### 物理代写|电磁学代写electromagnetism代考|PHYC20014

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Basics of Electrostatics

Topics. The electric charge. The electric field. The superposition principle. Gauss’s law. Symmetry considerations. The electric field of simple charge distributions (plane layer, straight wire, sphere). Point charges and Coulomb’s law. The equations of electrostatics. Potential energy and electric potential. The equations of Poisson and Laplace. Electrostatic energy. Multipole expansions. The field of an electric dipole.

Units. An aim of this book is to provide formulas compatible with both SI (French: Système International d’Unités) units and Gaussian units in Chapters 1-6, while only Gaussian units will be used in Chapters 7-13. This is achieved by introducing some system-of-units-dependent constants.

The first constant we need is Coulomb’s constant, $k_{\mathrm{e}}$, which for instance appears in the expression for the force between two electric point charges $q_{1}$ and $q_{2}$ in vacuum, with position vectors $\mathbf{r}{1}$ and $\mathbf{r}{2}$, respectively. The Coulomb force acting, for instance, on $q_{1}$ is
$$\mathbf{f}{1}=k{\mathrm{e}} \frac{q_{1} q_{2}}{\left|\mathbf{r}{1}-\mathbf{r}{2}\right|^{2}} \hat{\mathbf{r}}{12},$$ where $k{\mathrm{e}}$ is Coulomb’s constant, dependent on the units used for force, electric charge, and length. The vector $\mathbf{r}{12}=\mathbf{r}{1}-\mathbf{r}{2}$ is the distance from $q{2}$ to $q_{1}$, pointing towards $q_{1}$, and $\hat{r}{12}$ the corresponding unit vector. Coulomb’s constant is $$k{\mathrm{e}}= \begin{cases}\frac{1}{4 \pi \varepsilon_{0}} 8.987 \cdots \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} \cdot \mathrm{C}^{-2} \simeq 9 \times 10^{9} \mathrm{~m} / \mathrm{F} & \text { SI } \ 1 & \text { Gaussian. }\end{cases}$$
Constant $\varepsilon_{0} \simeq 8.854187817620 \cdots \times 10^{-12} \mathrm{~F} / \mathrm{m}$ is the so-called “dielectric permittivity of free space”, and is defined by the formula

$$\varepsilon_{0}=\frac{1}{\mu_{0} c^{2}}$$
where $\mu_{0}=4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}$ (by definition) is the vacuum magnetic permeability, and $c$ is the speed of light in vacuum, $c=299792458 \mathrm{~m} / \mathrm{s}$ (this is a precise value, since the length of the meter is defined from this constant and the international standard for time).

Basic equations The two basic equations of this Chapter are, in differential and integral form,
$$\begin{array}{ll} \boldsymbol{\nabla} \cdot \mathbf{E}=4 \pi k_{\mathrm{e}} \varrho, & \oint_{S} \mathbf{E} \cdot \mathrm{d} \mathbf{S}=4 \pi k_{\mathrm{e}} \int_{V} \varrho \mathrm{d}^{3} r \ \boldsymbol{\nabla} \times \mathbf{E}=0, & \oint_{C} \mathbf{E} \cdot \mathrm{d} \ell=0 \end{array}$$
where $\mathbf{E}(\mathbf{r}, t)$ is the electric field, and $\varrho(\mathbf{r}, t)$ is the volume charge density, at a point of location vector $\mathbf{r}$ at time $t$. The infinitesimal volume element is $\mathrm{d}^{3} r=\mathrm{d} x \mathrm{~d} y \mathrm{~d} z$. In (1.4) the functions to bẻ iñtēgrâtèd arré evalluateed ovèr añ arbbitrāry volume $V$, or over the surface $S$ enclosing the volume $V$. The function to be integrated in (1.5) is evaluated over an arbitrary closed path $C$. Since $\boldsymbol{\nabla} \times \mathbf{E}=0$, it is possible to define an electric potential $\varphi=\varphi(\mathbf{r})$ such that
$$\mathbf{E}=-\boldsymbol{\nabla} \varphi$$

## 物理代写|电磁学代写electromagnetism代考|Overlapping Charged Spheres

We assume that a neutral sphere of radius $R$ can be regarded as the superposition of two “rigid” spheres: one of uniform positive charge density $+\varrho_{0}$, comprising the nuclei of the atoms, and a second sphere of the same radius, but of negative uniform charge density $-\varrho_{0}$, comprising the electrons. We further assume that its is possible to shift the two spheres relative to each other by a quantity $\delta$, as shown in Fig. 1.1, without perturbing the internal structure of either sphere.

a) in the “inner” region, where the two spheres overlap,
b) in the “outer” region, i.e., outside both spheres, discussing the limit of small displacements $\delta \ll R$.

A sphere of radius $a$ has uniform charge density $\varrho$ over all its volume, excluding a spherical cavity of radius $b<a$, where $\varrho=0$. The center of the cavity, $O_{b}$ is located at a distance d, with $|\mathbf{d}|<(a-b)$, from the center of the sphere, $O_{a}$. The mass distribution of the sphere is proportional to its charge distribution.
a) Find the electric field inside the cavity.
Now we apply an external, uniform electric field $\mathbf{E}_{0}$. Find
b) the force on the sphere,

c) the torque with respect to the center of the sphere, and the torque with respect to the center of mass.

## 物理代写|电磁学代写electromagnetism代考|Energy of a Charged Sphere

A total charge $Q$ is distributed uniformly over the volume of a sphere of radius $R$. Evaluate the electrostatic energy of this charge configuration in the following three alternative ways:
a) Evaluate the work needed to assemble the charged sphere by moving successive infinitesimals shells of charge from infinity to their final location.
b) Evaluate the volume integral of $u_{\mathrm{E}}=|\mathbf{E}|^{2} /\left(8 \pi k_{\mathrm{e}}\right)$ where $\mathbf{E}$ is the electric field [Eq. (1.10)].
c) Evaluate the volume integral of $\rho \phi / 2$ where $\rho$ is the charge density and $\phi$ is the electrostatic potential [Eq. (1.11)]. Discuss the differences with the calculation made in b).

A square metal slab of side $L$ has thickness $h$, with $h \ll L$. The conduction-electron and ion densities in the slab are $n_{\mathrm{e}}$ and $n_{i}=n_{\mathrm{e}} / Z$, respectively, $Z$ being the ion charge.

An external electric field shifts all conduction electrons by the same amount $\delta$, such that $|\delta| \ll h$, perpendicularly to the base of the slab. We assume that both $n_{\mathrm{e}}$ and $n_{i}$ are constant, that the ion lattice is unperturbed by the external field, and that boundary effects are negligible.
a) Evaluate the electrostatic field generated by the displacement of the electrons.
b) Evaluate the electrostatic energy of the system.
Fig. 1.3
Now the external field is removed, and the “electron slab” starts oscillating around its equilibrium position.
c) Find the oscillation frequency, at the small displacement limit $(\delta \ll h)$.

## 物理代写|电磁学代写electromagnetism代考|Basics of Electrostatics

F1=ķ和q1q2|r1−r2|2r^12,在哪里ķ和是库仑常数，取决于力、电荷和长度的单位。向量r12=r1−r2是距离q2至q1, 指向q1， 和r^12对应的单位向量。库仑常数为

ķ和={14圆周率e08.987⋯×109 ñ⋅米2⋅C−2≃9×109 米/F 和  1 高斯。

e0=1μ0C2

∇⋅和=4圆周率ķ和ϱ,∮小号和⋅d小号=4圆周率ķ和∫在ϱd3r ∇×和=0,∮C和⋅dℓ=0

## 物理代写|电磁学代写electromagnetism代考|Overlapping Charged Spheres

a）在“内部”区域，两个球体重叠，
b）在“外部”区域，即在两个球体之外，讨论小位移的限制d≪R.

a) 求空腔内的电场。

b) 球体上的力，

c) 相对于球心的扭矩，以及相对于质心的扭矩。

## 物理代写|电磁学代写electromagnetism代考|Energy of a Charged Sphere

a) 通过将连续的无穷小电荷壳从无穷远移动到它们的最终位置来评估组装带电球体所需的功。
b) 评估体积积分在和=|和|2/(8圆周率ķ和)在哪里和是电场 [Eq. (1.10)]。
c) 评估体积积分ρφ/2在哪里ρ是电荷密度和φ是静电势 [Eq. (1.11)]。讨论与 b) 中计算的差异。

a) 评估由电子位移产生的静电场。
b) 评估系统的静电能量。

c) 找出小位移极限处的振荡频率(d≪H).

## 有限元方法代写

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## MATLAB代写

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