物理代写|电磁学代写electromagnetism代考|ELEC2300

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物理代写|电磁学代写electromagnetism代考|Electrical Charges

There exist several simple experiments to demonstrate the existence of electrical charges and forces. For example,

1. When we comb our hair on a dry day, we find that the comb attracts pieces of paper.
2. The same effect of attracting pieces of paper occurs when materials such as glass or rubber are rubbed with silk or fur.

As a general rule, for every material behaving in that way, we can say that it is electrified, or it becomes electrically charged.

Benjamin Franklin (1706-1790) found that there exist two types of electric charges, namely positive and negative. The following experiment can be used to demonstrate his finding. Suppose that we rubber with fur a hard rubber rod. In addition, we rub a glass rod with silk material. Then, if the glass rod is brought near

the rubber rod, we will observe that the two attract each other. However, if we bring near each other two charged rubber rods or two charged glass rods, then the two repel each other. This experiment indicates the existence of two different states of electrification for the rubber and glass. Furthermore, it finds that like charges repel each other and unlike charges attract each other.

By convention, the electric charge on the glass rod is positive, and that on the rubber rod is negative. Based on that convention, any charged object repelled by another charged object must have the same sign of charge with it, and any charged object attracted by another charged object must have an opposite sign of charge. It is important to note that the electricity model of Franklin implies that electric charge is always conserved. That is, an electrified state (positive or negative) is due to the charge transfer from one object to the other. In other words, when an object gains some amount of positive/negative charge, then the other gains an equal amount of the electric charge of the opposite sign.

Robert Millikan (1868-1953), in 1909, discovered that electric charge always appears as a multiple integer of a fundamental amount of charge, called $e$ such that the electric charge $q$, which is a standard symbol for the charge, is quantized as
$$q=N e$$
Here, $N$ is an integer number, $N=0, \pm 1, \pm 2, \ldots$

物理代写|电磁学代写electromagnetism代考|Coulomb’s Law

Based on an experiment performed by Coulomb, the electric force between two charged particles at rest is proportional to the inverse of the square of distance $r$ between them and directed along the line joining the two particles. In addition, the electric force is proportional to the charges $q_{1}$ and $q_{2}$ on each particle. Also, the electric force is attractive if the charges are of opposite sign and repulsive if the charges have the same sign. That is known as Coulomb’s Law.

Definition 1.1 Force is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, the law may be written as
$$F=k_{e} \frac{\left|q_{1}\right|\left|q_{2}\right|}{r^{2}}$$
In Eq. (1.2), $k_{e}$ is the Coulomb constant. Note that, in SI, the unit of charge is the coulomb (C). Therefore, the Coulomb constant $k_{e}$ in SI units has the value
$$k_{c}=8.9875 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}$$
Often, the constant is written as

where $\epsilon_{0}$ is the permittivity of free space given by
$$\epsilon_{0}=8.8542 \times 10^{-12} \mathrm{C}^{2} / \mathrm{N} \cdot \mathrm{m}^{2}$$
Coulomb’s force is a vector; hence it has a magnitude expressed by Eq. (1.2) and a direction. Therefore, the Coulomb’s law can be expressed in vector form concerning the electric force, $\mathbf{F}{12}$, exerted by the charge $q{1}$ (positive or negative) on another charge $q_{2}$ (positive or negative) as
$$\mathbf{F}{12}=k{e} \frac{q_{1} q_{2}}{r^{2}} \hat{\mathbf{r}}$$
In Eq.(1.6), $\hat{\mathbf{r}}$ denotes a unit vector pointing from $q_{1}$ to $q_{2}$. Note that based on the Newton’s third law, the electric force, $\mathbf{F}{21}$, exerted by a charge $q{2}$ (positive or negative) on a second charge $q_{2}$ (positive or negative) is
$$\mathbf{F}{21}=-\mathbf{F}{12}$$
Figure $1.1$ illustrates graphically the direction of Coulomb’s force vectors for different combinations of the pairs of positive and negative charges, namely negativenegative, positive-positive, and negative-positive charge-charge interactions.

物理代写|电磁学代写electromagnetism代考|Force Fields

The field forces act through space, producing an effect even when no physical contact between the objects occurs. As an example, we can mention the gravitational field. Michael Faraday developed a similar approach to electric forces. That is, an electric field exists in the region of space around any charged body, and when another charged body is inside this region of the electric field, an electric force acts on it.

Definition 1.2 The electric field $\mathbf{E}$ at a point in space is defined as the electric force $\mathbf{F}{e}$ acting on a positive test charge $q{0}$ placed at that point divided by the magnitude of the test charge:
$$\mathbf{E}=\frac{\mathbf{F}{e}}{q{0}}$$

The vector $\mathbf{E}$ has the SI units of newtons per coulomb (N/C). Figure $1.3$ illustrates the electric field $\mathbf{E}$ created by a positively charged sphere with total charge $Q$ at the positive test charge $q_{0}$. Here, we have assumed that the test charge $q_{0}$ is small enough that it does not disturb the charge distribution of the sphere responsible for the electric field.

Note that $\mathbf{E}$ is the field produced by some charge external to the test charge, and it is not the field produced by the test charge itself. Also, note that the existence of an electric field is a property of its source. For example, every electron comes with its electric field. An electric field exists at a point if a test charge at rest at that point experiences an electric force. The electric field direction is the direction of the force on a positive test charge placed in the field. Once we know the magnitude and direction of the electric field at some point, the electric force exerted on any charged particle (either positive or negative) placed at that point can be calculated. The electric field exists at some point space, including the free space, independent of the existence of another test charge at that point.

To determine the direction of electric field, consider a point charge $q$ loeated some distance $r$ from a test positive charge $q_{0}$ located at a point $P$, as shown in Fig. 1.4. fiel Coulomb’s law defines the force exerted by $q$ on $q_{0}$ as
$$\mathbf{F}{e}=k{e} \frac{q q_{0}}{r^{2}} \hat{\mathbf{r}}$$
where $\hat{\mathbf{r}}$ represents the usual unit vector directed from $q$ toward $q_{0}$ (see Fig. 1.4).

物理代写|电磁学代写electromagnetism代考|Electrical Charges

1. 当我们在干燥的日子梳理头发时，我们发现梳子会吸引纸片。
2. 当用丝绸或毛皮摩擦玻璃或橡胶等材料时，也会出现吸引纸片的相同效果。

q=ñ和

物理代写|电磁学代写electromagnetism代考|Coulomb’s Law

F=ķ和|q1||q2|r2

ķC=8.9875×109 ñ⋅米2/C2

ε0=8.8542×10−12C2/ñ⋅米2

F12=ķ和q1q2r2r^

F21=−F12

F和=ķ和qq0r2r^

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