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# 物理代写|电磁学代写Electromagnetism代考|PHYS404 Magnetization

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## 物理代写|电磁学代写Electromagnetism代考|Magnetization

Several times already, we came to the conclusion that based on our current knowledge there are no “magnetic charges”, which is the reason why $\mathbf{B}$ is sourcefree or solenoidal:
$$\nabla \cdot \mathbf{B}=0 \text {. }$$
However, for convenience we will introduce the formal concept of fictitious magnetic charges. The reason is that it will simplify certain problems. Nevertheless, the physical source of static magnetic fields is always found in currents, i.e., moving electric charges (disregarding the spin of particles). We have already found that all such fields can be thought of as being created by superposition of suitable dipole fields. The spin of elementary particles causes a magnetic moment, which has no explanation in classical physics. Consequently, we can say that all static magnetic fields are ultimately caused by magnetic dipoles. Magnetic dipoles are also fundamental in connection with the question about interaction between matter and magnetic fields. We will need to discuss this too. In doing so, we will again find a formal, broad analogy between electric and magnetic effects (see Sect. 5.5). First, we will deal with the field of a volume distribution of magnetic dipoles. It is convenient to define the magnetization as
$$\mathbf{M}=\frac{\mathrm{d} \mathbf{m}}{\mathrm{d} \tau} .$$

## 物理代写|电磁学代写Electromagnetism代考|Forces on Dipoles in Magnetic Fields

A moving charge in a magnetic field experiences the force
$$\mathbf{F}=Q \mathbf{v} \times \mathbf{B} \text {. }$$
If we observe the motion of a charge density distribution $\rho(\mathbf{r})$, then the force per unit volume, also called force density, is
$$\mathbf{f}=\rho \mathbf{v} \times \mathbf{B} \text {. }$$

Since
$$\rho \mathbf{v}=\mathbf{g}$$
is just the current density, we write
$$\mathbf{f}=\mathbf{g} \times \mathbf{B} .$$
Integrating this equation over the cross section of a current carrying wire gives the force per unit length at a location of the wire
$$\frac{\mathbf{F}(\mathbf{r})}{l}=\mathbf{I}(\mathbf{r}) \times \mathbf{B}(\mathbf{r}),$$
where $\mathbf{I}$ is a vector quantity pointing in the direction of a wire element, having the magnitude of the total current $I$ (Fig. 5.31). First, consider a dipole, i.e., a current loop within a uniform magnetic field (Fig. 5.32). Clearly, all forces cancel – there is no net force. What remains is a force pair with a torque, trying to orient $\mathbf{m}$ into the direction of $\mathbf{B}$. It shall be noted without proof that this torque is
$$\frac{1}{\mu_0} \mathbf{m} \times \mathbf{B} \text {. }$$

## 物理代写|电磁学代写Electromagnetism代考|Magnetization

$$\nabla cdot \mathbf{B}=0$$

$$\mathbf{M}=\frac{\mathrm{d} \mathbf{m}}{\mathrm{d} \tau} 。$$

## 物理代写|电磁学代写|电磁学代考|磁场中双极子的作用力

$$\Δmathbf{F}=Q Δmathbf{v} Δtimes Δmathbf{B} 。$$

$$\mathbf{f}=rho\mathbf{v}\times\mathbf{B}。$$

$$\rho\mathbf{v}=\mathbf{g}。$$

$$\mathbf{f}=\mathbf{g} \times\mathbf{B} 。$$

$$\frac{\mathbf{F}(\mathbf{r})}{l}=\mathbf{I}(\mathbf{r}) \times \mathbf{B}(\mathbf{r})$$

$$\frac{1}{mu_0}。\mathbf{m} \times `mathbf{B} 。$$

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