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

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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|Superposition Principle

According to superposition principle, at any point $P$, the total electric field due to a set of discrete point charges, $q_{1}, q_{2}, \ldots, q_{N}$, positive and negative charges, is equal to the sum of the individual charge electric field vectors (see Fig. 1.5). Mathematically, we sun wrile
$$\mathbf{E}(\mathbf{r})=\sum_{i=1}^{N} \mathbf{E}{i}=\sum{i=1}^{N} k_{e} \frac{q_{i}}{\left|\mathbf{r}-\mathbf{r}{i}\right|^{2}} \hat{\mathbf{r}}{i}$$
In Eq. (1.12), $\left|\mathbf{r}-\mathbf{r}{i}\right|$ is the distance from $q{i}$ to the point $P$ (the location of a test charge), where $\mathbf{r}$ is the position vector of the point $P$ with respect to some reference frame, as indicated in Fig. 1.5, and $\mathbf{r}{i}$ is the position vector of the charge $i$ in that reference frame. Furthermore, $\hat{\mathbf{r}}{i}$ is a unit vector directed from $q_{i}$ toward $P$.

Note that in Eq. (1.12) the dependence of $\mathbf{E}$ on only position vector of point $P, \mathbf{r}$, assumes a static configuration of the charges in space. That is, for some other configuration distribution of charges in space, $\mathbf{E}$ at the same point $P$ may be different. Note that often for convenience, Eq. (1.12) is also written as

$$\mathbf{E}(\mathbf{r})=\sum_{i=1}^{N} k_{e} \frac{q_{i}\left(\mathbf{r}-\mathbf{r}{i}\right)}{\left|\mathbf{r}-\mathbf{r}{i}\right|^{3}}$$
where
$$\hat{\mathbf{r}}{i}=\frac{\mathbf{r}-\mathbf{r}{i}}{\left|\mathbf{r}-\mathbf{r}_{i}\right|}$$
If the distances between charges in a set of charges are much smaller, compare with the distance of the set from a point where the electric field is to be calculated, then charge distribution is continuous.

To calculate the net electric field created by a continuous charge distribution in some volume $V$, we follow these steps. First, we divide the charge distribution into macroscopically small elements with small charge $\Delta q_{i}$, as shown in Fig. 1.6a. $\Delta q_{i}=\rho_{i} \Delta V$, where $\rho_{i}$ is seen from a microscopic viewpoint as a uniform charge density within the volume element $i$, which represents one of the possible configurations of microscopic description. It is important to note that with “macroscopically small” we should understand a small volume in space with a characteristic microscopic configuration of the charges inside it that can, on average, macroscopically be represented as a point-like charge, $\Delta q_{i}$. Then, we calculate the electric field due to one of these macroscopically point charges, $\Delta q_{i}$, at some point $P$ at distance $\left|\mathbf{r}-\mathbf{r}{i}\right|$ from the charge element, $\Delta q{i}$, as
$$\Delta \mathbf{E}\left(\mathbf{r}, \mathbf{r}{i}\right)=k{e} \frac{\Delta q_{i}}{\left|\mathbf{r}-\mathbf{r}{i}\right|^{2}} \hat{\mathbf{r}}{i}$$
where $\hat{\mathbf{r}}{i}$ is a unit vector directed from the charge element $\Delta q{i}$ toward $P$. Here, $\mathbf{r}$ is position vector of point $P$ in some reference frame, and $\mathbf{r}{i}$ is the position vector of the macroscopically point charge $\Delta q{i}$.

## 物理代写|电磁学代写electromagnetism代考|Electric Field Lines

By definition, electric field lines are drawn to follow the same direction as the electric field vector at any point. Furthermore, the electric field vector is tangent to the line at every point along the field line.
The electric field lines are such that $\mathbf{E}$ is tangent to the electric field line at each point. The number of lines per unit surface area passing a surface perpendicular to the lines is proportional to the magnitude $|\mathbf{E}|$ in that region. Furthermore, the lines are directed radially away from the positive point charge. Moreover, the lines are directed radially toward the negative point charge.
In Fig. 1.7, we show the electric field lines of a negative and positive point charge. It can be seen that for a negative point charge, $-q$, the electric field lines are drawn toward the charge (see Fig. 1.7a). On the other hand, for a positive point charge, $+q$, electric field lines are leaving the charge, as shown in Fig. 1.7b.

The lines start from a positive charge and end on a negative charge. Also, the number of lines drawn, leaving a positive charge, or approaching a negative charge is proportional to the magnitude of the charge. Moreover, no two field lines can cross.
In Fig. 1.8, we show the electric field vector for a positive point charge $+q$ located at the point $(0,3,0)$ (Fig. 1.8b) and a negative point charge $-q$ located at $(0,-3,0)$ (Fig. 1.8a), colored according to the magnitude of the electric field $\mathbf{E}$ using a color scaling, as depicted in Fig. 1.8. Besides, the electric field lines of the resultant electric field are shown in Fig. $1.8 \mathrm{c}$.

## 物理代写|电磁学代写electromagnetism代考|Motion in Uniform Electric Field

Suppose a charge particle of mass $m$ and charge $q$ is moving in a uniform electric field $\mathbf{E}$. Electric field $\mathbf{E}$ exerts on a particle placed in it the force
$$\mathbf{F}=q \mathbf{E}$$

If that force is equal to the resultant force exerted on the particle, it causes the particle to accelerate, based on Newton’s second law:
$$m \mathbf{a}=q \mathbf{E}$$
The acceleration gained by the charge is given as
$$\mathbf{a}=\frac{q}{m} \mathbf{E}$$
Therefore, if $\mathbf{E}$ is uniform (that is, constant in magnitude and direction), then $\mathbf{a}$ is constant. Furthermore, if the particle has a positive charge, then its acceleration is in the direction of the electric field. On the other hand, if the particle has a negative charge, then its acceleration is in the direction opposite the electric field.

## 物理代写|电磁学代写electromagnetism代考|Superposition Principle

r^一世=r−r一世|r−r一世|

Δ和(r,r一世)=ķ和Δq一世|r−r一世|2r^一世

F=q和

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## MATLAB代写

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