物理代写|电磁学代写electromagnetism代考|PHYS1002

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• 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|电磁学代写electromagnetism代考|Parallel-Plate Capacitors

Now, let us consider a capacitor composed of two parallel conductor plates of equal area $A$, which are at a distance $d$, see also Fig. 4.3. One of the plates carries a charge $+Q$, and the other $-Q$. Note that charges of like sign repel one another and that charges of opposite signs attract one another (see also Chap. 1). As a battery is charging a capacitor, electrons flow into the negative plate and out of the positive plate (see Fig. 4.2).

Note that the electric field between the plates of a parallel-plate capacitor is uniform near the center but nonuniform near the edges. When the capacitor plates are large, the accumulated charges can distribute themselves over a substantial area, and hence the amount of charge stored on each plate $Q$, for a given potential difference $\Delta V$, increases as the plate area increases to ensure a constant surface charge density $\sigma$. A simple argument can be used for that: because the electric field just outside the conductor is perpendicular to the surface of the conductor and with magnitude $E=\sigma / \epsilon_{0}$, where $E$ is proportional to constant $\Delta V$, then $\sigma$ is constant. Thus, we $\mathrm{~ m e x p a s i ~ t h e ~ ต i p u s i l}$

Above we derived a relationship between the electric field between the plates and magnitude of potential difference, given as
$$E=\frac{\Delta V}{d}$$
From Eq. (4.9), we see that when $d$ decreases, $E$ increases, for fixed $\Delta V$. If we move the plates closer together (that is, $d$ decreases), We also consider the situation before the charges have moved in response to that change, such that no charges have moved. Hence, the electric field between the plates is the same but extends over a shorter distance between plates. That situation corresponds to a new capacitor with a potential difference between the plates that is different from the terminal voltage of the battery. Now, across the wires connecting the battery to the capacitor exists a potential difference (see also Fig. $4.2$ for an illustration).

Based on the arguments that we discussed for a situation in Fig. 4.2, that potential difference creates an electric field in the wires that drives more charges onton the plates, which in turn increases the potential difference between the plates of the capacitor. When it becomes equal to the potential difference between the terminals of the battery (Fig. 4.2), the potential difference across the wires falls back to zero. Ihen, the flow of charge stops.

物理代写|电磁学代写electromagnetism代考|Parallel Combination

Figure $4.5$ presents a combination of two capacitors connected in parallel. Also, we show a circuit diagram for this combination of capacitors, as often seen in an electric circuit. Note from Fig. $4.5$ that the left plates of the capacitors connect to the positive terminal of the battery using conducting wires; therefore, those plates, after equilibrium of the electric potential establishes, are at the same electric potential as the positive terminal of the battery. For the same reason, the right plate connecting to the negative terminal of the battery has equal electric potential with the negative terminal after the equilibrium of the electric potential establishes. As a result, the potential differences across each capacitor connected in parallel are the same and equal to the voltage applied to the battery; that is, $\Delta V_{1}=\Delta V_{2}=\Delta V$.

Applying the model described above in Fig. 4.2, when two capacitors are initially connected in a circuit, as shown in Fig.4.5, electrons migrate between the wires and the plates. As a result, the left plates charge positively, and the right plates

negatively. In other words, the internal chemical energy stored in the battery is the source of that migration; that is, the internal chemical energy of the battery converts into electric potential energy associated with the surface charges in the plates of the capacitors at a separation $d$. During the process of the electrons migration, the voltage across the capacitors becomes equal to that across the battery terminals and then charge transfer stops. When that establishes in the circuit, the capacitors load to their maximum charge capacity.

In the following, we show a few steps to calculate the equivalent capacitance, $C_{e q}$, of the combinations of $C_{1}$ and $C_{2}$. For that, we denote by $Q_{1}$ and $Q_{2}$ the maximum charges on each capacitor, respectively, and by $Q$ the total charge stored by the two capaciturs:
$$Q=Q_{1}+Q_{2}$$
$Q$ is also the charge stored in the capacitor $C_{e q}$. The voltages applied across each capacitor are the same, see also Fig. $4.5$, and hence the charges in each capacitor are
\begin{aligned} &Q_{1}=C_{1} \Delta V \ &Q_{2}=C_{2} \Delta V \end{aligned}

物理代写|电磁学代写electromagnetism代考|Series Combination

Next, we consider an electric circuit in which two capacitors are combined in series, as shown in Fig. 4.6. That is known as a series combination of capacitors. In that combination, the left plate of capacitor 1 connects to one of the terminals of a battery (for example, the positive terminal in Fig.4.6) and the right plate of capacitor 2 connects to the other terminal (for example, the negative terminal in Fig. 4.6). Furthermore, the other two plates, from each capacitor, connect each other via a conducting wire and to nothing else, as shown in Fig. 4.6. Two capacitors connected that way form an isolated conductor that is initially uncharged and must continue to have a net charge zero.

In the following, we will analyze the combination of two capacitors in series. When the two capacitors are initially uncharged and just connect to a battery in the circuit, then the electrons transfer from the left plate of $C_{1}$ and into the right plate of $C_{2}$. That is, during the process, a negative charge (electrons) stores on the right plate of $C_{2}$ and the same amount of negative charge leaves the left plate of $C_{2}$ as electrons migrating from that plate to the conducting wire leave behind the left plate having an excess positive charge. Therefore, we can say that the negative charge leaving the left plate of $C_{2}$ transfers via the conducting wire and stores on the right plate of $C_{1}$. As a result, the right plates, when the equilibrium establishes, accumulate a charge $-Q$, and the left plates a charge $+Q$. That indicates that the charges on capacitors connected as in Fig. $4.6$ are the same.

It can be seen that the $\Delta V$ across the battery terminals is split between two capacitors:
$$\Delta V=\Delta V_{1}+\Delta V_{2}$$
In Eq. (4.21), $\Delta V_{1}$ and $\Delta V_{2}$ are the potential across $C_{1}$ and $C_{2}$, respectively. In general, the total potential difference across any number of capacitors connected in series is the sum of the potential differences across the individual capacitors. Now, consider an equivalent capacitor, $C_{e q}$, with same effect on the circuit as the series combination of the capacitors. After it is fully charged, the equivalent capacitor must have a charge of $-Q$ on its right plate and a charge of $+Q$ on its left plate. Using the definition of capacitance to the equivalent circuit in Fig.

Δ在=Δ在1+Δ在2

有限元方法代写

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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。