//開學之后是相當忙，完整的《電磁》英文筆記大概是更不動了

//這個文集以后的內(nèi)容將是一些小專題，不再對所有理論進行嚴格的引入與推導。

I Abstract

In this section we will discuss polarize and magnetize. We will discuss how electromagnetic field affect and be affected by matters. Dielectrics, or insulators, have an important factor?to describe the intensity they interact with electric field, the dielectric constant?. There's also a factor called magnetic permeability,?, which describes the effect of magnetic field.

We know charges and magnetic dipoles in molecules and atoms are not uniformly distributed. So when there's external electric field, it will cause slight displacement of opposite charges. Molecular or atomic magnetic dipole can also be affected by external magnetic field. That's the principle of Polarize and Magnetize.

II Dielectric Constant

Suppose a charged capacitor consisting two close conductive plains and dielectrics between them, with surface charge density? respectively. We already know the electric field should be

But this time with dielectric between, the molecular or atomic charge will be affective, and positive charge will be pulled slightly to one side, and negative to the other side. The result of this is polarized charge?on the surface of dielectric, with density?. The previous field will be reduced because opposite charges attract eachother.

where? is called dielectric constant, and? is called relative dielectric constant. The constant is only determined by the property of the material.

III Polarization Density

Consider another way to describe the phenomena, use the polarization density vector?. It is defined as total dipole caused by polarization per unit volume:

Consider a simple example, the nuclear of an atom can be regarded as a point charge , and the "electron cloud" can be regarded as uniform negative charge. When there's external field, there will be relative?displacement between nuclear and electron. Attraction between them will balance the force from external field, and the attraction is proportional to their relative displacement, so

which means?. Actually, it's easy to prove the following?using the capacitor we mentioned above:

Now we are going to find polarized charge density. Suppose the direction of?external?field is along x axis, and if we try to find total polarized charge in a cube as below:

so we can find

And the same is true for? direction. Therefore we find total?polarized charge density:

IV Electric Displacement

Now we've already known

where? is the total charge, including free charge and polarized charge. But in real cases, we may only know free charge distribution, so we need to find the equation of free charge. Because?, we can find

And if we define electric displacement vector:

The Gauss's law can be written as

which only consider free charge. There's a simpler form of?:

V Tensor

In some cases there might exist anisotropism (各向異性) matters, which means its dielectric constant might be different in different directions. Here the dielectric constant is a Tensor:

or

Of course there's?special directions of the axes, which can diagonalize the tensor:

(Same thing happens in mechanics of rigid bodies when we consider its?rotation.)

VI Magnetize

Like in electric field, matters in magnetic field can also be influenced. We can also define similar concept called magnetize.

There are three types of matter, responding differently to external magnetic field. They are: diamagnetic materials, which slightly repel a magnet; paramagnetic materials, which are slightly attracted by a magnet; and ferromagnetic materials, which are strongly attracted. (抗磁性、順磁性、鐵磁性) In fact, there do exist more amazing materials called superconductors, which don't allow any magnetic field pass?through it. This is one of its main properties?known as?Meissner effect (邁斯納效應).

Also like in electric field, magnetic moment?appear in materials due to the influence of external field. We can define magnetic polarization, or magnetization vector, as magnetic moment per unit volume:

There has been assumption that the basic principle of this is molecular or atomic current circuit, which forms magnetic moment naturally. These magnetic moments are pointing different directions originally,?but?external field changes their direction distribution, making them pointing specific direction, and therefore we can find total moment due to magnetize. Actually, the principle is much more complicated, relative to quantum theory, including spin of particles.

VII Bound currents due to magnetize

We mentioned the assumtion of molecular current. Now as the material is magnetized, the molecular and atomic current will form?macroscopic bound current. We can prove that the bound current is determined as follow:

The proof is kind of complicated, if you want to know exactly why...Well I'm not good at explaining these principles in English...Go and refer to your textbook! This is only a note where I practice my English skill...

VIII Maxwell's equations and vector?

Now we've known how to calculate bound current due to magnetize. Actually we can also find bound current due to polarize:

and we know total current consist of all three parts, including free current:

And back to maxwell's equations, we know

If we define another field,

then we can simplify the maxwell's equation, and that's just why?we define vector?.

This is the maxwell's equation considering polarize and magnetize. This equation, together with?, are the equations where we only need to consider free charge and that's a really practical way to simplify our problem.

References

[1] Edward M. Purcell, David J. Morin. Electricity and Magnetism (Third Edition)[M]. Cambridge Univercity Press.