### 物理代写|电磁学代写electromagnetism代考|PHYS3040

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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|Mie Oscillations

Now, instead of a the metal slab of Problem 1.4, consider a metal sphere of radius $R$. Initially, all the conduction electrons ( $n_{\mathrm{e}}$ per unit volume) are displaced by $-\delta$ (with $\delta \ll R$ ) by an external electric field, analogously to Problem 1.1.
a) At time $t=0$ the external field is suddenly removed. Describe the subsequent motion of the conduction electrons under the action of the self-consistent electrostatic field, neglecting the boundary effects on the electrons close to the surface of the sphere.
b) At the limit $\delta \rightarrow 0$ (but assuming $e n_{\mathrm{e}} \delta=\sigma_{0}$ to remain finite, $\mathrm{i}{2} \mathrm{e}{2}$, the charge distribution is a surface density), find the electrostatic energy of the sphere as a function of $\delta$ and use the result to discuss the electron motion as in point $\mathbf{a})$.

## 物理代写|电磁学代写electromagnetism代考|Coulomb explosions

At $t=0$ we have a spherical cloud of radius $R$ and total charge $Q$, comprising $N$ point-like particles. Each particle has charge $q=Q / N$ and mass $m$. The particle density is uniform, and all particles are at rest.

a) Evaluate the electrostatic potential energy of a charge located at a distance $r0$. Consider the particles located in the infinitesimal spherical shell $r_{0}<r_{\mathrm{s}}<r_{0}+\mathrm{d} r$, with $r_{0}+\mathrm{d} r<R$, at $t=0$. Show that the equation of motion of the layer is
$$m \frac{\mathrm{d}^{2} r_{\mathrm{s}}}{\mathrm{d} t^{2}}=k_{\mathrm{e}} \frac{q Q}{r_{\mathrm{s}}^{2}}\left(\frac{r_{0}}{R}\right)^{3}$$
c) Find the initial position of the particles that acquire the maximum kinetic energy during the cloud expansion, and determinate the value of such maximum energy.
d) Find the energy spectrum, i.e., the distribution of the particles as a function of their final kinetic energy. Compare the total kinetic energy with the potential energy initially stored in the electrostatic field.
e) Show that the particle density remains spatially uniform during the expansion.

## 物理代写|电磁学代写electromagnetism代考|Plane and Cylindrical Coulomb Explosions

Particles of identical mass $m$ and charge $q$ are distributed with zero initial velocity and uniform density $n_{0}$ in the infinite slab $|x|0$ the slab expands because of the electrostatic repulsion between the pairs of particles.
a) Find the equation of motion for the particles, its solution, and the kinetic energy acquired by the particles.
b) Consider the analogous problem of the explosion of a uniform distribution having cylindrical symmetry.

Two rigid spheres have the same radius $R$ and the same mass $M$, and opposite charges $\pm Q$. Both charges are uniformly and rigidly distributed over the volumes of the two spheres. The two spheres are initially at rest, at a distance $x_{0} \gg R$ between their centers, such that their interaction energy is negligible compared to the sum of their “internal” (construction) energies.
a) Evaluate the initial energy of the system.
The two spheres, having opposite charges, attract each other, and start moving at $t=0$.
b) Evaluate the velocity of the spheres when they touch each other (i.e. when the distance between their centers is $x=2 R$ ).
c) Assume that, after touching, the two spheres penetrate each other without friction. Evaluate the velocity of the spheres when the two centers overlap $(x=0)$.

An electrically neutral metal sphere of radius $a$ contains $N$ conduction electrons. A fraction $f$ of the conduction electrons $(0<f<1)$ is removed from the sphere, and the remaining $(1-f) N$ conduction electrons redistribute themselves to an equilibrium configurations, while the $N$ lattice ions remain fixed.
a) Evaluate the conduction-electron density and the radius of their distribution in the sphere.

Now the conduction-electron sphere is rigidly displaced by $\boldsymbol{\delta}$ relatively to the ion lattice, with $|\delta|$ small enough for the conduction-electron sphere to remain inside the ion sphere.
b) Evaluate the electric field inside the conduction-electron sphere.
c) Evaluate the oscillation frequency of the conduction-electron sphere when it is released.

## 物理代写|电磁学代写electromagnetism代考|Mie Oscillations

a) 有时吨=0外场突然被移除。描述导电电子在自洽静电场作用下的后续运动，忽略靠近球体表面的电子的边界效应。
b) 在极限d→0（但假设和n和d=σ0保持有限，一世2和2，电荷分布是表面密度），找到球体的静电能量作为函数d并使用结果来讨论电子运动一个).

## 物理代写|电磁学代写electromagnetism代考|Coulomb explosions

a) 评估位于远处的电荷的静电势能r0. 考虑位于无穷小球壳中的粒子r0<rs<r0+dr， 和r0+dr<R， 在吨=0. 证明层的运动方程为

c) 找出在云膨胀过程中获得最大动能的粒子的初始位置，并确定该最大能量的值。
d) 找出能谱，即粒子的分布作为其最终动能的函数。将总动能与最初存储在静电场中的势能进行比较。
e) 表明粒子密度在膨胀过程中保持空间均匀。

## 物理代写|电磁学代写electromagnetism代考|Plane and Cylindrical Coulomb Explosions

a) 找出粒子的运动方程、它的解以及粒子获得的动能。
b) 考虑具有圆柱对称性的均匀分布爆炸的类似问题。

a) 评估系统的初始能量。

b) 评估球体相互接触时的速度（即，当它们的中心之间的距离为X=2R）。
c) 假设两个球体接触后相互穿透，没有摩擦。计算两个中心重叠时球体的速度(X=0).

a) 评估传导电子密度及其在球体中的分布半径。

b) 评估传导电子球内的电场。
c) 评估传导电子球释放时的振荡频率。

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## MATLAB代写

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