物理代写|电磁学代写electromagnetism代考|PHYC20014

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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|Equivalent Reformulation of Maxwell’s Equations

Starting from the integral form of Maxwell’s equations (1.1-1.4), one can reformulate them in a differential form, ${ }^3$ with the help of Stokes and Ostrogradsky formulas
$$\int_S \operatorname{curl} \boldsymbol{F} \cdot \boldsymbol{d} S=\int_{\partial S} \boldsymbol{F} \cdot \boldsymbol{d} \boldsymbol{l} \text { and } \int_V \operatorname{div} \boldsymbol{F} d V=\int_{\partial V} \boldsymbol{F} \cdot \boldsymbol{d} S .$$
One easily derives the differential Maxwell equations (system of units SI):
\begin{aligned} \frac{\partial \boldsymbol{D}}{\partial t}-\operatorname{curl} \boldsymbol{H} &=-\boldsymbol{J}, \ \frac{\partial \boldsymbol{B}}{\partial t}+\operatorname{curl} \boldsymbol{E} &=0, \ \operatorname{div} \boldsymbol{D} &=\varrho, \ \operatorname{div} \boldsymbol{B} &=0 . \end{aligned}
The differential charge conservation equation can be expressed as
$$\frac{\partial \varrho}{\partial t}+\operatorname{div} \boldsymbol{J}=0 .$$
However, the above set of equations is not equivalent to the integral set of equations. As a matter of fact, two notions are missing.

The first one is related to the behavior of the fields across an interface between two different media. Let $\Sigma$ be such an interface.

Starting from the volumic integral equations (1.3)-(1.4), we consider thin volumes $V_\epsilon$ crossing the interface. As $\epsilon$ goes to zero, their height goes to zero, and so does the area of their top and bottom faces (parallel to the interface), with proper scaling. The top and bottom faces are disks whose radius is proportional to $\epsilon$, while the height is proportional to $\epsilon^2$. As a consequence, the area of the lateral surface is proportional to $\epsilon^3$ and its contribution is negligible as $\epsilon$ goes to zero.

物理代写|电磁学代写electromagnetism代考|Constitutive Relations

Maxwell’s equations are insufficient to characterize the electromagnetic fields completely. The system has to be closed by adding relations that describe the properties of the medium in which the electromagnetic fields propagate. These are the so-called constitutive relations, relating, for instance, $\boldsymbol{D}$ and $\boldsymbol{B}$ to $\boldsymbol{E}$ and $\boldsymbol{H}$, namely
$$\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{H}) \quad \text { and } \quad \boldsymbol{B}=\boldsymbol{B}(\boldsymbol{E}, \boldsymbol{H})$$
(We could also choose $a$ priori to use such a relation as $\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{B})$, etc.)

These constitutive relations can be very complex. For this reason, we will make a number of assumptions on the medium (listed below), which lead to generic expressions of the constitutive relations. This will yield three main categories of medium, which are, from the more general to the more specific:

1. the chiral medium, a linear and bi-anisotropic medium;
2. the perfect medium, a chiral, non-dispersive and anisotropic medium;
3. the inhomogeneous medium, a perfect and isotropic medium, and its subcategory, the homogeneous medium, which is, in addition, spatially homogeneous.

In what follows, $\boldsymbol{E}(t)$ (or $\boldsymbol{B}(t)$, etc.) denotes the value of the electric field on $\mathbb{R}^3$ at time $t: \boldsymbol{x} \mapsto \boldsymbol{E}(t, \boldsymbol{x})$. Let us now list the assumptions about the medium.

• The medium is linear. This means that its response is linear with respect to electromagnetic inputs (also called excitations later on). In addition, it is expected that when the inputs are small, the response of the medium is also small.
• The medium satisfies a causality principle. In other words, the value of $(\boldsymbol{D}(t), \boldsymbol{B}(t))$ depends only on the values of $(\boldsymbol{E}(s), \boldsymbol{H}(s))$ for $s \leq t$.
• The medium satisfies a time-invariance principle. Let $\tau>0$ be given. II the response to $t \mapsto(\boldsymbol{E}(t), \boldsymbol{H}(t))$ is $t \mapsto(\boldsymbol{D}(t), \boldsymbol{B}(t))$, then the response to $t \mapsto$ $(\boldsymbol{E}(t-\tau), \boldsymbol{H}(t-\tau))$ is $t \mapsto(\boldsymbol{D}(t-\tau), \boldsymbol{B}(t-\tau))$.

物理代写|电磁学代写electromagnetism代考|Equivalent Reformulation of Maxwell’s Equations

$$\int_S \operatorname{curl} \boldsymbol{F} \cdot \boldsymbol{d} S=\int_{\partial S} \boldsymbol{F} \cdot \boldsymbol{d} \boldsymbol{l} \text { and } \int_V \operatorname{div} \boldsymbol{F} d V=\int_{\partial V} \boldsymbol{F} \cdot \boldsymbol{d} S .$$

$$\frac{\partial \boldsymbol{D}}{\partial t}-\operatorname{curl} \boldsymbol{H}=-\boldsymbol{J}, \frac{\partial \boldsymbol{B}}{\partial t}+\operatorname{curl} \boldsymbol{E}=0, \operatorname{div} \boldsymbol{D}=\varrho, \operatorname{div} \boldsymbol{B}=0 .$$

$$\frac{\partial \varrho}{\partial t}+\operatorname{div} \boldsymbol{J}=0$$

物理代写|电磁学代写electromagnetism代考|Constitutive Relations

$$\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{H}) \quad \text { and } \quad \boldsymbol{B}=\boldsymbol{B}(\boldsymbol{E}, \boldsymbol{H})$$
(我们也可以选择 $a$ 先验地使用这样的关系 $\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{E}, \boldsymbol{B})$ ， ETC。)

1. 手性介质，一种线性和双各向异性介质；
2. 完美介质，手性、非色散和各向异性介质；
3. 非均质介质，一种完美的各向同性介质，及其子类别，均质介质，此外，它在空间上是均质的。
在接下来的内容中， $\boldsymbol{E}(t)$ (或者 $\boldsymbol{B}(t)$ 等) 表示电场的值 $\mathbb{R}^3$ 有时 $t: \boldsymbol{x} \mapsto \boldsymbol{E}(t, \boldsymbol{x})$. 现在让我们列出关于介质的 假设。
• 介质是线性的。这意味着它的响应相对于电磁输入 (后面也称为激励) 是线性的。此外，预计当输入较小 时，介质的响应也较小。
• 媒介满足因果关系原则。换句话说，价值 $(\boldsymbol{D}(t), \boldsymbol{B}(t))$ 仅取决于的值 $(\boldsymbol{E}(s), \boldsymbol{H}(s))$ 为了 $s \leq t$.
• 该介质满足时不变原理。让 $\tau>0$ 被给予。二、回应 $t \mapsto(\boldsymbol{E}(t), \boldsymbol{H}(t))$ 是 $t \mapsto(\boldsymbol{D}(t), \boldsymbol{B}(t))$ ，然后响应 $t \mapsto(\boldsymbol{E}(t-\tau), \boldsymbol{H}(t-\tau))$ 是 $t \mapsto(\boldsymbol{D}(t-\tau), \boldsymbol{B}(t-\tau)) .$

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