物理代写|电磁学代写electromagnetism代考|PHYS3040

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

In a similar manner, a static formulation can be written for the magnetic induction $\boldsymbol{B}^{\text {stat }}$. By applying the curl operator to equation $\operatorname{curl}\left(\mu^{-1} \boldsymbol{B}^{\text {stat }}\right)=\boldsymbol{J}$, we obtain
$$\operatorname{curl} \operatorname{curl}\left(\mu^{-1} B^{\text {stat }}\right)=\operatorname{curl} J \text {. }$$
In a homogeneous medium (for instance, in vacuum $\mu=\mu_0 \rrbracket_3$ ), and using the identity (1.36) again, we obtain the magnetostatic problem
$$-\Delta \boldsymbol{B}^{\text {stat }}=\mu_0 \text { curl } \boldsymbol{J}, \quad \operatorname{div} \boldsymbol{B}^{\text {stat }}=0,$$
whose solution, $\boldsymbol{B}^{\text {stat }}$, is called the magnetostatic field. This is a vector Poisson equation, i.e., an elliptic PDE (left Eq.), with a constraint (right Eq.). Again, this formulation leads to problems that are easier to solve than the complete set of Maxwell’s equations.

Note also that one has $\boldsymbol{B}^{\text {stat }}=\operatorname{curl} \boldsymbol{A}^{\text {stat }}$ (see (1.35)). If, moreover, the Coulomb gauge is chosen to remove the indetermination on the vector potential $\boldsymbol{A}^{\text {stat }}$, one finds the alternate magnetostatic problem
$$-\Delta \boldsymbol{A}^{\text {stat }}=\mu_0 \boldsymbol{J}, \quad \operatorname{div} \boldsymbol{A}^{\text {stat }}=0,$$
with $\boldsymbol{A}^{\text {stat }}$ as the unknown. Then, one sets $\boldsymbol{B}^{\text {stat }}=\operatorname{curl} \boldsymbol{A}^{\text {stat }}$ to recover the magnetostatic field.

物理代写|电磁学代写electromagnetism代考|A Scaling of Maxwell’s Equations

In order to define an approximate model, one has to neglect one or several terms in Maxwell’s equations. The underlying idea is to identify parameters, whose value can bee small (and thus. possibly négligiblé). To dérive a hiêrarchy of approximate models, one can perform an asymptotic analysis of those equations with respect to the parameters. This series of models is called a hierarchy, since considering a supplementary term in the asymptotic expansion leads to a new approximate model. An analogous principle is used, for instance, to build approximate (paraxial) models when simulating data migration in geophysics modelling (cf. among others $[41,85])$. From a numerical point of view, the approximate models are useful, first and foremost, if they coincide with a physical framework, and second, because in general, they efficiently solve the problem at a lower computational cost.

In the sequel, let us show how to build such approximate models formally (i.e., without mathematical justifications), recovering, in the process, static models, but also other intermediate ones.

Let us consider Maxwell’s equations in vacuum (1.26-1.29). As a first step, we introduce a scaling of these equations based on the following characteristic values:
$\bar{l}:$ characteristic length,
$\bar{t}$ : characteristic time,
$\bar{v}:$ characteristic velocity, with $\bar{v}=\bar{l} / \bar{t}$,
$\bar{E}, \bar{B}$ : scaling for $\boldsymbol{E}$ and $\boldsymbol{B}$,
$\bar{\varrho}, \bar{J}$ : scaling for $\varrho$ and $\boldsymbol{J}$.
In order to build dimensionless Maxwell equations, we set
\begin{aligned} \boldsymbol{x} &=\bar{l} \boldsymbol{x}^{\prime} \quad \Rightarrow \frac{\partial}{\partial x_i}=\frac{1}{\bar{l}} \frac{\partial}{\partial x_i^{\prime}} \ t &=\bar{t} t^{\prime} \quad \Rightarrow \frac{\partial}{\partial t}=\frac{1}{\bar{t}} \frac{\partial}{\partial t^{\prime}} \ \boldsymbol{E} &=\bar{E} \boldsymbol{E}^{\prime}, \text { etc. } \end{aligned}

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

$$\operatorname{curl} \operatorname{curl}\left(\mu^{-1} B^{\text {stat }}\right)=\operatorname{curl} J .$$

$$-\Delta \boldsymbol{B}^{\text {stat }}=\mu_0 \operatorname{curl} \boldsymbol{J}, \quad \operatorname{div} \boldsymbol{B}^{\text {stat }}=0,$$

$$-\Delta A^{\text {stat }}=\mu_0 \boldsymbol{J}, \quad \operatorname{div} A^{\text {stat }}=0,$$

物理代写|电磁学代写electromagnetism代考|A Scaling of Maxwell’s Equations

(因此。可能是 négligiblé) 。为了导出近似模型的层次结构，可以对这些方程关于参数进行渐近分析。 这一系列模型称为层次结构，因为在渐近展开中考虑补充项会导致新的近似模型。例如，在模拟地球物理 建模中的数据迁移时，使用类似的原理来构建近似 (近轴) 模型 (参见除其他外) $[41,85]$ ). 从数值的角 度来看，近似模型是有用的，首先，如果它们与物理框架一致，其次，因为一般来说，它们以较低的计算 成本有效地解决了问题。

$\bar{l}$ :特征长度，
$\bar{t}$ : 特征时间,
$\bar{v}:$ 特征速庻，与 $\bar{v}=\bar{l} / \bar{t}$,
$\bar{E}, \bar{B}:$ 缩放 $\boldsymbol{E}$ 和 $\boldsymbol{B}$,
$\bar{\varrho}, \bar{J}:$ 缩放 [email protected]$和$J\$.

$$\boldsymbol{x}=\bar{l} \boldsymbol{x}^{\prime} \quad \Rightarrow \frac{\partial}{\partial x_i}=\frac{1}{\bar{l}} \frac{\partial}{\partial x_i^{\prime}} t \quad=\bar{t} t^{\prime} \quad \Rightarrow \frac{\partial}{\partial t}=\frac{1}{\bar{t}} \frac{\partial}{\partial t^{\prime}} \boldsymbol{E}=\bar{E} \boldsymbol{E}^{\prime} \text {, etc. }$$

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