hand side of this equation is the magnetic flux that flows around the toroid. The second term is similar to our formula for the capacitance of a parallel plate capacitor, Equation (2.35). As we saw in Section 2.9, we can regard capacitance as a measure of the resistance to the flow of the flux. So, could we take this second term as a measure of resistance to magnetic flux?

## 物理代写|电动力学代写electromagnetism代考|THE SOLENOID

$$\mathrm{d} H_p=\mathrm{d} l \frac{N}{l} \frac{I \sin ^3 \beta}{2 r}$$

$$\mathrm{d} l \sin \beta=\mathrm{d} \beta \sqrt{r^2+R^2}$$

$$\mathrm{d} l=\frac{\mathrm{d} \beta}{\sin \beta} \sqrt{r^2+R^2}$$

$$\mathrm{d} H_p=\frac{\mathrm{d} \beta}{\sin \beta} \sqrt{r^2+R^2} \frac{N}{l} \frac{I \sin ^3 \beta}{2 r}=\frac{\sqrt{r^2+R^2}}{2 r} \frac{N}{l} I \sin ^2 \beta \mathrm{d} \beta$$

$$\mathrm{d} H_p=\frac{N I}{2 l} \sin \beta \mathrm{d} \beta$$

## 物理代写|电动力学代写electromagnetism代考|THE TOROIDAL COIL, RELUCTANCE AND MAGNETIC POTENTIAL

$$\boldsymbol{H}=\frac{N I}{l} \mathrm{~A} \mathrm{~m}^{-1}$$

$$\boldsymbol{B}=\mu \frac{N I}{l} \mathrm{~Wb} \mathrm{~m}^{-1}$$

$$\frac{\phi}{\text { area }}=\mu \frac{N I}{l}$$

$$N I=\phi \times \frac{1}{\mu \times \text { area }}$$

## 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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|THE FORCE BETWEEN CURRENT-CARRYING WIRES – THE DEFINITION OF THE AMPERE

We can now define the ampere. Readers may think that this is not worth considering since the ampere is simply a measure of current set down by international treaty. After all, we do not often have to concern ourselves with the definition of a metre.

However, as we will soon see, the definition of the ampere introduces us to the force between two current-carrying wires, and that is of some practical benefit.

Figure $3.8$ shows the situation we are to analyze. Two current-carrying wires run parallel to each other, separated by a distance $r$. These wires each carry a current of $I$ amperes. As we have already seen, current-carrying wires produce magnetic fields. As each wire carries the same current, the magnetic field produced by the left-hand conductor will exactly balance the field produced by the right-hand conductor at the point mid-way between the two conductors. Thus, the field at this point will be zero, resulting in the field distribution of Figure $3.8 \mathrm{c}$. The weakening of the field between two wires shows that they attract each other.

Now, in Section $3.4$ we met the force on an isolated north pole due to a currentcarrying element. In this example, we do not have an isolated pole; instead we must consider a current element in the right-hand wire.

To find the force on the right-hand conductor, we need to find the magnetic flux density produced by the left-hand conductor. By applying Ampere’s law, Equation (3.21), we can write
$$I=\oint H \mathrm{~d} l$$

## 物理代写|电动力学代写electromagnetism代考|THE MAGNETIC FIELD OF A CIRCULAR CURRENT ELEMENT

In electrical engineering, we often come across wound components – transformers and coils. Section $7.4$ deals with transformers in detail. Here, we concern ourselves with the field produced by a circular piece of wire carrying a current. This will help us when we come to consider coils and solenoids in the next section.

Figure $3.9$ shows a simple single-turn coil. We require to study the distribution of the magnetic field along (for simplicity) the axis of the coil. To analyze this situation, we will use the Biot-Savart law to find the field produced by a small section of the loop and then integrate around the loop to find the total field.

Let us consider a simple current element of length $\mathrm{d}$. Now, from the Biot-Savart law, the magnitude of the magnetic field strength at point $P$ is given by
$$\mathrm{d} H=\frac{I \mathrm{~d} l}{4 \pi x^2} \sin \theta$$
As the angle $\theta$ is $\pi / 2$ in this instance, we can write
$$\mathrm{d} H=\frac{I \mathrm{~d} l}{4 \pi x^2}$$

Now, $\mathrm{d} H$ makes an angle to the horizontal of $\pi / 2-\beta$, and so we can resolve $\mathrm{d} H$ into vertical and horizontal components. When we integrate around the current loop, we find that the vertical component is zero due to the symmetry of the situation. (Interested readers can try this for themselves.) Thus, we need only consider the horizontal component of $\mathrm{d} H_3, \mathrm{i}{\cdot} \mathrm{e}_w, \mathrm{~d} H{p^*}$ So,
\begin{aligned} \mathrm{d} H_p &=\mathrm{d} H \cos (\pi / 2-\beta) \ &=\mathrm{d} H \sin \beta \ &=\frac{I \mathrm{~d} l}{4 \pi x^2} \sin \beta \end{aligned}

## 物理代写|电动力学代写electromagnetism代考|THE MAGNETIC FIELD OF A CIRCULAR CURRENT ELEMENT

$$\mathrm{d} H=\frac{I \mathrm{~d} l}{4 \pi x^2} \sin \theta$$

$$\mathrm{d} H=\frac{I \mathrm{~d} l}{4 \pi x^2}$$

$$\mathrm{d} H_p=\mathrm{d} H \cos (\pi / 2-\beta) \quad=\mathrm{d} H \sin \beta=\frac{I \mathrm{~d} l}{4 \pi x^2} \sin \beta$$

## 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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

• the radial unit vector
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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Relativity in Newtonian mechanics

Newton’s laws of motion were long assumed to be valid for all inertial reference frames. In Newton’s model, an observer in one reference frame measures the position $x$ of an object at various times $t$. An observer in a second reference frame moves with speed $v$ relative to the first frame, with identical, synchronized clocks and metre sticks. Time intervals and lengths are assumed to be same for both observers.

The second observer sees the first observer move away at speed $v$. The distance between the two observers at a time $t^{\prime}$ is given by $v t^{\prime}$. Hence, the second observer can use the measurements of the first observer, provided the following changes are made:
$$\begin{array}{r} x^{\prime}=x-v t \ t^{\prime}=t \end{array}$$
Equations (26) and (27) are known as a Galilean transformation. It is easy to see that if Newton’s second law holds for one observer, it automatically holds for the other. For an object moving at speed $u$ we find that
$$u^{\prime} \equiv \frac{\mathrm{d} x^{\prime}}{\mathrm{d} t^{\prime}}=\frac{\mathrm{d} x^{\prime}}{\mathrm{d} t}=\frac{\mathrm{d} x}{\mathrm{~d} t}-v=u-v,$$
so we get
$$a^{\prime}=\frac{\mathrm{d}^{2} x^{\prime}}{\mathrm{d} t^{\prime 2}}=\frac{\mathrm{d}^{2} x^{\prime}}{\mathrm{d} t^{2}}=\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}=a .$$
Hence, in both reference frames, the accelerations are the same, and hence the forces are the same, too.

## 物理代写|电动力学代写electromagnetism代考|The wave equation in two inertial reference frames

A problem occurs when we consider light waves. The transformation (28) implies that, in a rest frame travelling at the speed of light $c$ with respect to an emitter, light would be at rest it is not clear how that could be.

To put this problem on a firmer mathematical footing, we derive the general linear transformation of the wave equation; we then substitute in the Galilean transformation. For an electromagnetic wave, the electric field $E$ satisfies, in one reference frame,
$$\frac{\partial^{2} E}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}}=0 .$$
We can express the derivative with respect to $x$ in terms of variables used in another reference frame, $x^{\prime}$ and $t^{\prime}$, by using the chain rule:
$$\frac{\partial E}{\partial x}=\frac{\partial E}{\partial x^{\prime}} \frac{\partial x^{\prime}}{\partial x}+\frac{\partial E}{\partial t^{\prime}} \frac{\partial t^{\prime}}{\partial x} .$$
The second derivative contains five terms:
$$\frac{\partial^{2} E}{\partial x^{2}}=\frac{\partial^{2} E}{\partial x^{\prime 2}}\left(\frac{\partial x^{\prime}}{\partial x}\right)^{2}+2 \frac{\partial^{2} E}{\partial x^{\prime} \partial t^{\prime}} \frac{\partial x^{\prime}}{\partial x} \frac{\partial t^{\prime}}{\partial x}+\frac{\partial^{2} x^{\prime}}{\partial x^{2}} \frac{\partial E}{\partial x^{\prime}}+\frac{\partial^{2} E}{\partial t^{\prime 2}}\left(\frac{\partial t^{\prime}}{\partial x}\right)^{2}+\frac{\partial^{2} t^{\prime}}{\partial x^{2}} \frac{\partial E}{\partial t^{\prime}} .$$

## 物理代写|电动力学代写electromagnetism代考|Relativity in Newtonian mechanics

$$x^{\prime}=x-v t t^{\prime}=t$$

$$u^{\prime} \equiv \frac{\mathrm{d} x^{\prime}}{\mathrm{d} t^{\prime}}=\frac{\mathrm{d} x^{\prime}}{\mathrm{d} t}=\frac{\mathrm{d} x}{\mathrm{~d} t}-v=u-v,$$

$$a^{\prime}=\frac{\mathrm{d}^{2} x^{\prime}}{\mathrm{d} t^{\prime 2}}=\frac{\mathrm{d}^{2} x^{\prime}}{\mathrm{d} t^{2}}=\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}=a .$$

## 物理代写|电动力学代写electromagnetism代考|The wave equation in two inertial reference frames

$$\frac{\partial^{2} E}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}}=0$$

$$\frac{\partial E}{\partial x}=\frac{\partial E}{\partial x^{\prime}} \frac{\partial x^{\prime}}{\partial x}+\frac{\partial E}{\partial t^{\prime}} \frac{\partial t^{\prime}}{\partial x} .$$

$$\frac{\partial^{2} E}{\partial x^{2}}=\frac{\partial^{2} E}{\partial x^{\prime 2}}\left(\frac{\partial x^{\prime}}{\partial x}\right)^{2}+2 \frac{\partial^{2} E}{\partial x^{\prime} \partial t^{\prime}} \frac{\partial x^{\prime}}{\partial x} \frac{\partial t^{\prime}}{\partial x}+\frac{\partial^{2} x^{\prime}}{\partial x^{2}} \frac{\partial E}{\partial x^{\prime}}+\frac{\partial^{2} E}{\partial t^{\prime 2}}\left(\frac{\partial t^{\prime}}{\partial x}\right)^{2}+\frac{\partial^{2} t^{\prime}}{\partial x^{2}} \frac{\partial E}{\partial t^{\prime}} .$$

## 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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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 field due to a uniformly charged hollow cylinder

The net field at any point $P$ follows from superposition. We use a righthanded Cartesian coordinate system where the positive $y$-axis points up and the positive $z$-axis points out of the page. When comparing the contributions from the right half of the cylinder to the electric field with those from the left half, it is clear by symmetry that the $y$-components are equal and add, while the $x$-components are equal and subtract to yield zero. Hence
$$E=2 \int_{-\pi / 2}^{\pi / 2} \mathrm{~d} E_{y}=\frac{\sigma R}{\pi \epsilon_{0}} \int_{-\pi / 2}^{\pi / 2} \frac{\cos \theta}{r} \mathrm{~d} \phi$$
The integrand in (12) contains 3 variables, $r, \phi$, and $\theta$. We may write $r$ and $\cos \theta$ in terms of $\phi$ and constants:
$$\left{\begin{array}{l} r=\sqrt{(R \cos \phi)^{2}+\left(R \sin \phi-y_{0}\right)^{2}}=\sqrt{R^{2}+y_{0}^{2}-2 R y_{0} \sin \phi} \ \cos \theta-\frac{y_{0}-R \sin \phi}{r} \end{array}\right.$$
hence
$$E=\frac{\sigma R}{\pi \epsilon_{0}} \int_{-\pi / 2}^{\pi / 2} \frac{y_{0}-R \sin \phi}{R^{2}+y_{0}^{2}-2 y_{0} R \sin \phi} \mathrm{d} \phi .$$
When entering the integral into the Mathematica online integrator (2011), the antiderivative is given as
$$\frac{-\arctan \left(\frac{R \cos x / 2-y_{0} \sin x / 2}{y_{0} \cos x / 2-R \sin x / 2}\right)}{2 y_{0}}+\frac{\arctan \left(\frac{y_{0} \sin x / 2-R \cos x / 2}{y_{0} \cos x / 2-R \sin x / 2}\right)}{2 y_{0}}+\left.\frac{x}{2 y_{0}}\right|{-\pi / 2} ^{\pi / 2},$$ which is admittedly ugly, but not difficult to use. Since arctan is an odd function, the first two terms are identical, and the antiderivative simplifies to $$\left.\frac{1}{y{0}}\left[\arctan \left(\frac{y_{0} \sin x / 2-R \cos x / 2}{y_{0} \cos x / 2-R \sin x / 2}\right)+\frac{x}{2}\right]\right|_{-\pi / 2} ^{\pi / 2} .$$

## 物理代写|电动力学代写electromagnetism代考|Magnetic fields and current-carrying wires

The flow of charge is called current. To be more precise, define a cross sectional area $A$ through which a charge $\mathrm{dQ}$ flows in a time interval $\mathrm{d} t$. The current $I$ through this area is defined as
$$I \equiv \frac{\mathrm{d} Q}{\mathrm{~d} t} .$$
It is often convenient to define a current density $\mathrm{J}$, which is the current per unit cross sectional area $A$ :
$$J \equiv I / A \text {. }$$
A steady current flowing through a homogeneous wire can be modeled as a linear charge density $\lambda$ moving at constant drift speed $v_{d}$. In that case, the total charge flowing through a cross sectional area in a time interval $\Delta t$ is given by $\lambda v_{d} \Delta t$, and
$$I=\lambda v_{d} .$$

In this chapter, we will only concern ourselves with magnetic effects due to straight current-carrying wires. Oersted found experimentally that a magnet (compass needle) gets deflected when placed near a current-carrying wire (Shamos, 1987b). As in electrostatics, we model this behaviour by invoking a field: the current in the wire creates a magnetic field $B$ that acts on the magnet.

In subsequent decades, experiments showed that moving charged objects are affected by magnetic fields. The magnetostatic force (so called because the source of the magnetic field is steady; it is also often called the Lorentz force) is proportional to the charge $q$, the speed $v$, the field $B$, and the sine of the angle $\phi$ between $v$ and $B$; it is also perpendicular to $v$ and $B$. In vector notation,
$$\vec{F}{m}=q \vec{v} \times \vec{B} ;$$ in scalar notation, $$F{m}=q v B \sin \phi .$$
As a corollary, two parallel currents exert a magnetostatic force on each other, as the charges in each wire move in the magnetic field of the other wire.

## 物理代写|电动力学代写electromagnetism代考|Electric field due to a uniformly charged hollow cylinder

$$E=2 \int_{-\pi / 2}^{\pi / 2} \mathrm{~d} E_{y}=\frac{\sigma R}{\pi \epsilon_{0}} \int_{-\pi / 2}^{\pi / 2} \frac{\cos \theta}{r} \mathrm{~d} \phi$$
(12) 中的被积函数包含 3 个变量， $r, \phi$ ，和 $\theta$. 我们可以写 $r$ 和 $\cos \theta$ 按照 $\phi$ 和常量:
$\$ \$$Veft {$$
r=\sqrt{(R \cos \phi)^{2}+\left(R \sin \phi-y_{0}\right)^{2}}=\sqrt{R^{2}+y_{0}^{2}-2 R y_{0} \sin \phi} \cos \theta-\frac{y_{0}-R \sin \phi}{r}
$$正确的。 hence ## 物理代写|电动力学代写electromagnetism代考|Magnetic fields and current-carrying wires 电荷的流动称为电流。更准确地说，定义横截面积 A 通过它收费 \mathrm{dQ} 在一个时间间隔内流动 \mathrm{d} t. 目前的 I 通过该区 域定义为$$
I \equiv \frac{\mathrm{d} Q}{\mathrm{~d} t} .
$$定义电流密度通常很方便 \mathrm{J} ，即单位截面积的电流 A :$$
J \equiv I / A .
$$流过均匀导线的稳定电流可以建模为线性电荷密度 \lambda 以恒定的漂移速度移动 v_{d}. 在这种情况下，在一个时间间隔内 流过一个横截面积的总电荷 \Delta t 是 (准) 给的 \lambda v_{d} \Delta t ，和$$
I=\lambda v_{d} .
$$在本章中，我们将只关注由直通电流导线引起的磁效应。奥斯特通过实验发现，磁铁 (罗盘针) 在靠近载流导线 时会发生偏转 (Shamos，1987b) 。就像在静电学中一样，我们通过调用一个场来模拟这种行为：电线中的电流 会产生一个磁场 B 作用在磁铁上。 在随后的几十年中，实验表明移动的带电物体会受到磁场的影响。静磁力（之所以这样称呼，是因为磁场的来源 是稳定的；通常也称为洛伦兹力 ) 与电荷成正比 q ，速度 v ， 场 B ，和角度的正弦 \phi 之间 v 和 B; 它也垂直于 v 和 B. 在矢量符号中，$$
\vec{F} m=q \vec{v} \times \vec{B} ;
$$在标量符号中，$$
F m=q v B \sin \phi .
$$In SI units, the constant of proportionality is given as 1 / 4 \pi \epsilon_{0} for convenience in calculations. The constant \epsilon_{0} is called the permittivity of vacuum. It is often useful to define the charge per unit length, called the linear charge density (symbol: \lambda ); the charge per unit (surface) area, symbol: \sigma; and the charge per unit volume, symbol \rho. ## 物理代写|电动力学代写electromagnetism代考|An infinite line charge Imagine an infinitely long line of uniform linear charge density \lambda. Take a segment of length \mathrm{d} z, a horizontal distance z from point P which has a perpendicular distance r to the line charge. By Coulomb’s Law, the magnitude of the electric field at P due this line segment is$$
\mathrm{d} E=\frac{\lambda \mathrm{d} z}{4 \pi \epsilon_{0}\left(r^{2}+z^{2}\right)} .
$$A second segment of the same length \mathrm{d} z a distance z from P (see Fig. 1b) gives rise to an electric field of the same magnitude, but pointing in a different direction. The z components cancel, leaving only the r component:$$
\mathrm{d} E_{r}=\frac{\lambda \mathrm{d} z \sin \phi}{4 \pi \epsilon_{0}\left(r^{2}+z^{2}\right)} .
$$To find the net field at P, we add the contributions due to all line segments. This net field is thus an infinite sum, given by the integral$$
E=\frac{\lambda}{4 \pi \epsilon_{0}} \int_{-\infty}^{\infty} \frac{\mathrm{d} z \sin \phi}{r^{2}+z^{2}} .
$$## 电动力学代考 ## 物理代写|电动力学代写electromagnetism代考|Coulomb’s Law 18 世纪后期，库仑用扭力天平表明，两个带电的小球相互施加一个与球心之间距离的平方反比成正比的力，并沿 连接中心的直线作用 (沙莫斯，1987a) 。他还表明，作为平方反比定律的结果，导体上的所有电荷都必须驻留 在表面上。此外，根据壳定理（维基百科，2011），两个完美球形空心壳之间的力就像所有电荷都集中在每个球 体的中心一样。这种情况非常接近于两个通过摩擦带电的球形绝缘体，这种偏差是由非常小的极化效应引起的。 库仑也是第一个量化电荷的人。例如，在完成一次测量后，他通过将球体与相同的球体接触，将球体上的电荷减 半。当球体恢复到扭力平衡时，他测量到球体之间的力减半 (Arons，1996) 。当他用天平上的另一个球体重复 这个过程时，球体之间的力变成了原来值的四分之一。 在现代符号中，库仑因此找到了以他的名字命名的定律：静电力 \vec{F} E 两个点状物体之间的距离 r 分开，收费 Q 和 q 分别由下式给出$$
\vec{F} E=\frac{1}{4 \pi \epsilon_{0}} \frac{Q q}{r^{2}} \hat{r}
$$在 \mathrm{SI} 单位中，比例常数为 1 / 4 \pi \epsilon_{0} 为了计算方便。常数 \epsilon_{0} 称为真空的介电常数。 定义每单位长度的电荷通常很有用，称为线性电荷密度（符号: \lambda ); 每单位 (表面积) 面积的电荷，符号： \sigma; 以 及每单位体积的电荷，符号 \rho. ## 物理代写|电动力学代写electromagnetism代考|An infinite line charge 想象一条无限长的均匀线性电荷密度线 \lambda. 取一段长度 \mathrm{d} z, 水平距离 z 从点 P 具有垂直距离 r 到线费。根据库仑定 律，电场强度为 P 由于这条线段是$$
\mathrm{d} E=\frac{\lambda \mathrm{d} z}{4 \pi \epsilon_{0}\left(r^{2}+z^{2}\right)}
$$相同长度的第二段 \mathrm{d} z 一段距离 z 从 P (见图 1b) 产生相同大小的电场，但指向不同的方向。这 z 组件取消，只留 下 r 零件:$$
\mathrm{d} E_{r}=\frac{\lambda \mathrm{d} z \sin \phi}{4 \pi \epsilon_{0}\left(r^{2}+z^{2}\right)}
$$在以下位置找到网络场 P ，我们将所有线段的贡献相加。因此，这个净场是一个无限和，由积分给出$$
E=\frac{\lambda}{4 \pi \epsilon_{0}} \int_{-\infty}^{\infty} \frac{\mathrm{d} z \sin \phi}{r^{2}+z^{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。

## 物理代写|电动力学代写electromagnetism代考|PHYSICS 2534

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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|The significance of using auxiliary sources

Consider an electromagnetic environment (Figure 1.5) comprised of one (or several) components whose behavior is described in the system V-I, E-J or even E-Js.

This element may be replaced by a source, the shutdown for which is formed by the electromagnetic environment.

The closed matrix may be established across two given sources; $E_{0}$ being the device power source and $E_{\mathrm{A}}$ being the auxiliary source.

As Figure $1.5$ shows, this component may be replaced with a source (stage 1), the shutdown for which may be constituted by this component (stage 2).

The calculation produces impedance (and the potential source) in view of this source, from the electromagnetic environment. This will operate within the spectral domain.

Stage (2) involves stating that the source of stage (1) (the arbitrary source) is shutdown on the impedance of the output circuit.

## 物理代写|电动力学代写electromagnetism代考|Description of the environment

This consists of a system $Q$, both fed by a source $S_{0}$ and closed by an impedance $Z$. In general $Q$ constitutes the center of the electromagnetic field E.M. To examine this system, we can break it down into two parts, each separate and distinct from the other and fed by a source known as the auxiliary source. System (I) describes the behavior of the impedance $Z$. System (II) describes the main source within its environment; E.M. $Z$ can generally be defined at any point, and forms the spatial sphere. On the other hand, $Q$ is often described within the environment E.M. $Q$ is defincd by its impedance or diffraction matrix. It is necessary to resort to the spectral sphere. Using this method, the calculation of impedance from the angle of $S_{0}$ is not achieved directly but it is first necessary to calculate a quadrupole with the help of an auxiliary source. This will subsequently be replaced by the localized impedance within the actual issue (allowing for its potential source). This latter operation is known as an operation in the spectral domain.

## 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。