### 物理代写|几何光学代写Geometrical Optics代考|PHYSICS134A

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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|几何光学代写Geometrical Optics代考|Energy Stored in Capacitor

The energy stored in the absence of the dielectric is
$$U_0=\frac{Q_0^2}{2 C_0}$$
After the battery is removed and the dielectric inserted, the charge on the capacitor remains the same. Hence, the energy stored in the presence of the dielectric is
$$U=\frac{Q_0^2}{2 C}$$
Using the relation $C=\varepsilon C_0$, then
$$U=\frac{Q_0^2}{2 \varepsilon C_0}$$
or
$$U=\frac{U_0}{\varepsilon}$$
Because $\varepsilon>1$, the final energy is less than the initial energy (see also Eq. (4.48)) $\Delta U=U-U_0<0$. We can account for the “missing” energy by noting that the dielectric, when inserted, gets pulled into the device. An external agent must do negative work to keep the dielectric from accelerating.
This work is simply the difference
$$W_a=U-U_0$$
Alternatively, the positive work done on the external agent by the system is
$$W=-W_a=U_0-U$$

## 物理代写|几何光学代写Geometrical Optics代考|Electric Polarization

Consider an electric field applied to a medium made up of a large number of particles, such as atoms or molecules. The charges bound in molecules will then respond to the external electric field, and they will follow the perturbed motion to align with the external field. Thus, the charge density within the molecules will be distorted. The dipole moments ${ }^3$ of each molecule will be different in comparison to the dipole moments in the absence of the applied electric field. That is, in the absence of the external field, the average dipole moments over all molecules of the substance are zero because the dipole vectors are oriented randomly. In contrast, in the presence of the applied electric field, the net dipole moment of the substance is different from zero. Therefore, in the medium, there is an average dipole moment per unit volume, which is called electric polarization $\mathbf{P}$, given as
$$\mathbf{P}(\mathbf{r})=\sum_i n_i\left\langle\mathbf{p}_i\right\rangle$$
In Eq. (4.51), $\mathbf{p}_i$ is the dipole moment of the molecule type $i$ in the medium, $\langle\cdots\rangle$ denotes the average over a small volume around $\mathbf{r}$, and $n_i$ is the average number per unit volume of the molecule type $i$ at the position $\mathbf{r}$.

If the net charge of the molecule $i$ is $Q_i$, and there is a macroscopic excess or free charge, the charge density at the macroscopic level is
$$\rho(\mathbf{r})=\sum_i n_i\left\langle Q_i\right\rangle+\rho_{\text {free }}$$
Note that, in general, average charge of a molecule $i$ is zero, $\left\langle Q_i\right\rangle=0$, and hence, the charge density $\rho$ is equal to the macroscopic excess or free charge, $\rho_{\text {free }}$.

In the following, we will consider the case of a continuous charge distribution, as in Fig. $3.6$ (Chap. 3), and see the medium from a macroscopic viewpoint. The potential at some point $P$ at the position $\mathbf{r}$ from a macroscopic small volume element $d V$ at the position $\mathbf{r}^{\prime}$ is the sum of the potential created by the charge of $d V, d q-\rho\left(\mathbf{r}^{\prime}\right) d V$ and the dipole moment of $d V$ is $\mathbf{P}\left(\mathbf{r}^{\prime}\right) d V$, assuming that there are no higher macroscopic multipole moment densities:
$$d \phi\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=k_e\left(\frac{\rho\left(\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}+\frac{\mathbf{P}\left(\mathbf{r}^{\prime}\right) \cdot\left(\mathbf{r}-\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^3}\right)$$

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Energy Stored in Capacitor

$$U_0=\frac{Q_0^2}{2 C_0}$$

$$U=\frac{Q_0^2}{2 C}$$

$$U=\frac{Q_0^2}{2 \varepsilon C_0}$$

$$U=\frac{U_0}{\varepsilon}$$

$$W_a=U-U_0$$

$$W=-W_a=U_0-U$$

## 物理代写|几何光学代写Geometrical Optics代考|Electric Polarization

$$\mathbf{P}(\mathbf{r})=\sum_i n_i\left\langle\mathbf{p}i\right\rangle$$ 在等式中。(4.51), $\mathbf{p}_i$ 是分子类型的偶极矩 $i$ 在媒体中， $\langle\cdots\rangle$ 表示周围小体积的平均值 $\mathbf{r}$ ，和 $n_i$ 是分子类型 每单位体积的平均数 $i$ 在那个位置r. 如果分子的净电荷 $i$ 是 $Q_i$ ，并且存在宏观过剩或自由电荷，宏观层面的电荷密度为 $$\rho(\mathbf{r})=\sum_i n_i\left\langle Q_i\right\rangle+\rho{\text {free }}$$

$$d \phi\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=k_e\left(\frac{\rho\left(\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}+\frac{\mathbf{P}\left(\mathbf{r}^{\prime}\right) \cdot\left(\mathbf{r}-\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^3}\right)$$

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## MATLAB代写

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