## 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|The Extended Charge Model

In the linear approximation the momentum is a constant of the motion, $\mathbf{p}_0$, so that we need only consider the equation of motion for the coordinate. We make the following substitutions in the linear part of the vector potential, $\mathbf{A}^L$
$$x=\left(\frac{2 a}{\pi}\right) k, \quad y=\left(\frac{\pi c}{2 a}\right)\left(t-t^{\prime}\right) .$$
18 This idea is due to Dirac in a slightly different context [45].

The equation of motion derived from (3.317) in this approximation with $\Delta m$ as before is then
$$m \dot{\mathbf{q}}(t)=\mathbf{p}_0-\Delta m \int_0^{\infty} \int_0^{\infty} x \chi_a^2\left(\frac{\pi x}{2 a}\right) \sin (x y) \dot{\mathbf{q}}\left(t-\frac{2 a y}{\pi c}\right) \mathrm{d} y \mathrm{~d} x$$
which is a linear integro-differential equation with a delay. We define a linear operator $\mathrm{L}$ by the relation
$$\mathrm{L}(\phi(t))=m \phi(t)+\Delta m I_a(\phi(t))$$
where
$$I_a(\phi(t))=\int_0^{\infty} \int_0^{\infty} x \chi_a^2\left(\frac{\pi x}{2 a}\right) \sin (x y) \phi\left(t-\frac{2 a y}{\pi c}\right) \mathrm{d} y \mathrm{~d} x$$
so that (3.356) is concisely expressed as
$$\mathrm{L}(\dot{\mathbf{q}}(t))=\mathbf{p}_0$$
If we can solve this equation, the orbit of the particle will again be (3.351).
The linear equation (3.356) can be solved by the method of characteristic functions [43]. The characteristic equation of $L$ is found directly by studying its action on the exponential function $e^{s t}$, where $s$ is a parameter that will determine the solutions, if any exist; in general $s$ will be a complex number. Consider then
$$\mathrm{L}\left(e^{s t}\right)=m e^{s t}+\Delta m I_a\left(e^{s t}\right)$$
The $y$ integration is elementary and there results
$$\mathrm{L}\left(e^{s t}\right)=e^{s t}\left[m+\Delta m \int_0^{\infty} \frac{x^2 \chi_a^2\left(\frac{\pi x}{2 a}\right)}{x^2+\left(\frac{2 a s}{\pi c}\right)^2} \mathrm{~d} x\right]$$

## 物理代写|电动力学代写electromagnetism代考|Classical Hamiltonian Electrodynamics Revisited

A fundamental result in the Hamiltonian formulation of mechanics is that the time evolution of the system can be regarded as the unfolding of a sequence of infinitesimal canonical transformations for which the Hamiltonian itself is the generator. Recall that if $G$ is the generator of such a transformation, the change in any dynamical variable $\Omega$, a function of the canonical variables, is given by the P.B. relation
$$\delta \Omega={\Omega, G}$$

If we choose $H \mathrm{~d} t$ as the generator, we get the following relations for the result of transformation of the basic phase space variables $\left(q(t)n, p(t)_n\right)$, \begin{aligned} & Q(t)_n=q(t)_n+\frac{\partial H}{\partial p_n} \mathrm{~d} t=q(t)_n+\dot{q}_n \mathrm{~d} t=q(t+\mathrm{d} t)_n \ & P(t)_n=p(t)_n-\frac{\partial H}{\partial q_n} \mathrm{~d} t=p(t)_n+\dot{p}_n \mathrm{~d} t=p(t+\mathrm{d} t)_n \end{aligned} corresponding to the ‘passive’ interpretation (LHS) in terms of transformation to new variables, and an ‘active’ interpretation (RHS) in terms of the time evolution of $q(t)_n, p(t)_n$. A transformation from old $\left(q_n, p_n\right)$ to new $\left(Q_n, P_n\right)$ variables is canonical if the P.B. relations are preserved by the transformation, \begin{aligned} & \left{q_n, q_m\right}=\left{p_n, p_m\right}=0, \quad\left{q_n, p_m\right}=\delta{n m} \ & \quad \rightarrow\left{Q_n, Q_m\right}=\left{P_n, P_m\right}=0, \quad\left{Q_n, P_m\right}=\delta_{n m} \end{aligned}
Now it is easily seen that if we take the Hamiltonian for a charge interacting with its own electromagnetic field, the above relations are not satisfied. The velocity in (3.316) is the gauge-invariant quantity defined by (3.244) which by (3.259) has components which no longer have vanishing P.B.s with each other. $\mathbf{q}$ also occurs in the infinitesimally time-translated field variables, and so the field and particle variables will have some non-zero P.B.s, contrary to the original assumptions. There is therefore a fundamental problem with the conventional classical Hamiltonian formulation which amounts to an incomplete specification of the set of dynamical variables; in other words, we need to identify additional variables such that we can make independent variations in the action integral. In the point particle limit the vector potential for the interacting system is proportional to the particle acceleration [42]. If such a Hamiltonian is to be derived from a Lagrangian, it too must involve the particle acceleration.

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|The Extended Charge Model

$$x=\left(\frac{2 a}{\pi}\right) k, \quad y=\left(\frac{\pi c}{2 a}\right)\left(t-t^{\prime}\right)$$
18 这个想法是由于狄拉克在稍微不同的背景下提出的 [45]。

$$m \dot{\mathbf{q}}(t)=\mathbf{p}_0-\Delta m \int_0^{\infty} \int_0^{\infty} x \chi_a^2\left(\frac{\pi x}{2 a}\right) \sin (x y) \dot{\mathbf{q}}\left(t-\frac{2 a y}{\pi c}\right) \mathrm{d} y \mathrm{~d} x$$

$$\mathrm{L}(\phi(t))=m \phi(t)+\Delta m I_a(\phi(t))$$

$$I_a(\phi(t))=\int_0^{\infty} \int_0^{\infty} x \chi_a^2\left(\frac{\pi x}{2 a}\right) \sin (x y) \phi\left(t-\frac{2 a y}{\pi c}\right) \mathrm{d} y \mathrm{~d} x$$

$$\mathrm{L}(\dot{\mathbf{q}}(t))=\mathbf{p}_0$$

$$\mathrm{L}\left(e^{s t}\right)=m e^{s t}+\Delta m I_a\left(e^{s t}\right)$$

$$\mathrm{L}\left(e^{s t}\right)=e^{s t}\left[m+\Delta m \int_0^{\infty} \frac{x^2 \chi_a^2\left(\frac{\pi x}{2 a}\right)}{x^2+\left(\frac{2 a s}{\pi c}\right)^2} \mathrm{~d} x\right]$$

## 物理代写|电动力学代写electromagnetism代考|Classical Hamiltonian Electrodynamics Revisited

$$\delta \Omega=\Omega, G$$

$$Q(t)_n=q(t)_n+\frac{\partial H}{\partial p_n} \mathrm{~d} t=q(t)_n+\dot{q}_n \mathrm{~d} t=q(t+\mathrm{d} t)_n \quad P(t)_n=p(t)_n-\frac{\partial H}{\partial q_n} \mathrm{~d} t=p(t)_n$$

## 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|Mass Renormalisation

The relationship between the mechanical mass $m$ and the observed mass $m^{\text {obs }}$ is the basis for mass renormalisation. We take account explicitly of the contribution to the mass of the charged particle due to the electromagnetic self interaction, so that the ‘structure’ parameter does not appear in the equations of motion. For the theory based on an extended charge distribution, this is achieved by extracting from the equation of motion a term simply proportional to $\dot{\mathbf{q}}(t)$ and identifying its coefficient as the mass correction due to the self-interaction. Clearly this is not possible if the point charge limit is taken first; historically, mass renormalisation was devised within the point charge model and had to proceed by quite different means [44].

We use integration by parts on the $t^{\prime}$ integration in (3.317), choosing the ‘ $\mathrm{d} v$ ‘ factor as $\sin \left[k c\left(t-t^{\prime}\right)\right]$. The boundary term is easily evaluated since it vanishes in the far past and the exponential and cosine terms simply give 1 at $t^{\prime}=t$. Hence, after the remaining elementary integration over $\mathbf{k}$, this contribution to the vector potential reduces to
$$u v \mid=\left(\frac{\Delta m}{e}\right) \dot{\mathbf{q}}(t) \text {. }$$
The integrated part does not simplify and can probably only be usefully evaluated in some approximation. The renormalised equation of motion for the coordinate $\mathbf{q}$ is therefore
\begin{aligned} & m^{\mathrm{obs}} \dot{\mathbf{q}}=\mathbf{p} \ & -\left(\frac{e C}{c}\right) \iint_{-\infty}^t\left(\frac{\chi_a^2(k)}{k^2}\right) \cos \left[k c\left(t-t^{\prime}\right)\right] \frac{\mathrm{d} \boldsymbol{\varepsilon}\left(\mathbf{k}, t^{\prime}\right)}{\mathrm{d} t^{\prime}} \mathrm{d} t^{\prime} \mathrm{d}^3 \mathbf{k} \end{aligned}
where we have put
$$m^{\mathrm{obs}}=m+\Delta m,$$
and
$$\boldsymbol{\varepsilon}\left(\mathbf{k}, t^{\prime}\right)=\left(\left(1+K_{\mathbf{q}}\left(\mathbf{k}, t^{\prime}\right)\right)(\mathbf{1}-\hat{\mathbf{k}} \hat{\mathbf{k}}) \cdot \dot{\mathbf{q}}\left(t^{\prime}\right)\right)$$

## 物理代写|电动力学代写electromagnetism代考|The Point Charge Model

An important limiting case of the calculation just described is the point charge limit with $\chi_0(k)=1$. In this limit we have $\mathbf{Q}_{t^{\prime} t}=0$, and the coefficient of $\Delta m$ is simply proportional to $\ddot{\mathbf{q}}$ [42]. Strictly speaking, we can no longer take the particle momentum to be constant in time, since the homogeneous field $\mathbf{A}(\mathbf{q}, t)_h$ contributes $^{17}$ also to (3.315), so we write the equation of motion for a point charge as
$$\ddot{\mathbf{q}}(t)-\omega_0 \dot{\mathbf{q}}(t)=-\left(\frac{\omega_0}{m}\right) \mathbf{p}(t)+\left(\frac{e}{m}\right) \mathbf{A}(\mathbf{q}, t)_h,$$
where
$$\omega_0=\left(\frac{c}{2 a}\right)\left(\frac{m}{\Delta m}\right)$$

Let
$$\dot{\mathbf{q}}(t)=\mathbf{z}(t),$$
so that
$$\mathbf{q}(t)=\int^t \mathbf{z}\left(t^{\prime}\right) \mathrm{d} t^{\prime}+\mathbf{q}0$$ where $\mathbf{q}_0$ is an integration constant. The solution for the velocity is $$\mathbf{z}(t)=e^{\omega_0 t}\left[e^{-\omega_0 t_0} \mathbf{z}\left(t_0\right)+\int{t_0}^t e^{-v \omega_0}\left(\left(\frac{\omega_0}{m}\right) \mathbf{p}(v)+\left(\frac{e}{m}\right) \mathbf{A}(\mathbf{q}, v)_h\right) \mathrm{d} v\right],$$
which in general shows runaway behaviour, $\mathbf{z}(+\infty)=\infty$; the omission of the free-field vector potential does not alter this conclusion. Since $\omega_0$ contains $e^{-2}$, the coordinate has an essential singularity at $e=0$, so this is a non-perturbative solution.

The situation can be ‘saved’ if we allow the specification of a particular value for the velocity $\mathbf{z}$ at the instant $t_0$ as an extra initial condition. This is contrary to the spirit of Hamilton’s equations which are a pair of coupled first-order differential equations to be solved with initial data $\mathbf{q}\left(t_0\right), \mathbf{p}\left(t_0\right)$. We chose $\mathbf{z}\left(t_0\right)$ so that ${ }^{18}$
$$e^{-\omega_0 t_0} \mathbf{z}\left(t_0\right)=\int_{t_0}^{\infty} e^{-\omega_0 v}\left(\frac{\omega_0}{m} \mathbf{p}(v)-\left(\frac{e}{m}\right) \mathbf{A}(\mathbf{q}, v)_h\right) \mathrm{d} v .$$
Substitution of this choice for $\mathbf{z}\left(t_0\right)$ in (3.352) yields the velocity as
\begin{aligned} \mathbf{z}(t) & =\int_t^{\infty} e^{\omega_0(t-v)}\left(\frac{\omega_0}{m} \mathbf{p}(v)-\left(\frac{e}{m}\right) \mathbf{A}(\mathbf{q}, v)_h\right) \mathrm{d} v \ & =\int_0^{\infty} e^{-\omega_0 \tau}\left(\frac{\omega_0}{m} \mathbf{p}(\tau+t)-\left(\frac{e}{m}\right) \mathbf{A}(\mathbf{q}, \tau+t)_h\right) \mathrm{d} \tau . \end{aligned}

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Mass Renormalisation

$$u v \mid=\left(\frac{\Delta m}{e}\right) \dot{\mathbf{q}}(t)$$

$$m^{\mathrm{obs}} \dot{\mathbf{q}}=\mathbf{p} \quad-\left(\frac{e C}{c}\right) \iint_{-\infty}^t\left(\frac{\chi_a^2(k)}{k^2}\right) \cos \left[k c\left(t-t^{\prime}\right)\right] \frac{\mathrm{d} \varepsilon\left(\mathbf{k}, t^{\prime}\right)}{\mathrm{d} t^{\prime}} \mathrm{d} t^{\prime} \mathrm{d}^3 \mathbf{k}$$

$$m^{\text {obs }}=m+\Delta m$$

$$\varepsilon\left(\mathbf{k}, t^{\prime}\right)=\left(\left(1+K_{\mathbf{q}}\left(\mathbf{k}, t^{\prime}\right)\right)(\mathbf{1}-\hat{\mathbf{k}} \hat{\mathbf{k}}) \cdot \dot{\mathbf{q}}\left(t^{\prime}\right)\right)$$

## 物理代写|电动力学代写electromagnetism代考|The Point Charge Model

$$\mathbf{z}(t)=\int_t^{\infty} e^{\omega_0(t-v)}\left(\frac{\omega_0}{m} \mathbf{p}(v)-\left(\frac{e}{m}\right) \mathbf{A}(\mathbf{q}, v)_h\right) \mathrm{d} v \quad=\int_0^{\infty} e^{-\omega_0 \tau}\left(\frac{\omega_0}{m} \mathbf{p}(\tau+t)-\left(\frac{e}{m}\right)\right.$$

## 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|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|Molecular Structure and Chemical Bonds

Having sorted out ideas about elements and compounds in terms of atoms and molecules, attention shifted to synthesis – the making of new compounds – and progress thereafter was rapid, especially in the chemistry of compounds containing the element carbon, what we call organic chemistry. It seems pertinent to recognise that the synthesis of new substances has been the principal experimental activity of chemists for more than 200 years. The number of known pure organic and inorganic substances has grown from a few hundred in 1800 to several hundred million today, with a doubling time of about 13 years that had been remarkably constant over the whole span of two centuries [16]. In order to keep track of the growth of experimental results, more and more transformations of compounds into other compounds, some kind of theoretical framework was needed. In the nineteenth century, the only known forces of attraction that might hold atoms together were the electromagnetic and gravitational forces, but these were seen to be absolutely useless for chemistry and so were given up in favour of a basic structural principle. The development of the interpretation of chemical experiments in terms of molecular structure was a highly original step for chemists to take since it had nothing to do with the then known physics based on the Newtonian ideal of the mathematical specification of the forces responsible for the observed motions of matter. It was one of the most far-reaching steps ever taken in science. G. N. Lewis once wrote [17]

No generalization of science, even if we include those capable of exact mathematical statement, has ever achieved a greater success in assembling in a simple way a multitude of heterogeneous observations than this group of ideas which we call structural theory.

In the 1850 s the idea of atoms having autonomous valencies had developed, and this led Frankland to his conception of a chemical bond [18], [19]. He wrote [20]

By the term bond, I intend merely to give a more concrete expression to what has received various names from different chemists, such as atomicity, an atomic power, and an equivalence. A monad is represented as an element having one bond, a dyad as an element having two bonds, etc. It is scarcely necessary to remark by this term I do not intend to convey the idea of a material connection between the elements of a compound, the bonds actually holding the atoms of a chemical compound being, as regards their nature much more like those which connect the members of our solar system.

The idea of representing a bond as a straight line joining atomic symbols is probably due to Crum Brown. Frankland, with due acknowledgement, adopted Crum Brown’s representation which put circles round the atom symbols, but by 1867 the circles had been dropped and more or less modern chemical notation became widespread.

## 物理代写|电动力学代写electromagnetism代考|Atomic Structure and Chemistry

The first tentative steps towards a theory of the chemical bond followed Thomson’s discovery of the electron in the late 1890 s and his claim that the electron was a universal constituent of atoms. There were several independent measurements of the charge/mass ratio of cathode rays contemporary with Thomson’s announcement in 1897; crucially, however, he was the first to measure the charge on the electron in an experiment with his student Rutherford using the Wilson cloud chamber device invented in Cambridge [28]. Thomson initially favoured a uniform distribution of positive charge inside an ‘atomic sphere’ with solely negatively charged electrons – the so-called ‘plum pudding model’ of an atom. He had found that the mass of the electron was about 1/1700 of the mass of the hydrogen atom, and since he assumed the positive charge distribution contributed no mass to the atom, this implied that atoms must contain thousands of electrons [29].

In his Romanes Lecture (1902), Lodge suggested that chemical combination must be the result of the pairing of oppositely charged ions, for (quoted in Stranges, [30])
It becomes a reasonable hypothesis to surmise that the whole of the atom may be built up of positive and negative electrons interleaved together, and of nothing else; an active or charged ion having one negative electron in excess or defect, but the neutral atom having an exact number of pairs.

The notion of positive and negative electrons was an early ‘solution’ to the evident problem of the electroneutrality of the atom, and also its stability since a positive charge is needed to keep the electrons together [31]. Earnshaw’s theorem in classical electrostatics implies that a collection of charges interacting purely through Coulomb’s inverse square law cannot have an equilibrium configuration, and so must be moving [32]; on the other hand, classical electrodynamics implies that moving charges must generally lose energy by radiation. ${ }^5$

In 1906, Thomson showed that the number of electrons in an atom is of similar magnitude to the relative atomic mass of the corresponding substance, and that the mass of the carriers of positive electricity could not be small compared to the total mass of the atomic electrons. These conclusions came from three independent theoretical results: firstly, a formula he derived for the refractive index of a monatomic gas; secondly, his formula for the absorption of $\beta$-particles in matter; and thirdly, the cross section, ${ }^6 \sigma$, for the scattering of X-rays by gases [33]:
$$\sigma=\frac{8 \pi}{3}\left(\frac{1}{4 \pi \varepsilon_0} \frac{e^2}{m_e c^2}\right)^2$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Atomic Structure and Chemistry

1906年，汤姆逊证明原子中的电子数与相应物质的相对原子质量具有相似的数量级，正电载流子的质量与原子电子的总质量相比不能小. 这些结论来自三个独立的理论结果：第一，他推导出的单原子气体折射率公式；其次，他的吸收公式b-物质中的粒子；第三，横截面，6p，对于气体对 X 射线的散射 [33]：

p=8π3(14π电子0这是2米这是C2)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|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|The Origins of Chemistry

Chemistry is concerned with the composition and properties of matter, and with the transformations of matter that can occur spontaneously or under the action of heat, radiation or other sources of energy. It emerged as a science in recognisably modern form at the end of the eighteenth century. From the results of chemical experiments, the chemist singles out a particular class of materials that have characteristic and invariant properties. This is done through the use of the classical separation procedures crystallisation, distillation, sublimation and so on – that involve a phase transition. Such materials are called pure substances and may be of two kinds: elements and compounds. A pure substance is an idealisation since perfect purity is never achieved in practice.

Formally, elements may be defined as substances which have not been converted either by the action of heat, radiation or chemical reaction with other substances, or small electrical voltages, into any simpler substance. Compounds are formed from the chemical combination of the elements, and have properties that are invariably different from the properties of the constituent elements; they are also homogeneous. These statements derive from antiquity; thus from Aristotle [1]:

An element, we take it, is a body into which other bodies may be analysed, present in them potentially or in actuality (which of these, is still disputable), and not itself divisible into bodies different in form.

Similar statements can be found in Boyle and in Lomonosov, for example; they gain significance when the notion of ‘simpler’ substance is explicated. A substantial account of the history and philosophy of these ideas can be found in a recent Handbook [2].

In the seventeenth century, a scientific attitude emerged that is recognisably ‘modern’; it aimed to describe the physical aspects of the natural world through analytical procedures of classification and systematisation in order to find explanations of natural phenomena in purely naturalistic terms [3]. The underlying mechanical philosophy ${ }^1$ was grounded firmly in a picture of a world of physical objects endowed with well-defined fixed properties that can be described in mathematical terms – shape, size, position, number and so on. It can be seen as a return to the mathematical ideals of the Pythagoreans and of Plato, and a renewal of the ideas of the early Greek atomists, for example Democritus. There was quite explicitly a movement against the still prevailing Aristotelian system of the scholastic philosophers which was closely connected with the religious authorities. The prime movers of this revolution were Galileo and Descartes; both sought a quantitative approach to physics through the use of mathematics applied to mechanical or corpuscular models that would replace a philosophical tradition that had originated in antiquity.

## 物理代写|电动力学代写electromagnetism代考|Stoichiometry and Atoms

Measurements of changes in weight – stoichiometry $^4$ – are a characteristic feature of the quantitative study of chemical reactions; such measurements reveal one of the most important facts about the chemical combination of substances, namely that it generally involves fixed and definite proportions by weight of the reacting substances. These changes in weight are found to be subject to two fundamental laws:
Law of conservation of mass: (A. Lavoisier, 1789)
L1 No change in the total weight of all the substances taking part in any chemical process has ever been observed in a closed system.
Law of definite proportions: (J. L. Proust, 1799)
L2 A particular chemical compound always contains the same elements united together in the same proportions by weight.

The chemical equivalent (or equivalent weight) of an element is the number of parts by weight of it which combines with, or replaces eight parts by weight of oxygen or the chemical equivalent of any other element; the choice of eight parts by weight of oxygen is purely conventional. By direct chemical reaction and the careful weighing of reagents and products, one can determine accurate equivalents directly. Depending on the physical conditions under which reactions are carried out, one may find significantly different equivalent weights for the same element corresponding to the formation of several chemically distinct pure substances. These findings are summarised in the laws of chemical combination [10]:
Law of multiple proportions: (J. Dalton, 1803)
L3 If two elements combine to form more than one compound the different weights of one which combine with the same weight of the other are in the ratio of simple whole numbers.

Let $E[A, n]$ be the equivalent weight of element $A$ in compound $n[11]$; if we consider the different binary compounds formed by elements $A$ and $B$, the Law of Multiple Proportions implies
$$\frac{E[A, i]}{E[B, i]}=\omega_{i j} \frac{E[A, j]}{E[B, j]},$$
where $\omega_{i j}$ is a simple fraction.

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Stoichiometry and Atoms

L1 在任何化学过程中从末观察到参与任何化学过程的所有物质的总重量没有变化封闭系统。

L2一种特定的化合物总是包含以相同重量比例结合在一起的相同元素。

$L 3$ 如果两种元素结合形成一种以上的化合物，则一种元素的不同重量与另一种元素的相同重量组合成简 单整数之比。

$$\frac{E[A, i]}{E[B, i]}=\omega_{i j} \frac{E[A, j]}{E[B, j]}$$

## 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|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
• (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
• (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
• (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。