Posted on Categories:General Relativity, 广义相对论, 物理代写, 相对论

# 物理代写|相对论代写Theory of relativity代考|Stat131 The Electromagnetic Field Tensor

avatest™

## 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

•200+ 英语母语导师

•70分以下全额退款

## 物理代写|相对论代写Theory of relativity代考|Electromagnetic Field Tensor

The Electromagnetic Field Tensor. Another interesting quantity in special relativity is the electromagnetic field tensor, whose components are
$$\left[\begin{array}{llll} F^{t t} & F^{t x} & F^{y y} & F^{t z} \ F^{x t} & F^{x x} & F^{x y} & F^{x z} \ F^{y t} & F^{y x} & F^{y y} & F^{y z} \ F^{y t} & F^{z x} & F^{z y} & F^{z z} \end{array}\right]=\left[\begin{array}{cccc} 0 & E_x & E_y & E_z \ -E_x & 0 & B_z & -B_y \ -E_y & -B_z & 0 & B_x \ -E_z & B_y & -B_x & 0 \end{array}\right]$$
where $\left[E_x, E_y, E_z\right]$ are the components of the electric field vector $\vec{E}$ and $\left[B_x, B_y, B_z\right]$ are the same for the magnetic field vector $\vec{B}$ (in GR units, both vectors have the same units of $\mathrm{kg} \cdot \mathrm{C}^{-1} \mathrm{~m}^{-1}$ : see box $4.2$ ). Using this tensor, we can write a relativistically valid version of the Lorentz force law (which describes the total electromagnetic force acting on particle with charge $q$ moving through an electromagnetic field),
$$\frac{d p^\mu}{d \tau}=q F^{\mu v} \eta_{v \alpha} u^\alpha$$
and Gauss’s law and the Ampere-Maxwell relation become the single equation
$$\frac{\partial F^{\mu \nu}}{\partial x^\nu}=4 \pi k J^\mu$$
where $k$ is the Coulomb constant (in GR units), and $J^t=\rho, J^x=\rho v_x, f^y=\rho v_y$, and $J^2=\rho v_z$ are the components of the four-current $J$ of charge flowing at the event $(\rho$ is the density of charge at the event in question and $v_x, v_y$, and $v_z$ are the usual velocity components of the flowing charge). In equation 4.16, the superscript $\nu$ in the denominator of the derivative is considered to be equivalent to a subscript in the numerator, so there is an implicit sum over $\nu$. See box $4.3$ for a discussion of these equations. However, you can see how the abstract component notation here yields very compact versions of these important equations.

## 物理代写|相对论代写Theory of relativity代考|Free and Bound Indices

Free and Bound Indices. Consider the following example equation regarding the total four-momentum of a two-particle system:
$$p_{\text {tot }}^{\prime \mu}=p_1^{\prime \mu}+p_t^{\prime \mu}=\Lambda_\nu^\mu p_1^\nu+\Lambda_\alpha^\mu p_2^\alpha$$
If $\mu=t$, this equation tells us that the primed-frame $t$ component of the system’s fourmomentum $\boldsymbol{p}_{\text {tot }}$ is the sum of the primed-frame $t$ components of particle 1’s four-momentum $\boldsymbol{p}_1$ and particle 2’s four-momentum $\boldsymbol{p}_2$, which in turn is the sum of the $t$ rows of the matrix equations for the Lorentz transformation of $\boldsymbol{p}_1$ ‘s and $\boldsymbol{p}_2$ ‘s unprimed-frame components. If $\mu=x$, the equation makes the corresponding statement about the $x$ components and so on. This abstract equation therefore actually represents four equations about the four components of the total momentum. Note that there are implicit sums over the indices $\nu$ and $\alpha$.

What I want you to focus on at the moment is the structure of this equation. The index $\mu$ appears in every term of the equation, and it is not summed in any term. We are free to assign any value to $\mu$ that we like in order to specify which component of the equation we are talking about. We therefore call $\mu$ a free index.

In contrast, the sole purpose of the pair of indices labeled $\nu$ in the term $\Lambda^\mu{ }_\nu p_1^v$ is to indicate a sum. We are not free to specify a value for these indices: they must together take on all four possible values and the four resulting terms summed if the equation is to make any sense. We call $\nu$ a bound index (some texts call this a dummy index). The index $\alpha$ in the next term is also a bound index.

## 物理代写|相对论代写Theory of relativity代考|Electromagnetic Field Tensor

$\left[\begin{array}{llllllllllllllllllll}F^{t t} & F^{t x} & F^{y y} & F^{t z} & F^{x t} & F^{x x} & F^{x y} & F^{x z} F^{y t} & F^{y x} & F^{y y} & F^{y z} & F^{y t} & F^{z x} & F^{y y} & F^{z z}\end{array}\right]=\left[\begin{array}{lllllllll}0 & E_x & E_y & E_z-E_x & 0 & B_z & -B_y-E_y & -B_z\end{array}\right.$

$\mathrm{kg} \cdot \mathrm{C}^{-1} \mathrm{~m}^{-1}$ :见框 $4.2$ ). 使用这个张量，我们可以写出洛伦兹力定律的相对论有效版本（它描述了作用在带电粒子上的总电磁力 $q$ 通过电碰场移动)，
$$\frac{d p^\mu}{d \tau}=q F^{\mu v} \eta_{v \alpha} u^\alpha$$

$$\frac{\partial F^{\mu \nu}}{\partial x^\nu}=4 \pi k J^\mu$$

## 物理代写|相对论代写Theory of relativity代考|Free and Bound Indices

$$p_{\text {tot }}^{\prime \mu}=p_1^{\prime \mu}+p_t^{\prime \mu}=\Lambda_\nu^\mu p_1^\nu+\Lambda_\alpha^\mu p_2^\alpha$$

## 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。