### 物理代写|几何光学代写Geometrical Optics代考|PHYS201

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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
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• 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 Storage in the Electric Field

To transfer an amount of charge from one plate of a capacitor to the other during the process of charging the capacitor, an external work is done against the electric field. That work stores in the capacitor in the form of the potential energy. For that, let $q$ be the charge on the capacitor at some instant during the charging process when the potential difference across the capacitor is $\Delta V=q / C$. At that instant, one of the plates is carrying a charge $+q$ and the other $-q$. To transfer an increment of charge $d q$ from the plate with charge $-q$ (which is at a lower electric potential) to the plate carrying charge $+q$ (which is at a higher electric potential) an elementary work is done against the electric field:
$$d W=\Delta V d q=\frac{q}{C} d q$$
To calculate the total work required to charge the capacitor from $q=0$ to final charge $Q$, we integrate Eq. (4.27) as follows:
$$W=\int_0^Q \frac{q}{C} d q=\frac{1}{2} \frac{Q^2}{C}$$

This work done to charge the capacitor stores in the capacitor as an electric potential energy $U$. Therefore, $U=W$. Also, we can express the potential energy $U$ in the following forms:
\begin{aligned} U & =\frac{1}{2} \frac{Q^2}{C} \ & =\frac{1}{2} Q \Delta V \ & =\frac{1}{2} C(\Delta V)^2 \end{aligned}
Note that all expressions given by Eqs. (4.29)-(4.31) are equivalent; that is, they can all be used to calculate the potential energy stored in a capacitor depending on what is known. We can consider the energy stored in a capacitor as being stored in the electric field created between the plates as the capacitor is charged. This description is reasonable from the viewpoint that the electric field is proportional to the charge $Q$ stored on a capacitor. For a capacitor of two parallel plates, the potential difference is related to the electric field through a simple relationship $\Delta V=E d$. Furthermore, its capacitance is $C=\epsilon_0 \frac{A}{d}$. Then, we obtain
$$U=\frac{1}{2}\left(\epsilon_0 \frac{A}{d}\right)(E d)^2=\frac{1}{2} \epsilon_0(A d) E^2$$
Since the volume is $A d$, then the energy density is given
$$u_E=\frac{U}{A d}=\frac{1}{2} \epsilon_0 E^2$$
This expression is generally valid. That is, the energy density in any electric field is proportional to the square of the magnitude of the electric field at a given point.

## 物理代写|几何光学代写Geometrical Optics代考|Electrostatics of Macroscopic Media and Dielectrics

There exist many materials that do not allow electric charges to move freely within them, or may allow such motion to occur only very slowly. Those materials are used to block the flow of electrical current, and to form the insulators. For example, they can create insulating layers between the plates of a capacitor. Those materials are known as dielectric materials. As an application, the use of the dielectric material for a capacitor reduces its size for a given capacitance or increases its working voltage. Note that a dielectric material subject to a high enough electric field becomes a conductor; that is, the dielectric material experiences a dielectric breakdown. Thus, there exists a maximum voltage for dielectric capacitors to work. For example, there is a maximum power that a coaxial cable can adequately function in high-power applications such as radio transmitters; similarly, for microcircuits there are maximum voltages, which can be handled.

To know about the differences between dielectric and conducting materials, we can consider their behavior in electric fields. In particular, we have shown in Fig. 4.7 a conducting and dielectric sheet between the parallel plates in which a potential difference exists. That is, there are an equal amount of opposite charges on the two plates.

In the conducting sheet, the conducting electrons are free to move, and they establish a surface charge which exactly cancels the electric field within the conductor, as shown in Fig.4.7. That is, the surface charge density of the plates and conducting sheet is the same but with opposite sign. On the other hand, the electrons in the dielectric material are bound to atoms, and the external electric field causes only a displacement of the electronic configuration of atoms (see Fig. 4.7). However, it is sufficient to produce some surface charge with density $\sigma_{\text {ind }}$ (called an induced charge). We say that the dielectric is polarized. Note that the surface charge is not able to cancel the external electric field within the sheet; however, it does reduce. In the following, we will introduce a simplified molecular theory of dielectrics to understand the behavior of dielectric materials in the presence of an external electric field. ${ }^1$ A more complicated, but more precise theory, will be introduced in the following sections, accounting for electric polarization of the ponderable media. ${ }^2$

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Energy Storage in the Electric Field

$$d W=\Delta V d q=\frac{q}{C} d q$$

$$W=\int_0^Q \frac{q}{C} d q=\frac{1}{2} \frac{Q^2}{C}$$

$$U=\frac{1}{2} \frac{Q^2}{C} \quad=\frac{1}{2} Q \Delta V=\frac{1}{2} C(\Delta V)^2$$

$$U=\frac{1}{2}\left(\epsilon_0 \frac{A}{d}\right)(E d)^2=\frac{1}{2} \epsilon_0(A d) E^2$$

$$u_E=\frac{U}{A d}=\frac{1}{2} \epsilon_0 E^2$$

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