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## 物理代写|电磁学代写electromagnetism代考|Effects of non-uniform resistivity

Two geometrically identical cylindrical conductors have both height $h$ and radius $a$, but different resistivities $\rho_{1}$ and $\rho_{2}$. The two cylinders are connected in series as in Fig. 4.6, forming a single conducting cylinder of height $2 h$ and cross section $S=\pi a^{2}$. The two opposite bases are connected to a voltage source maintaining a potential difference $V$ through the system, as shown in the figure.

a) Evaluate the electric fields, the electric current and the current densities flowing in the two cylinders in stationary conditions.
b) Evaluate the surface charge densities at the surface separating the two materials, and at the base surfaces connected to the voltage source.

A spherical capacitor has internal radius $a$ and external radius $b$. The spherical shell $a<$ $r<b$ is filled by a lossy dielectric medium of relative dielectric permittivity $\varepsilon_{\mathrm{r}}$ and conductivity $\sigma$. At time $t=0$, the charge of the capacitor is $Q_{0}$.
a) Evaluate the time constant for the discharge of the capacitor.
b) Evaluate the power dissipated by Joule heating inside the capacitor, and compare it with the temporal variation of the electrostatic energy.

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

Topics. Stationary magnetic field in vacuum. Lorentz force. Motion of an electric point charge in a magnetic field. The magnetic force on a current. The magnetic field of steady currents. “Mechanical” energy of a circuit in a magnetic field. Biot-Savart law. Ampères’ circuital law. The magnetism of matter. Volume and surface magnetization current densities (bound currents). Magnetic susceptibility. The “auxiliary” vector H. Magnetic field boundary conditions. Equivalent magnetic charge method.
Units. In order to write formulas compatible with both SI and Gaussian units, we introduce two new “system dependent” constants, $k_{\mathrm{m}}$ and $b_{\mathrm{m}}$, defined as
$$k_{\mathrm{m}}=\left{\begin{array}{ll} \frac{\mu_{0}}{4 \pi}, & \mathrm{Sl}, \ \frac{1}{c}, & \text { Gaussian, } \end{array} \quad b_{\mathrm{m}}= \begin{cases}1, & \text { SI } \ \frac{1}{c}, & \text { Gaussian },\end{cases}\right.$$
where, again, $\mu_{0}=4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}$ is the “magnetic permeability of vacuum”, and $c=29979245800 \mathrm{~cm} / \mathrm{s}$ is the light speed in vacuum.

Basic equations The two Maxwell equations for the magnetic field $\mathbf{B}$ relevant to this chapter are
\begin{aligned} \boldsymbol{\nabla} \cdot \mathbf{B} &=0 \ \boldsymbol{\nabla} \times \mathbf{B} &=4 \pi k_{\mathrm{m}} \mathbf{J} \end{aligned}
Equation (5.2) is always valid (in the absence of magnetic monopoles), while (5.3) is valid in the absence of time-dependent electric fields. It is thus possible to introduce a vector potential $\mathbf{A}$, such that
$$\mathbf{B}(\mathbf{r})=\boldsymbol{\nabla} \times \mathbf{A}(\mathbf{r}),$$

Imposing the gauge condition $\boldsymbol{\nabla} \cdot \mathbf{A}=0$, the vector potential satisfies
$$\nabla^{2} \mathbf{A}(\mathbf{r})=-4 \pi k_{\mathrm{m}} \mathbf{J}(\mathbf{r})$$
which is the vector analogous of Poisson’s equation (2.1). Thus,
$$\mathbf{A}(\mathbf{r})=k_{\mathrm{m}} \int_{V} \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \mathrm{d}^{3} r^{\prime}$$

## 物理代写|电磁学代写electromagnetism代考|The Rowland Experiment

This experiment by Henry A. Rowland (1876) aimed at showing that moving charges generate magnetic fields. A metallic disk or radius $a$ and thickness $b \ll a$ is electrically charged and kept in rotation with a constant angular velocity $\omega$.
a) The disk rotates between two conducting plates, one at a distance $h \simeq 0.5 \mathrm{~cm}$ above its upper surface, and the other at $h$ below its lower surface, as in Fig. 5.1. The two plates are connected to the same terminal of a voltage source maintaining a potential difference $V_{0}=10^{4} \mathrm{~V}$, while the other terminal is connected to the disk by a sliding contact. Evaluate the surface charge density on the disk surfaces.
b) Calculate the magnetic field $\mathbf{B}{\mathrm{c}}$ near the center of the disk and the magnetic field component $B{r}$ parallel and close to the disk surfaces, as a function of the distance $r$ from the axis. Typical experimental values were $a=10 \mathrm{~cm}$, and $\omega \simeq 2 \pi \times 10^{2} \mathrm{rad} / \mathrm{s}$ (period $T=2 \pi / \omega=10^{-2} \mathrm{~s}$ ).
c) The field component $B_{r}$ generated by the disk at $r=a$ can be measured by orienting the apparatus so that $\hat{\mathbf{r}}$ is perpendicular to the Earth’s magnetic field $\mathbf{B}{\oplus}$, of strength $B{\oplus} \simeq 5 \times 10^{-5} \mathrm{~T}$, and measuring the deviation of a magnetic needle when the disk rotates. Find the deviation angle of the needle.

## 物理代写|电磁学代写electromagnetism代考|Effects of non-uniform resistivity

a) 评估静止条件下两个圆柱体中流动的电场、电流和电流密度。
b) 评估分离两种材料的表面以及连接到电压源的基面的表面电荷密度。

a) 评估电容器放电的时间常数。
b) 评估电容㕷内部焦耳热耤散的功率，并将其与静电能量的时间变化进行比较。

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

$\$ \$$k_{-} \wedge mathrm \left.{\mathrm{m}}\right}=\backslash left { \frac{\mu_{0}}{4 \pi}, \quad \mathrm{Sl}, \frac{1}{c}, \quad Gaussian, \backslash quad b_气mathrm{m}}= \left{1, \quad\right. SI \frac{1}{c}, \quad Gaussian 正确的。 \ \$$

$$\boldsymbol{\nabla} \cdot \mathbf{B}=0 \boldsymbol{\nabla} \times \mathbf{B} \quad=4 \pi k_{\mathrm{m}} \mathbf{J}$$

$$\mathbf{B}(\mathbf{r})=\boldsymbol{\nabla} \times \mathbf{A}(\mathbf{r}),$$

$$\nabla^{2} \mathbf{A}(\mathbf{r})=-4 \pi k_{\mathrm{m}} \mathbf{J}(\mathbf{r})$$

$$\mathbf{A}(\mathbf{r})=k_{\mathrm{m}} \int_{V} \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \mathrm{d}^{3} r^{\prime}$$

## 物理代写|电磁学代写electromagnetism代考|The Rowland Experiment

a) 圆盘在两块导电板之间旋转，一块在一定距离处 $h \simeq 0.5 \mathrm{~cm}$ 在其上表面之上，另一个在 $h$ 低于其下表面，如图 $5.1$ 所示。两个板连接到 保持电位差的电压源的同一端子 $V_{0}=10^{4} \mathrm{~V}$ ，而另一个端子通过滑动触点连接到磁盘。评估磁盘表面的表面电荷密度。
b) 计算磁场 $\mathrm{Bc} \mathrm{c}$ 靠近磁盘中心和磁场分量 $B r$ 平行和靠近磁盘表面，作为距离的函数 $r$ 从轴。典型的实验值是 $a=10 \mathrm{~cm}$ ，和
$\omega \simeq 2 \pi \times 10^{2} \mathrm{rad} / \mathrm{s}$ (时期 $T=2 \pi / \omega=10^{-2} \mathrm{~s}$ )。
c) 字段組件 $B_{r}$ 由磁盘在 $r=a$ 可以通过调整设备的方向来测量全垂直于地球碰场 $\mathbf{B} \oplus$ ，强度 $B \oplus \simeq 5 \times 10^{-5} \mathrm{~T}$ ，并测量磁盘旋转时磁针的