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物理代写|电动力学作业代写Electrodynamics代考|Conservation Laws

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物理代写|电动力学作业代写Electrodynamics代考|Conservation and Invariance of the Electric Charge

As prototype of a local conservation law, i.e. a conservation law based on a continuity equation relative to a certain four-current $j^{\mu}$, we consider the conservation of the electric charge. If the charged matter consists of point-like particles the current is given by (2.16); if conversely the charge is spread out smoothly, as in macroscopic systems, the current will have a generic form. For what follows, the particular form of the current $j^{\mu}(x)$ will be irrelevant in that we will make reference only to the following assumptions:
(1) $j^{\mu}$ is a vector field;
(2) $j^{\mu}$ satisfies the continuity equation $\partial_{\mu} j^{\mu}=0$;
(3) $\lim {|\mathbf{x}| \rightarrow \infty}\left(|\mathbf{x}|^{3} j^{\mu}(t, \mathbf{x})\right)=0$. The meaning of the condition (3) is that for a fixed $t$ the current decreases at spatial infinity more rapidly than $1 /|\mathbf{x}|^{3}$, a property certainly possessed by the expressions (2.109) and (2.110). Assuming that $j^{\mu}$ satisfies the conditions (1)-(3), we now prove that there exists a conserved as well as Lorentz invariant total charge $Q$. The construction of $Q$ follows a standard procedure which consists in integrating the continuity equation over a generic spatial volume $V$ $$\int{V} \partial_{0} j^{0} d^{3} x=-\int_{V} \nabla \cdot \mathbf{j} d^{3} x$$

物理代写|电动力学作业代写Electrodynamics代考|Energy-Momentum Tensor and Four-Momentum

In Sect. 2.4.3 we will show how the conservation of energy and momentum – cornerstone of any fundamental physical theory – occurs in electrodynamics as a consequence of equations (2.20)-(2.22). In this section, before considering this particular case, we set the problem of the realization of these conservation laws in a generic relativistic theory. In a relativistic theory the energy constitutes the fourth component of a four-vector, namely of the four-momentum. Since in such a theory a Lorentz transformation mixes energy and momentum, it is natural to expect that the conservation of the first cannot take place without the simultaneous conservation of the second. So we are looking for four constants of motion, grouped in a four-momentum $P^{\nu}$ whose time component $P^{0}=\varepsilon$ represents the total energy of the system. In line with the conservation paradigm of the electric charge exposed in Sect. 2.4.1, we envisage local conservation laws also for the four-momentum. We require, therefore, each of the four components of $P^{\nu}$ to be associated with a conserved four-current $j^{\mu(\nu)}(x)$. These four currents altogether form a double tensor called energy-momentum tensor – which commonly is denoted by the symbol
$$T^{\mu \nu}=j^{\mu(\nu)}$$
As for the electric charge, we postulate that in a relativistic theory the conservation of the four-momentum occurs as consequence of the existence of an energymomentum tensor $T^{\mu \nu}(x)$, possessing the following properties:
(1) $T^{\mu \nu}$ is a tensor field;
(2) $T^{\mu \nu}$ satisfies the continuity equation $\partial_{\mu} T^{\mu \nu}=0$;
(3) $\lim _{|\mathbf{x}| \rightarrow \infty}\left(|\mathbf{x}|^{3} T^{\mu \nu}(t, \mathbf{x})\right)=0$.

物理代写|电动力学作业代写ELECTRODYNAMICS代考|Energy-Momentum Tensor in Electrodynamics

In this section we provide a constructive proof of the existence in the electrodynamics of point-like particles of an energy-momentum tensor $T^{\mu \nu}$ with the properties (1)-(3) postulated in Sect. 2.4.2. We first derive heuristically the form of the energy density $T^{00}$, and then we resort to Lorentz invariance to reconstruct the whole tensor. We begin recalling the known expression of the energy density of the electromagnetic field
$$T_{\mathrm{em}}^{00}=\frac{1}{2}\left(E^{2}+B^{2}\right) .$$
Obviously the conserved total energy cannot simply be given by the integral of $T_{\mathrm{em}}^{00}$, because the electromagnetic field exchanges energy with the charged particles. To quantify this exchange we calculate the time derivative of $T_{\mathrm{em}}^{00}$, using the Maxwell equations in the form (2.52)-(2.55):
\begin{aligned} \frac{\partial T_{\mathrm{em}}^{00}}{\partial t} &=\mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t}+\mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t}=\mathbf{E} \cdot(\boldsymbol{\nabla} \times \mathbf{B}-\mathbf{j})-\mathbf{B} \cdot \boldsymbol{\nabla} \times \mathbf{E} \ &=-\mathbf{j} \cdot \mathbf{E}-\boldsymbol{\nabla} \cdot(\mathbf{E} \times \mathbf{B}) \end{aligned}

物理代写|电动力学作业代写ELECTRODYNAMICS代考|CONSERVATION AND INVARIANCE OF THE ELECTRIC CHARGE

(1) $j^{\mu}$ is a vector field;
(2) $j^{\mu}$ satisfies the continuity equation $\partial_{\mu} j^{\mu}=0$;
(3) $\lim {|\mathbf{x}| \rightarrow \infty}\left(|\mathbf{x}|^{3} j^{\mu}(t, \mathbf{x})\right)=0$. The meaning of the condition (3) is that for a fixed $t$ the current decreases at spatial infinity more rapidly than $1 /|\mathbf{x}|^{3}$, a property certainly possessed by the expressions (2.109) and (2.110). Assuming that $j^{\mu}$ satisfies the conditions (1)-(3), we now prove that there exists a conserved as well as Lorentz invariant total charge $Q$. The construction of $Q$ follows a standard procedure which consists in integrating the continuity equation over a generic spatial volume $V$ $$\int{V} \partial_{0} j^{0} d^{3} x=-\int_{V} \nabla \cdot \mathbf{j} d^{3} x$$

物理代写|电动力学作业代写ELECTRODYNAMICS代考|ENERGY-MOMENTUM TENSOR AND FOUR-MOMENTUM

(1) $T^{\mu \nu}$ is a tensor field;
(2) $T^{\mu \nu}$ satisfies the continuity equation $\partial_{\mu} T^{\mu \nu}=0$;
(3) $\lim _{|\mathbf{x}| \rightarrow \infty}\left(|\mathbf{x}|^{3} T^{\mu \nu}(t, \mathbf{x})\right)=0$.

物理代写|电动力学作业代写ELECTRODYNAMICS代考|ENERGY-MOMENTUM TENSOR IN ELECTRODYNAMICS

\begin{aligned} \frac{\partial T_{\mathrm{em}}^{00}}{\partial t} &=\mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t}+\mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t}=\mathbf{E} \cdot(\boldsymbol{\nabla} \times \mathbf{B}-\mathbf{j})-\mathbf{B} \cdot \boldsymbol{\nabla} \times \mathbf{E} \ &=-\mathbf{j} \cdot \mathbf{E}-\boldsymbol{\nabla} \cdot(\mathbf{E} \times \mathbf{B}) \end{aligned}

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。