### 物理代写|电磁学代写electromagnetism代考|ELEC3104

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## 物理代写|电磁学代写electromagnetism代考|Solvability of Maxwell’s Equations

What about the proof of the existence of electromagnetic fields on $\mathbb{R}^3$ ?
To begin with, there exist many “experimental proofs” of the existence of electromagnetic fields! These experiments actually led to the definition of the equations that govern electromagnetic phenomena, and of the related electromagnetic fields, by Maxwell and many others during the nineteenth and twentieth centuries. So, it is safe to assume that these fields exist, the challenge being mathematical and computational nowadays…

Where does the theory originate? Let us give a brief account of one of the more elementary (mathematically speaking!) results on charged particles at rest (results have also been obtained for circuits, involving currents).

The fundamental experimental results we report here were obtained by Charles Augustin de Coulomb in 1785, when he studied repulsive or attractive forces between charged bodies, small elder balls. In the air-a homogeneous medium respective positions are $x_1$ and $\boldsymbol{x}$, whereas their respective electric charges are $q_1$ and $q$. In short, Coulomb’s results (now known as Coulomb’s law) state that the two particles interact electrically ${ }^7$ with one another, in the following way. The force $\boldsymbol{F}$ acting on particle part and originating from particle part $_1$ is such that:

• it is repulsive if $q_1 q>0$, and attractive if $q_1 q<0$;
• its direction is parallel to the line joining the two particles;
• its modulus is proportional to $\left|x-x_1\right|^{-2}$;
• its modulus is also proportional to $q_1$ and $q$.

## 物理代写|电磁学代写electromagnetism代考|Potential Formulation of Maxwell’s Equations

Let us introduce another formulation of Maxwell’s equations. For the sake of simplicity, we assume that we are in vacuum (in all space, $\mathbb{R}^3$ ), with Maxwell’s equations written in differential form as Eqs. (1.26-1.29). According to the divergencefree property of the magnetic induction $\boldsymbol{B}$, there exists a vector potential $\boldsymbol{A}$ such that
$$B=\operatorname{curl} A$$

Plugging this into Faraday’s law (1.27), we obtain
$$\operatorname{curl}\left(\frac{\partial \boldsymbol{A}}{\partial t}+\boldsymbol{E}\right)=0$$
Then, there exists a scalar potential $\phi$ such that
$$\frac{\partial A}{\partial t}+\boldsymbol{E}=-\operatorname{grad} \phi .$$
This allows us to introduce a formulation in the variables $(\boldsymbol{A}, \boldsymbol{\phi})$ – the vector potential and the scalar potential, respectively – since it holds there that
\begin{aligned} &\boldsymbol{E}=-\operatorname{grad} \phi-\frac{\partial \boldsymbol{A}}{\partial t} \ &\boldsymbol{B}=\operatorname{curl} \boldsymbol{A} \end{aligned}
This formulation requires only the four unknowns $\boldsymbol{A}$ and $\phi$. instead of the six unknowns for the $\boldsymbol{E}$ and $\boldsymbol{B}$-field formulation. Moreover, any couple $(\boldsymbol{E}, \boldsymbol{B})$ defined by Eqs. (1.34-1.35) automatically satisfies Faraday’s law and the absence of free magnetic monopoles. From this (restrictive) point of view, the potentials $\boldsymbol{A}$ and $\phi$ are independent of one another. Now, if one takes into account Ampère’s and Gauss’s laws, constraints appear in the choice of $\boldsymbol{A}$ and $\phi$ (see Eqs (1.37-1.38) below). Also, the vector potential $\boldsymbol{A}$ governed by Eq. (1.35) is determined up to a gradient of a scalar function: there lies an indetermination that has to be removed. On the other hand, for the scalar potential, the indetermination is up to a constant: it can be removed simply by imposing a vanishing limit at infinity.

## 物理代写|电磁学代写electromagnetism代考|Solvability of Maxwell’s Equations

• 如果 $q_1 q>0$, 并且如果 $q_1 q<0$;
• 它的方向平行于连接两个粒子的线;
• 它的模量与 $\left|x-x_1\right|^{-2}$;
• 它的模量也与 $q_1$ 和 $q$.

## 物理代写|电磁学代写electromagnetism代考|Potential Formulation of Maxwell’s Equations

$$B=\operatorname{curl} A$$

$$\operatorname{curl}\left(\frac{\partial \boldsymbol{A}}{\partial t}+\boldsymbol{E}\right)=0$$

$$\frac{\partial A}{\partial t}+\boldsymbol{E}=-\operatorname{grad} \phi$$

$$\boldsymbol{E}=-\operatorname{grad} \phi-\frac{\partial \boldsymbol{A}}{\partial t} \quad \boldsymbol{B}=\operatorname{curl} \boldsymbol{A}$$

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## MATLAB代写

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