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# 物理代写|电磁学代写Electromagnetism代考|PHYS2200 The Electric Current and Ampere’s Law

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## 物理代写|电磁学代写Electromagnetism代考|The Electric Current and Ampere’s Law

The discovery of electric forces between electrically charged bodies has led to the previously discussed electrostatic concepts. Besides those, another type of force has been known for a long time, the so-called magnetic force, whose close relationship to the electric forces is a rather late discovery.

The earth, for instance, is surrounded or penetrated by a strange field, which expresses itself by exerting forces on specific materials. This field or those forces, respectively, have peculiar characteristics. For instance, they exhibit a force on a magnetic needle by trying to align it into a specific direction, while they exert no or only a minor net force on the needle as a whole. The primary effect is a torque and to a lesser degree net forces, which may even vanish entirely.

Historically, these phenomena were explained in terms of “magnetic charges”, which were thought to be located in the magnetic poles of a magnet. This linguistic use is more confusing than helpful, and we will not introduce these concepts here in this way. Magnetic forces are – as much as we know today – of a different kind, as electrostatic ones which we have dealt with so far. We will refrain from using this only seemingly apparent analogy that suggests magnetic fields as the result of magnetic charges. Based on our current knowledge, there are no magnetic charges. The cause of magnetic fields is rather an electric current, i.e. moving electric charges. By experiment, one finds that a current carrying wire in the vicinity of a magnetic needle exhibits a magnetic field that influences the needle. Before we study this in more detail, we have to define the electric current and electric current density. Observe an infinitesimal area element $d \mathbf{A}$ that is perpendicular to the flow of the charge and through which in the time interval $d t$ flows the charge $d^2 Q$. Then the vector of the current density is defined as
$$\mathbf{g}=\frac{d^2 Q}{d t d A} \frac{d \mathbf{A}}{d A}$$
The flux of $\mathbf{g}$ through a surface $A$ is the electric current $I$.
$$I=\int_A \mathbf{g} \cdot d \mathbf{A}=\int_A \frac{d^2 Q}{d t}=\frac{d}{d t} \int_A d Q=\frac{d Q}{d t}$$

## 物理代写|电磁学代写Electromagnetism代考|The Principle of Charge Conservation and Maxwell’s First Equation

Consider an arbitrary volume. Charges contained in it may flow in, or out of it. This is the only way for the overall charge in the volume to change. The only other possibility would be that charges spontaneously appear or disappear. Our experience teaches us that this is not the case. This is the Principle of Conservation of Electric Charge.

Expressed in a more general way, we find that the overall charge in the universe is constant (probably zero). Although there exist processes where new charges are created, this does not change the overall charge balance, because always the same number of positive as negative charges are created. Our experience up to now is that naturally occurring charges always come in multiples of an elementary charge. For instance, in the negative charge of an electron or in the positive charge of its counterpart, the proton. It is possible that a photon creates a pair of particles with opposite charges (for example a particle, antiparticle pair; electron, positron or proton, antiproton). We need to mention particles with charges that are one third or two thirds of the elementary charge (quarks) which, however, do not change the principle of charge conservation.
This principle is mathematically formulated as follows:
$$\oint \mathbf{g} \bullet d \mathbf{A}=-\frac{\partial}{\partial t} \int \rho \cdot d \tau=\int \nabla \bullet \mathbf{g} \cdot d \tau .$$
Consequently
$$\nabla \bullet \mathrm{g}+\frac{\partial \rho}{\partial t}=0 \text {. }$$
This is the continuity equation. It is an expression of charge conservation. On the other hand
$$\rho=\nabla \bullet \mathbf{D}$$
and therefore
$$\nabla \cdot\left(\mathbf{g}+\frac{\partial \mathbf{D}}{\partial t}\right)=0 .$$
This means that the vector sum $\mathbf{g}+\partial \mathbf{D} / \partial t$ is source free. Therefore, it is possible to express it as the curl of a suitable vector field, as according to (1.40) the divergence of any curl vanishes:
$$\nabla \times \mathbf{a}=\mathbf{g}+\frac{\partial \mathbf{D}}{\partial t}$$

## 物理代写电磁学代写Electromagnetism代考|The Electric Current and Ampere’s Law

$$\mathbf{g}=\frac{d^2 Q}{d t d A} \frac{d \mathbf{A}}{d A}$$

$$I=\int_A \mathbf{g} \cdot d \mathbf{A}=\int_A \frac{d^2 Q}{d t}=\frac{d}{d t} \int_A d Q=\frac{d Q}{d t}$$

## 物理代写|电磁学代写Electromagnetism代考|The Principle of Charge Conservation and Maxwell’s First Equation

$$\oint \mathbf{g} \bullet d \mathbf{A}=-\frac{\partial}{\partial t} \int \rho \cdot d \tau=\int \nabla \bullet \mathbf{g} \cdot d \tau$$

$$\nabla \bullet \mathrm{g}+\frac{\partial \rho}{\partial t}=0 .$$

$$\rho=\nabla \bullet \mathbf{D}$$

$$\nabla \cdot\left(\mathbf{g}+\frac{\partial \mathbf{D}}{\partial t}\right)=0$$

$$\nabla \times \mathbf{a}=\mathbf{g}+\frac{\partial \mathbf{D}}{\partial t}$$

## MATLAB代写

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