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## 物理代写|电动力学代考Electrodynamics代写|Paramagnetism

What happens if there is a permanent intrinsic moment,
$$\boldsymbol{\mu}_0 \neq \mathbf{0} \text { ? }$$

This situation is analogous to that of permanent electric dipole moments, discussed in Section 5.4. Due to thermal motion, the average magnetization will be zero if there is no external field. In the presence of a magnetic field, we obtain the thermal averaged magnetic moment from (5.64) by replacing $d$ by $\mu_0$ and E by $\mathbf{B}$. The obvious high temperature limit, from (5.68), is
$$\left\langle\boldsymbol{\mu}0\right\rangle_T=\frac{\mu_0^2}{3 k T} \mathbf{B}$$ corresponding to the magnetization $$\mathbf{M}=\frac{n \mu_0^2}{3 k T} \mathbf{B} .$$ This is appropriate to the weak field circumstance $$\mu_0 B \ll k T \text {. }$$ Inasmuch as the typical magnitudes of $\mu_0 / d$ are of order $v / c \sim 10^{-2}-10^{-3}$, the upper limit to $B$, at room temperature, for $(6.42)$ to be valid is in the range of millions of gauss, or hundreds of Teslas. Note that unlike in diamagnetism, the magnetization here is parallel to the magnetic field. The permeability is [cf. $(6.38)$ ] $$\mu=\frac{1}{1-4 \pi n \frac{\mu_0^2}{3 k T}} \approx 1+4 \pi n \frac{\mu_0^2}{3 k T}>1, \quad \chi_m=n \frac{\mu_0^2}{3 k T},$$ since, again, the magnetization is small. Substances with positive magnetic susceptibilities are called paramagnetic. For this class of materials, the permeability is greater than one. The simple models indicate that the ratio of paramagnetic to diamagnetic susceptibilities is of the order $$\frac{\chi{m, \text { para }}}{\chi_{m, \text { dia }}} \sim \frac{m v^2}{k T} \sim 100 \text { at room temperature, }$$
where $m v^2$ is related to the magnitude of energies in the atom. The estimate in (6.45) is in general agreement with the observation that paramagnetic gaseous oxygen at standard pressure and room temperature has a positive susceptibility about one fifth the susceptibility of water, although the molecular density of the oxygen is less than a thousandth of that of water. The susceptibilities of paramagnetic substances are still so small compared with unity (for liquid oxygen, $\chi_m=3 \times 10^{-4}$ ) that the approximation of neglecting the distinction between $\mathbf{B}$ and $\mathbf{H}$ in (6.44) is well justified. The inverse dependence on temperature displayed there was discovered experimentally by Pierre Curie (1859-1906).

## 物理代写|电动力学代考Electrodynamics代写|Ferromagnetism

The history of magnetism did not begin with the phenomena of paramagnetism and diamagnetism, which were first recognized by Faraday in 1845 . The ancients were familiar with the remarkable properties of Magnesian stone, the iron oxide $\mathrm{Fe}3 \mathrm{O}_4$. The term ferromagnetism refers to the property of such substances, primarily members of the iron group, of exhibiting permanent magnetization. A simple model of this effect was introduced by Pierre Weiss (1865-1940), who effectively postulated that the driving magnetic field within ferromagnets is not $(6.50)$, but rather $$\mathbf{B}{\text {driving }}=\mathbf{H}+\lambda \mathbf{M} \text {, }$$
where $\lambda \gg 1$. In terms of $\mathbf{B}_{\text {driving }}$ we wish to calculate the thermal average of the intrinsic magnetic moment, $\left\langle\boldsymbol{\mu}_0\right\rangle_T$. Rather than use a classical distribution (but see Problem 6.3), it is simpler and more accurate quantum mechanically to suppose that the atomic magnetic moment $\boldsymbol{\mu}0$ is either lined up parallel or anti-parallel to $\mathbf{B}{\text {driving }}$, which defines the $z$ axis. Since the interaction energies, for the two possibilities, are
$$-\mu_0 \cdot \mathbf{B}{\text {driving }}=\mp \mu_0 B{\text {driving }},$$
the Boltzmann weighting of states yields
$$\left\langle\mu_{0 z}\right\rangle_T=\frac{\mu_0 e^x-\mu_0 e^{-x}}{e^x+e^{-x}}=\mu_0 \tanh x,$$
with
$$x=\frac{\mu_0}{k T}(H+\lambda M) .$$
The resulting magnetization has magnitude
$$M=n \mu_0 \tanh \frac{\mu_0}{k T}(H+\lambda M) .$$
The possible existence of a magnetization in the absence of the field $H$ is implied by the equation
$$\frac{M}{n \mu_0}=\tanh \left(\frac{T_c}{T} \frac{M}{n \mu_0}\right)$$
in which
$$T_c \equiv \frac{n \mu_0^2}{k} \lambda$$

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Paramagnetism

$$\boldsymbol{\mu}_0 \neq \mathbf{0} \text { ? }$$

$$\left\langle\boldsymbol{\mu}0\right\rangle_T=\frac{\mu_0^2}{3 k T} \mathbf{B}$$对应磁化强度$$\mathbf{M}=\frac{n \mu_0^2}{3 k T} \mathbf{B} .$$这适用于弱场情况$$\mu_0 B \ll k T \text {. }$$由于$\mu_0 / d$的典型数量级为$v / c \sim 10^{-2}-10^{-3}$，在室温下，$(6.42)$有效的上限为$B$，在数百万高斯或数百特斯拉的范围内。注意，不像抗磁性，这里的磁化平行于磁场。磁导率为[cf. $(6.38)$] $$\mu=\frac{1}{1-4 \pi n \frac{\mu_0^2}{3 k T}} \approx 1+4 \pi n \frac{\mu_0^2}{3 k T}>1, \quad \chi_m=n \frac{\mu_0^2}{3 k T},$$，因为磁化强度也很小。具有正磁化率的物质称为顺磁性的。对于这类材料，磁导率大于1。简单模型表明，顺磁磁化率与抗磁磁化率之比约为$$\frac{\chi{m, \text { para }}}{\chi_{m, \text { dia }}} \sim \frac{m v^2}{k T} \sim 100 \text { at room temperature, }$$

## 物理代写|电动力学代考Electrodynamics代写|Ferromagnetism

$$x=\frac{\mu_0}{k T}(H+\lambda M) .$$

$$M=n \mu_0 \tanh \frac{\mu_0}{k T}(H+\lambda M) .$$

$$\frac{M}{n \mu_0}=\tanh \left(\frac{T_c}{T} \frac{M}{n \mu_0}\right)$$

$$T_c \equiv \frac{n \mu_0^2}{k} \lambda$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Electrodynamics, 物理代写, 电动力学

## avatest™帮您通过考试

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## 物理代写|电动力学代考Electrodynamics代写|Conductivity

We start by considering a simple model of a metal in which the current is linearly related to the electric field. The model is to be considered as suggestive only but it does lead to a qualitative understanding of the important phenomena of conduction. Of course, an accurate description requires quantum mechanics.
First consider a free electric charge (an electron) moving under the influence of an external electric field, and subject to collisions with the atoms of the substance. The electric field accelerates the charge, and the collisions slow it down. Our model represents the effects of the collisions by a frictional force that is proportional-and opposed-to the velocity. The equation of motion for the particle, having charge $e$ and mass $m$, is
$$m \frac{d}{d t} \mathbf{v}(t)=-m \gamma \mathbf{v}(t)+e \mathbf{E}(t), \quad \gamma>0$$
or
$$\frac{d}{d t} \mathbf{v}(t)=-\gamma \mathbf{v}(t)+\frac{e}{m} \mathbf{E}(t) .$$
(The variation of the electric field with position is ignored here-the velocities of interest are of very small magnitude compared with $c$.) The frictional constant $\gamma$ is given a physical interpretation by considering the situation for $\mathbf{E}=\mathbf{0}$ :
$$\frac{d}{d t} \mathbf{v}(t)=-\gamma \mathbf{v}(t), \quad \mathbf{v}(t)=\mathbf{v}_0 e^{-\gamma t}$$
any initial velocity decreases exponentially in time, due to collisions with atoms, with $1 / \gamma$ supplying the characteristic decay time. The general solution to (5.2) is found by first rewriting it as
$$\frac{d}{d t}\left[e^{\gamma t} \mathbf{v}(t)\right]=\frac{e}{m} e^{\gamma t} \mathbf{E}(t),$$
and then integrating from $t^{\prime}=-\infty$ (a time before any field has been applied), to $t$ (the time of observation),
$$\mathbf{v}(t)=\frac{e}{m} \int_{-\infty}^t d t^{\prime} e^{-\gamma\left(t-t^{\prime}\right)} \mathbf{E}\left(t^{\prime}\right)$$

## 物理代写|电动力学代考Electrodynamics代写|Dielectric Constant

We now modify the above model in order to discuss bound charge by including an additional binding force term in (5.1). We will take as the simplest model of such binding a harmonic oscillator force, which turns out, for the most part, to give qualitatively correct results. That is, we will adopt, taking the origin to be the center of the force,
$$m \frac{d}{d t} \mathbf{v}=-m \omega_0^2 \mathbf{r}-m \gamma \mathbf{v}+e \mathbf{E}, \quad \mathbf{v}=\frac{d \mathbf{r}}{d t},$$
as the new equation of motion. Here $\omega_0$ is the natural (angular) frequency of the electron bound in the atom, while $\gamma$ is a damping constant, primarily due to electromagnetic radiation. (More about this in Chapter 35.)

For a harmonic time dependence of the driving electric field, (5.9), the above force equation becomes
$$\frac{d^2}{d t^2} \mathbf{r}+\omega_0^2 \mathbf{r}+\gamma \frac{d}{d t} \mathbf{r}=\frac{e}{m} \operatorname{Re}\left(\mathbf{E}(\omega) e^{-i \omega t}\right) .$$
This implies that the steady-state solution for the position vector will also exhibit harmonic time variation, that is,
$$\mathbf{r}(t)=\frac{e}{m} \operatorname{Re}\left[\frac{\mathbf{E}(\omega) e^{-i \omega t}}{-\omega^2+\omega_0^2-i \gamma \omega}\right]$$

Under the usual circumstance of $\gamma \ll \omega_0$, the amplitude of the induced oscillation becomes very large for $\omega=\omega_0$, the condition of resonance.

It is now immediate to calculate the polarization (4.35) in terms of the induced electric dipole moment and the density of bound electrons, $n_b$,
$$\mathbf{P}=n_b e \mathbf{r},$$
or, explicitly in terms of the electric field,
\begin{aligned} \mathbf{P}(t) & =\frac{n_b e^2}{m} \operatorname{Re}\left[\frac{\mathbf{E}(\omega) e^{-i \omega t}}{-\omega^2+\omega_0^2-i \gamma \omega}\right] \ & =\operatorname{Re}\left[\chi_e(\omega) \mathbf{E}(\omega) e^{-i \omega t}\right], \end{aligned}
where $\chi_e$ is the (frequency-dependent) electric susceptibility,
$$\chi_e(\omega)=\frac{n_b e^2}{m} \frac{1}{-\omega^2-i \omega \gamma+\omega_0^2},$$
which satisfies
$$\chi_e(\omega)=\chi_e(-\omega)^*$$

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Conductivity

$$m \frac{d}{d t} \mathbf{v}(t)=-m \gamma \mathbf{v}(t)+e \mathbf{E}(t), \quad \gamma>0$$

$$\frac{d}{d t} \mathbf{v}(t)=-\gamma \mathbf{v}(t)+\frac{e}{m} \mathbf{E}(t) .$$
(此处忽略电场随位置的变化——与$c$相比，感兴趣的速度是非常小的量级。)考虑$\mathbf{E}=\mathbf{0}$的情况，给出摩擦常数$\gamma$的物理解释:
$$\frac{d}{d t} \mathbf{v}(t)=-\gamma \mathbf{v}(t), \quad \mathbf{v}(t)=\mathbf{v}0 e^{-\gamma t}$$ 由于与原子的碰撞，任何初始速度随时间呈指数递减，$1 / \gamma$提供特征衰变时间。(5.2)的通解可以先将其重写为 $$\frac{d}{d t}\left[e^{\gamma t} \mathbf{v}(t)\right]=\frac{e}{m} e^{\gamma t} \mathbf{E}(t),$$ 然后从$t^{\prime}=-\infty$(任何字段应用之前的时间)到$t$(观察时间)进行积分， $$\mathbf{v}(t)=\frac{e}{m} \int{-\infty}^t d t^{\prime} e^{-\gamma\left(t-t^{\prime}\right)} \mathbf{E}\left(t^{\prime}\right)$$

## 物理代写|电动力学代考Electrodynamics代写|Dielectric Constant

$$m \frac{d}{d t} \mathbf{v}=-m \omega_0^2 \mathbf{r}-m \gamma \mathbf{v}+e \mathbf{E}, \quad \mathbf{v}=\frac{d \mathbf{r}}{d t},$$

$$\frac{d^2}{d t^2} \mathbf{r}+\omega_0^2 \mathbf{r}+\gamma \frac{d}{d t} \mathbf{r}=\frac{e}{m} \operatorname{Re}\left(\mathbf{E}(\omega) e^{-i \omega t}\right) .$$

$$\mathbf{r}(t)=\frac{e}{m} \operatorname{Re}\left[\frac{\mathbf{E}(\omega) e^{-i \omega t}}{-\omega^2+\omega_0^2-i \gamma \omega}\right]$$

$$\mathbf{P}=n_b e \mathbf{r},$$

\begin{aligned} \mathbf{P}(t) & =\frac{n_b e^2}{m} \operatorname{Re}\left[\frac{\mathbf{E}(\omega) e^{-i \omega t}}{-\omega^2+\omega_0^2-i \gamma \omega}\right] \ & =\operatorname{Re}\left[\chi_e(\omega) \mathbf{E}(\omega) e^{-i \omega t}\right], \end{aligned}

$$\chi_e(\omega)=\frac{n_b e^2}{m} \frac{1}{-\omega^2-i \omega \gamma+\omega_0^2},$$

$$\chi_e(\omega)=\chi_e(-\omega)^*$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Electrodynamics, 物理代写, 电动力学

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 物理代写|电动力学代考Electrodynamics代写|Electrostatics

Our intention is to move toward the general picture as quickly as possible, starting with a review of electrostatics. We take for granted the phenomenology of electric charge, including the Coulomb law of force between charges of dimensions that are small in comparison with their separation. This is expressed by the interaction energy, $E$, of a system of such charges in otherwise empty space, a vacuum:
$$E=\frac{1}{2} \sum_{\substack{a, b \ a \neq b}} \frac{e_a e_b}{r_{a b}},$$
where $e_a$ is the charge of the ath particle while
$$r_{a b}=\left|\mathbf{r}_a-\mathbf{r}_b\right|$$
is the separation between the ath and $b$ th particles. (Throughout this book we use the Gaussian system of units. Connection with the SI units will be given in Appendix A.) As we shall see, this starting point, the Coulomb energy (1.1), summarizes all the experimental facts of electrostatics. The energy of interaction of an individual charge with the rest of the system can be emphasized by rewriting (1.1) as
$$E=\frac{1}{2} \sum_a e_a \sum_{b \neq a} \frac{e_b}{r_{a b}}=\frac{1}{2} \sum_a e_a \phi_a,$$
where we have introduced the electrostatic potential at the location of the ath charge that is due to all the other charges,
$$\phi_a=\sum_{b \neq a} \frac{e_b}{r_{a b}}$$

## 物理代写|电动力学代考Electrodynamics代写|Inference of Maxwell’s Equations

We introduce time dependence in the simplest way by assuming that all charges are in uniform motion with a common velocity $\mathbf{v}$ as produced by transforming a static arrangement of charges to a coordinate system moving with velocity -v. (We insist that the same physics applies in the two situations.) At first we will take $|\mathbf{v}|$ to be very small in comparison with a critical speed $c$, which will be identified with the speed of light. To catch up with the moving charges, one would have to move with their velocity, $\mathbf{v}$. Accordingly, the time derivative in the co-moving coordinate system, in which the charges are at rest, is the sum of explicit time dependent and coordinate dependent contributions,
$$\frac{d}{d t}=\frac{\partial}{\partial t}+\mathbf{v} \cdot \nabla$$
so, in going from the static system to the uniformly moving system, we make the replacement
$$\frac{\partial}{\partial t} \rightarrow \frac{d}{d t}=\frac{\partial}{\partial t}+\mathbf{v} \cdot \nabla$$
The equation for the eonstancy of the charge density in (1.39) becomes, in the moving system
$$0=\frac{\partial \rho}{\partial t} \rightarrow \frac{d \rho}{d t}=\frac{\partial \rho}{\partial t}+\mathbf{v} \cdot \nabla \rho$$
or, since $\mathbf{v}$ is constant,
$$\frac{\partial \rho}{\partial t}+\nabla \cdot(\mathbf{v} \rho)=0$$
We recognize here a particular example of the charge flux vector or the (electric) current density $\mathbf{j}$,
$$\mathbf{j}=\rho \mathbf{v}$$

The relation between charge density and current density,
$$\frac{\partial}{\partial t} \rho(\mathbf{r}, t)+\nabla \cdot \mathbf{j}(\mathbf{r}, t)=0$$
is the general statement of the conservation of charge. Conservation demands that the rate of decrease of the charge within an arbitrary volume $V$ must equal the rate at which the charge flows out of the bounding surface $S$, that is
$$-\frac{d}{d t} \int_V(d \mathbf{r}) \rho(\mathbf{r}, t)=\oint_S d \mathbf{S} \cdot \mathbf{j}(\mathbf{r}, t)=\int_V(d \mathbf{r}) \boldsymbol{\nabla} \cdot \mathbf{j}(\mathbf{r}, t) .$$
Since $V$ is arbitrary, the local conservation law, (1.45), follows. We also note that the expression for the current density, (1.44), continues to be valid even when $\mathbf{v}$ is dependent upon position, $\mathbf{v} \rightarrow \mathbf{v}(\mathbf{r}, t)$. (See Problem 1.4.)

We can perform a similar transformation on the equation for the electric field $\partial \mathbf{E} / \partial t=0$; namely,
$$\mathbf{0}=\frac{d}{d t} \mathbf{E}=\frac{\partial \mathbf{E}}{\partial t}+(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E} .$$
Making use of a vector identity, together with (1.26) and (1.44), ( $\mathbf{v}$ is constant),
\begin{aligned} \boldsymbol{\nabla} \times(\mathbf{v} \times \mathbf{E}) & =\mathbf{v}(\boldsymbol{\nabla} \cdot \mathbf{E})-(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E} \ & =\mathbf{v} 4 \pi \rho-(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E} \ & =4 \pi \mathbf{j}-(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E}, \end{aligned}
we find an equation relating $\mathbf{E}$ to the current density,
$$0=\frac{\partial \mathbf{E}}{\partial t}+4 \pi \mathbf{j}-\nabla \times(\mathbf{v} \times \mathbf{E})$$

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Electrostatics

$$E=\frac{1}{2} \sum_{\substack{a, b \ a \neq b}} \frac{e_a e_b}{r_{a b}},$$

$$r_{a b}=\left|\mathbf{r}a-\mathbf{r}_b\right|$$ 是ath和$b$粒子之间的距离。(在本书中，我们使用高斯单位制。与国际单位制单位的连接将在附录a中给出。我们将看到，库仑能(1.1)这个起点概括了静电学的所有实验事实。单个电荷与系统其余部分的相互作用能可以通过重写式(1.1)来强调 $$E=\frac{1}{2} \sum_a e_a \sum{b \neq a} \frac{e_b}{r_{a b}}=\frac{1}{2} \sum_a e_a \phi_a,$$

$$\phi_a=\sum_{b \neq a} \frac{e_b}{r_{a b}}$$

## 物理代写|电动力学代考Electrodynamics代写|Inference of Maxwell’s Equations

$$\frac{d}{d t}=\frac{\partial}{\partial t}+\mathbf{v} \cdot \nabla$$

$$\frac{\partial}{\partial t} \rightarrow \frac{d}{d t}=\frac{\partial}{\partial t}+\mathbf{v} \cdot \nabla$$

$$0=\frac{\partial \rho}{\partial t} \rightarrow \frac{d \rho}{d t}=\frac{\partial \rho}{\partial t}+\mathbf{v} \cdot \nabla \rho$$

$$\frac{\partial \rho}{\partial t}+\nabla \cdot(\mathbf{v} \rho)=0$$

$$\mathbf{j}=\rho \mathbf{v}$$

$$\frac{\partial}{\partial t} \rho(\mathbf{r}, t)+\nabla \cdot \mathbf{j}(\mathbf{r}, t)=0$$

$$-\frac{d}{d t} \int_V(d \mathbf{r}) \rho(\mathbf{r}, t)=\oint_S d \mathbf{S} \cdot \mathbf{j}(\mathbf{r}, t)=\int_V(d \mathbf{r}) \boldsymbol{\nabla} \cdot \mathbf{j}(\mathbf{r}, t) .$$

$$\mathbf{0}=\frac{d}{d t} \mathbf{E}=\frac{\partial \mathbf{E}}{\partial t}+(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E} .$$

\begin{aligned} \boldsymbol{\nabla} \times(\mathbf{v} \times \mathbf{E}) & =\mathbf{v}(\boldsymbol{\nabla} \cdot \mathbf{E})-(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E} \ & =\mathbf{v} 4 \pi \rho-(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E} \ & =4 \pi \mathbf{j}-(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{E}, \end{aligned}

$$0=\frac{\partial \mathbf{E}}{\partial t}+4 \pi \mathbf{j}-\nabla \times(\mathbf{v} \times \mathbf{E})$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Electrodynamics, 物理代写, 电动力学

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## 物理代写|电动力学代考Electrodynamics代写|Fourier Analysis

Every periodic function $f(x+L)=f(x)$ has a Fourier series representation
$$f(x)=\sum_{m=-\infty}^{\infty} \hat{f}_m e^{i 2 \pi m x / L} .$$
The Fourier expansion coefficients in (1.123) are given by
$$\hat{f}m=\frac{1}{L} \int_0^L d x f(x) e^{-i 2 \pi m x / L} .$$ For non-periodic functions, the sum over integers in (1.123) becomes an integral over the real line. When the integral converges, we find the Fourier transform pair: $$\begin{gathered} f(x)=\frac{1}{2 \pi} \int{-\infty}^{\infty} d k \hat{f}(k) e^{i k x} \ \hat{f}(k)=\int_{-\infty}^{\infty} d x f(x) e^{-i k x} . \end{gathered}$$
If $f(x)$ happens to be a real function, it follows from these definitions that
$$f(x)=f^*(x) \Rightarrow \hat{f}(k)=\hat{f}(-k)$$
In the time domain, it is conventional to write
$$g(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega \hat{g}(\omega) e^{-i \omega t} \quad \hat{g}(\omega)=\int_{-\infty}^{\infty} d t g(t) e^{i \omega t}$$
Thus, our convention for the Fourier transform and inverse Fourier transform of a function $f(\mathbf{r}, t)$ of time and all three spatial variables is
$$\begin{gathered} f(\mathbf{r}, t)=\frac{1}{(2 \pi)^4} \int d^3 k \int_{-\infty}^{\infty} d \omega \hat{f}(\mathbf{k} \mid \omega) e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)} \ \hat{f}(\mathbf{k} \mid \omega)=\int d^3 r \int_{-\infty}^{\infty} d t f(\mathbf{r}, t) e^{-i(\mathbf{k} \cdot \mathbf{r}-\omega t)} . \end{gathered}$$

## 物理代写|电动力学代考Electrodynamics代写|The Convolution Theorem

A function $h(t)$ is called the convolution of $f(t)$ and $g(t)$ if
$$h(t)=\int_{-\infty}^{\infty} d t^{\prime} f\left(t-t^{\prime}\right) g\left(t^{\prime}\right) .$$
The convolution theorem states that the Fourier transforms $\hat{h}(\omega), \hat{f}(\omega)$, and $\hat{g}(\omega)$ are related by
$$\hat{h}(\omega)=\hat{f}(\omega) \hat{g}(\omega)$$
We prove this assertion by using the left side of (1.128) to rewrite (1.132) as
$$h(t)=\int_{-\infty}^{\infty} d t^{\prime}\left[\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega \hat{f}(\omega) e^{-i \omega\left(t-t^{\prime}\right)}\right]\left[\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega^{\prime} \hat{g}\left(\omega^{\prime}\right) e^{-i \omega^{\prime} t^{\prime}}\right] .$$
Rearranging terms gives
$$h(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega e^{-i \omega t} \hat{f}(\omega) \int_{-\infty}^{\infty} d \omega^{\prime} \hat{g}\left(\omega^{\prime}\right)\left[\frac{1}{2 \pi} \int_{-\infty}^{\infty} d t^{\prime} e^{-i\left(\omega^{\prime}-\omega\right) t^{\prime}}\right] .$$
The identity (1.101) identifies the quantity in square brackets as the delta function $\delta\left(\omega-\omega^{\prime}\right)$. Therefore,
$$h(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega e^{-i \omega t} \hat{f}(\omega) \hat{g}(\omega)$$

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Fourier Analysis

$$f(x)=\sum_{m=-\infty}^{\infty} \hat{f}m e^{i 2 \pi m x / L} .$$ (1.123)的傅里叶展开系数由 $$\hat{f}m=\frac{1}{L} \int_0^L d x f(x) e^{-i 2 \pi m x / L} .$$对于非周期函数，式(1.123)中整数的和变成实线上的积分。当积分收敛时，我们找到傅里叶变换对:$$\begin{gathered} f(x)=\frac{1}{2 \pi} \int{-\infty}^{\infty} d k \hat{f}(k) e^{i k x} \ \hat{f}(k)=\int{-\infty}^{\infty} d x f(x) e^{-i k x} . \end{gathered}$$

$$f(x)=f^*(x) \Rightarrow \hat{f}(k)=\hat{f}(-k)$$

$$g(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega \hat{g}(\omega) e^{-i \omega t} \quad \hat{g}(\omega)=\int_{-\infty}^{\infty} d t g(t) e^{i \omega t}$$

$$\begin{gathered} f(\mathbf{r}, t)=\frac{1}{(2 \pi)^4} \int d^3 k \int_{-\infty}^{\infty} d \omega \hat{f}(\mathbf{k} \mid \omega) e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)} \ \hat{f}(\mathbf{k} \mid \omega)=\int d^3 r \int_{-\infty}^{\infty} d t f(\mathbf{r}, t) e^{-i(\mathbf{k} \cdot \mathbf{r}-\omega t)} . \end{gathered}$$

## 物理代写|电动力学代考Electrodynamics代写|The Convolution Theorem

$$h(t)=\int_{-\infty}^{\infty} d t^{\prime} f\left(t-t^{\prime}\right) g\left(t^{\prime}\right) .$$

$$\hat{h}(\omega)=\hat{f}(\omega) \hat{g}(\omega)$$

$$h(t)=\int_{-\infty}^{\infty} d t^{\prime}\left[\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega \hat{f}(\omega) e^{-i \omega\left(t-t^{\prime}\right)}\right]\left[\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega^{\prime} \hat{g}\left(\omega^{\prime}\right) e^{-i \omega^{\prime} t^{\prime}}\right] .$$

$$h(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega e^{-i \omega t} \hat{f}(\omega) \int_{-\infty}^{\infty} d \omega^{\prime} \hat{g}\left(\omega^{\prime}\right)\left[\frac{1}{2 \pi} \int_{-\infty}^{\infty} d t^{\prime} e^{-i\left(\omega^{\prime}-\omega\right) t^{\prime}}\right] .$$

$$h(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \omega e^{-i \omega t} \hat{f}(\omega) \hat{g}(\omega)$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Electrodynamics, 物理代写, 电动力学

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

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## 物理代写|电动力学代考Electrodynamics代写|Stokes’ Theorem

Stokes’ theorem applies to a vector function $\mathbf{F}(\mathbf{r})$ defined on an open surface $S$ bounded by a closed curve $C$. If $d \ell$ is a line element of $C$,
$$\int_S d \mathbf{S} \cdot \nabla \times \mathbf{F}=\oint_C d \ell \cdot \mathbf{F} .$$
The curve $C$ in (1.83) is traversed in the direction given by the right-hand rule when the thumb points in the direction of $d \mathbf{S}$. As with the divergence theorem, variations of (1.83) follow from the choices $\mathbf{F}=\mathbf{c} \psi$ and $\mathbf{F}=\mathbf{A} \times \mathbf{c}$ :
$$\begin{gathered} \int_S d \mathbf{S} \times \nabla \psi=\oint_C d \boldsymbol{\ell} \psi \ \oint_C d \boldsymbol{\ell} \times \mathbf{A}=\int_S d S_k \nabla A_k-\int_S d \mathbf{S}(\nabla \cdot \mathbf{A}) . \end{gathered}$$

## 物理代写|电动力学代考Electrodynamics代写|The Time Derivative of a Flux Integral

Leibniz’ Rule for the time derivative of a one-dimensional integral is
$$\frac{d}{d t} \int_{x_1(t)}^{x_2(t)} d x b(x, t)=b\left(x_2, t\right) \frac{d x_2}{d t}-b\left(x_1, t\right) \frac{d x_1}{d t}+\int_{x_1(t)}^{x_2(t)} d x \frac{\partial b}{\partial t}$$
This formula generalizes to integrals over circuits, surfaces, and volumes which move through space. Our treatment of Faraday’s law makes use of the time derivative of a surface integral where the surface $S(t)$ moves because its individual area elements move with velocity $\boldsymbol{v}(\mathbf{r}, t)$. In that case,
$$\frac{d}{d t} \int_{S(t)} d \mathbf{S} \cdot \mathbf{B}=\int_{S(t)} d \mathbf{S} \cdot\left[\boldsymbol{v}(\nabla \cdot \mathbf{B})-\nabla \times(\boldsymbol{v} \times \mathbf{B})+\frac{\partial \mathbf{B}}{\partial t}\right] .$$
Proof: We calculate the change in flux from
$$\delta\left[\int \mathbf{B} \cdot d \mathbf{S}\right]=\int \delta \mathbf{B} \cdot d \mathbf{S}+\int \mathbf{B} \cdot \delta(\hat{\mathbf{n}} d S) .$$
The first term on the right comes from time variations of $\mathbf{B}$. The second term comes from time variations of the surface. Multiplication of every term in (1.88) by $1 / \delta t$ gives
$$\frac{d}{d t} \int \mathbf{B} \cdot d \mathbf{S}=\int \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S}+\frac{1}{\delta t} \int \mathbf{B} \cdot \delta(\hat{\mathbf{n}} d S)$$
We can focus on the second term on the right-hand side of (1.89) because the first term appears already as the last term in (1.87). Figure 1.3 shows an open surface $S(t)$ with local normal $\hat{\mathbf{n}}(t)$ which moves and/or distorts to the surface $S(t+\delta t)$ with local normal $\hat{\mathbf{n}}(t+\delta t)$ in time $\delta t$.

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Stokes’ Theorem

Stokes定理适用于定义在开放曲面$S$上的向量函数$\mathbf{F}(\mathbf{r})$，该曲面以封闭曲线$C$为界。如果$d \ell$是$C$的线素，
$$\int_S d \mathbf{S} \cdot \nabla \times \mathbf{F}=\oint_C d \ell \cdot \mathbf{F} .$$
(1.83)中的曲线$C$在拇指指向$d \mathbf{S}$方向时沿着右手定则给出的方向遍历。与散度定理一样，(1.83)的变化由选项$\mathbf{F}=\mathbf{c} \psi$和$\mathbf{F}=\mathbf{A} \times \mathbf{c}$得出:
$$\begin{gathered} \int_S d \mathbf{S} \times \nabla \psi=\oint_C d \boldsymbol{\ell} \psi \ \oint_C d \boldsymbol{\ell} \times \mathbf{A}=\int_S d S_k \nabla A_k-\int_S d \mathbf{S}(\nabla \cdot \mathbf{A}) . \end{gathered}$$

## 物理代写|电动力学代考Electrodynamics代写|The Time Derivative of a Flux Integral

$$\frac{d}{d t} \int_{x_1(t)}^{x_2(t)} d x b(x, t)=b\left(x_2, t\right) \frac{d x_2}{d t}-b\left(x_1, t\right) \frac{d x_1}{d t}+\int_{x_1(t)}^{x_2(t)} d x \frac{\partial b}{\partial t}$$

$$\frac{d}{d t} \int_{S(t)} d \mathbf{S} \cdot \mathbf{B}=\int_{S(t)} d \mathbf{S} \cdot\left[\boldsymbol{v}(\nabla \cdot \mathbf{B})-\nabla \times(\boldsymbol{v} \times \mathbf{B})+\frac{\partial \mathbf{B}}{\partial t}\right] .$$

$$\delta\left[\int \mathbf{B} \cdot d \mathbf{S}\right]=\int \delta \mathbf{B} \cdot d \mathbf{S}+\int \mathbf{B} \cdot \delta(\hat{\mathbf{n}} d S) .$$

$$\frac{d}{d t} \int \mathbf{B} \cdot d \mathbf{S}=\int \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S}+\frac{1}{\delta t} \int \mathbf{B} \cdot \delta(\hat{\mathbf{n}} d S)$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。