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## 物理代写|电磁学代写electromagnetism代考|Magnetic Levitation

In a given region of space we have a static magnetic field, which, in a cylindrical reference frame $(r, \phi, z)$, is symmetric around the $z$ axis, i.e., is independent of $\phi$, and can be written $\mathbf{B}=\mathbf{B}(r, z)$. The field component along $z$ is $B_{z}(z)=B_{0} z / L$, where $B_{0}$ and $L$ are constant parameters.
a) Find the radial component $B_{r}$ close to the $z$ axis.
A particle of magnetic polarizability $\alpha$ (such that it acquires an induced magnetic dipole moment $\mathbf{m}=\alpha \mathbf{B}$ in a magnetic field $\mathbf{B})$, is located close to the $z$ axis.
b) Find the potential energy of the particle in the magnetic field.
c) Discuss the existence of equilibrium positions for the particle, and find the frequency of oscillations for small displacements from equilibrium either along $z$ or $r$ (let $M$ be the mass of the particle).

A magnetically “hard” cylinder of radius $R$ and height $h$, with $R \ll h$, carries a uniform magnetization $\mathbf{M}$ parallel to its axis.

a) Show that the volume magnetization current density $\mathbf{J}{\mathrm{m}}$ is zero inside the cylinder, while the lateral surface of the cylinder carries a surface magnetization current density $\mathbf{K}{\mathrm{m}}$, with $\left|\mathbf{K}{\mathrm{m}}\right|=|\mathbf{M}|$. b) Find the magnetic field $\mathbf{B}$ inside and outside the cylinder, at the limit $h \rightarrow \infty$. c) Now consider the opposite case of a “flat” cylinder, i.e., $h \ll R$, and evaluate the magnetic field $\mathbf{B}{0}$ at the center of the cylinder.
d) According to the result of c), $\lim {R / h \rightarrow \infty} \mathbf{B}{0}=0$. Obtain the same result using the equivalent magnetic charge method.

## 物理代写|电磁学代写electromagnetism代考|MCharged Particle in Crossed Electric and Magnetic Fields

A particle of electric charge $q$ and mass $m$ is initially at rest in the presence of a uniform electric field $\mathbf{E}$ and $\mathbf{a}$ ıniform magnetic field $\mathbf{R}$, perpendicular to $\mathbf{E}$.
a) Describe the subsequent motion of the particle.
b) Use the above result to discuss the following problem. We have a parallel-plate capacitor with surface $S$, plate separation $h$ and voltage $V$, as in Fig. 5.4. A uniform magnetic field $\mathbf{B}$ is applied to the capacitor, perpendicular to the capacitor electric field, i.e., parallel to the plates. Ultraviolet radiation causes the negative plate to emit electrons with zero initial velocity. Evaluate the minimum value of $\mathbf{B}$ for which the electrons cannot reach the positive plate.

An infinite cylindrical conductor of radius $a$ has a cylindrical cavity of radius $b$ bored parallel to, and centered at a distance $h<a-b$ from the cylinder axis as in Fig. 5.5, which shows a section of the conductor. The current density $\mathbf{J}$ is perpendicular to, and uniform over the section of the conductor (i.e., excluding the cavity!). The figure shows a section of the conductor. Evaluate the magnetic field $\mathbf{B}$, showing that it is uniform inside the cavity.

## 物理代写|电磁学代写electromagnetism代考|Conducting Cylinder in a Magnetic Field

A conducting cylinder of radius $a$ and height $h \gg a$ rotates around its axis at constant angular velocity $\omega$ in a uniform magnetic field $\mathbf{B}_{0}$, parallel to the cylinder axis.

a) Evaluate the magnetic force acting on the conduction electrons, assuming $\omega=2 \pi \times 10^{2} \mathrm{~s}^{-1}$ and $B=5 \times 10^{-5} \mathrm{~T}$ (the Earth’s magnetic field), and the ratio of the magnetic force to the centrifugal force.
Assume that the cylinder is rotating in stationary conditions. Evaluate
b) the electric field inside the cylinder, and the volume and surface charge densities;
c) the magnetic field $\mathbf{B}{1}$ generated by the rotation currents inside the cylinder, and the order of magnitude of $B{1} / B_{0}$ (assume $a \approx 0.1 \mathrm{~m}$ ).

The concentric cylindrical shells of a cylindrical capacitor have radii $a$ and $b>a$, respectively, and height $h \gg b$. The capacitor charge is $Q$, with $+Q$ on the inner shell of radius $a$, and $-Q$ on the outer shell of radius $b$, as in Fig. 5.7. The whole capacitor rotates about its axis with angular velocity $\omega=2 \pi / T$. Boundary effects are negligible.
a) Evaluate the magnetic field $\mathbf{B}$ generated by the rotating capacitor over the whole space.
b) Evaluate the magnetic forces on the charges of the two rotating cylindrical shells, and compare them to the electrostatic forces.

## 物理代写|电磁学代写electromagnetism代考|Magnetic Levitation

a) 找到径向分量乙r靠近和轴。

b) 求粒子在磁场中的势能。
c) 讨论粒子平衡位置的存在，并找出平衡点的小位移的振荡频率和或者r（让米是粒子的质量）。

a) 表明体积磁化电流密度Ĵ米在圆柱体内为零，而圆柱体的侧面带有表面磁化电流密度ķ米， 和|ķ米|=|米|. b) 找到磁场乙气缸内外，在极限处H→∞. c) 现在考虑“扁平”圆柱体的相反情况，即H≪R, 并评估磁场乙0在圆柱体的中心。
d) 根据 c) 的结果，林R/H→∞乙0=0. 使用等效磁荷法得到同样的结果。

## 物理代写|电磁学代写electromagnetism代考|MCharged Particle in Crossed Electric and Magnetic Fields

a) 描述粒子的后续运动。
b) 用上面的结果讨论下面的问题。我们有一个带表面的平行板电容器小号, 板分离H和电压在，如图 5.4 所示。均匀磁场乙施加到电容器上，垂直于电容器电场，即平行于极板。紫外线辐射使负极板以零初始速度发射电子。评估最小值乙电子无法到达正极板。

## 物理代写|电磁学代写electromagnetism代考|Conducting Cylinder in a Magnetic Field

a) 评估作用在传导电子上的磁力，假设ω=2圆周率×102 s−1和乙=5×10−5 吨（地球磁场），以及磁力与离心力的比值。

c) 磁场乙1由气缸内的旋转电流产生，数量级为乙1/乙0（认为一个≈0.1 米 ).

a) 评估磁场乙由旋转电容器在整个空间上产生。
b) 评估两个旋转圆柱壳的电荷上的磁力，并将它们与静电力进行比较。