### 物理代写|电磁学代写electromagnetism代考|PHYSICS2534

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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|Electrostatics of Conductors

Topics. The electrostatic potential in vacuum. The uniqueness theorem for Poisson’s equation. Laplace’s equation, harmonic functions and their properties. Boundary conditions at the surfaces of conductors: Dirichlet, Neumann and mixed boundary conditions. The capacity of a conductor. Plane, cylindrical and spherical capacitors. Electrostatic field and electrostatic pressure at the surface of a conductor. The method of image charges: point charges in front of plane and spherical conductors.
Basic equations Poisson’s equation is
$$\nabla^{2} \varphi(\mathbf{r})=-4 \pi k_{\mathrm{e}} \varrho(\mathbf{r})$$
where $\varphi(\mathbf{r})$ is the electrostatic potential, and $\varrho(\mathbf{r})$ is the electric charge density, at the point of vector position $\mathbf{r}$. The solution of Poisson’s equation is unique if one of the following boundary conditions is true

1. Dirichlet boundary condition: $\varphi$ is known and well defined on all of the boundary surfaces.
2. Neumann boundary condition: $\mathbf{E}=-\nabla \varphi$ is known and well defined on all of the boundary surfaces.
3. Modified Neumann boundary condition (also called Robin boundary condition): conditions where boundaries are specified as conductors with known charges.
4. Mixed boundary conditions: a combination of Dirichlet, Neumann, and modified Neumann boundary conditions:
Laplace’s equation is the special case of Poisson’s equation
$$\nabla^{2} \varphi(\mathbf{r})=0$$
which is valid in vacuum.

## 物理代写|电磁学代写electromagnetism代考|Metal Sphere in an External Field

A a metal sphere of radius $R$ consists of a “rigid” lattice of ions, each of charge $+Z e$, and valence electrons each of charge $-e$. We denote by $n_{\mathrm{i}}$ the ion density, and by $n_{\mathrm{e}}$ the electron density. The net charge of the sphere is zero, therefore $n_{\mathrm{e}}=Z n_{\mathrm{i}}$. The sphere is located in an external, constant, and uniform electric field $\mathbf{E}{0}$. The field causes a displacement $\delta$ of the “electron sea” with respect to the ion lattice, so that the total field inside the sphere, $\mathbf{E}$, is zero. Using Problem $1.1$ as a model, evaluate a) the displacement $\delta$, giving a numerical estimate for $E{0}=10^{3} \mathrm{~V} / \mathrm{m}$;
b) the field generated by the sphere at its exterior, as a function of $\mathbf{E}_{0}$;
c) the surface charge density on the sphere.

(b) Consider the configurations of
(c)
a) A charge $q$ is located at a distance $a$ from an infinite conducting plane.
b) Two opposite charges $+q$
Fig. $2.1$ and $-q$ are at a distance $d$ from distance $a$ from an infinite conducting plane.
c) A charge $q$ is at distances $a$ and $b$, respectively, from two infinite conducting half planes forming a right dihedral angle.

## 物理代写|电磁学代写electromagnetism代考|Fields Generated by Surface Charge Densities

Consider the case a) of Problem 2.2: we have a point charge $q$ at a distance $a$ from an infinite conducting plane.
a) Evaluate the surface charge density $\sigma$, and the total induced charge $q_{\text {ind }}$, on the plane.

b) Now assume to have a nonconducting plane with the same surface charge distribution as in point a). Find the electric field in the whole space.
c) A non conducting spherical surface of radius $a$ has the same charge distribution as the conducting sphere of Problem 2.4. Evaluate the electric field in the whole space.

A point charge $q$ is located at a distance $d$ from the center of a conducting grounded sphere of radius $a<d$. Evaluate
a) the electric potential $\varphi$ over the whole space;
b) the force on the point charge;
c) the electrostatic energy of the system.
Answer the above questions also in the case of an isolated, uncharged conducting sphere.

An electric dipole $\mathbf{p}$ is located at a distance $d$ from the center of a conducting sphere of radius $a$. Evaluate the electrostatic potential $\varphi$ over the whole space assuming that
a) $\mathbf{p}$ is perpendicular to the direction from $\mathbf{p}$ to the center of the sphere,
b) $\mathbf{p}$ is directed towards the center of the sphere.
c) $\mathbf{p}$ forms an arbitrary angle $\theta$ with respect to the straight line passing through the center of the sphere and the dipole location.

In all three cases consider the two possibilities of i) a grounded sphere, and ii) an electrically uncharged isolated sphere.

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

∇2披(r)=−4圆周率ķ和ϱ(r)

1. 狄利克雷边界条件：披是已知的并且在所有的边界表面上定义良好。
2. 纽曼边界条件：和=−∇披是已知的并且在所有的边界表面上定义良好。
3. 修正的 Neumann 边界条件（也称为 Robin 边界条件）：边界被指定为具有已知电荷的导体的条件。
4. 混合边界条件：Dirichlet、Neumann 和修正的 Neumann 边界条件的组合：
拉普拉斯方程是泊松方程的特例
∇2披(r)=0
这在真空中是有效的。

## 物理代写|电磁学代写electromagnetism代考|Metal Sphere in an External Field

A 一个半径为金属的球体R由离子的“刚性”晶格组成，每个电荷+从和, 和价电子，每个电荷−和. 我们表示n一世离子密度，并由n和电子密度。球体的净电荷为零，因此n和=从n一世. 球体位于一个外部的、恒定的、均匀的电场中和0. 该场导致位移d相对于离子晶格的“电子海”，因此球体内的总场，和, 为零。使用问题1.1作为模型，评估 a) 位移d, 给出一个数值估计和0=103 在/米;
b) 球体在其外部产生的场，作为以下函数的函数和0;
c) 球面上的表面电荷密度。

(b) 考虑
(c)
a) 电荷的配置q位于远处一个从一个无限的导电平面。
b) 两个相反的电荷+q

c) 收费q在远处一个和b，分别来自形成直二面角的两个无限导电半平面。

## 物理代写|电磁学代写electromagnetism代考|Fields Generated by Surface Charge Densities

a) 评估表面电荷密度σ, 和总感应电荷q工业 ， 在飞机上。

b) 现在假设有一个非导电平面，其表面电荷分布与 a) 点相同。求整个空间的电场。
c) 半径为非导电球面一个具有与问题 2.4 的导电球相同的电荷分布。评估整个空间中的电场。

a) 电位披覆盖整个空间；
b) 点电荷上的力；
c) 系统的静电能。

a)p垂直于从p到球心，
b)p指向球体的中心。
C）p形成任意角度θ关于通过球心和偶极子位置的直线。

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## MATLAB代写

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