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Anonymous Mon 24 Aug 13:16:00 2020 No.12042881 View Reply Original Report

Quoted By: >>12043146 >>12043398 >>12043423

Can someone explain Maxwell's equations in matter to me?

I don't understand what H and D mean. What is the causality there? Are they supposed to exist internally in matter and then create an external field? Or are they the reaction to an external field? In the latter case how can the magnetic permeability ever be 0? Then by H = 1/mu *B we'd have an infinite magnetic field in the material.

Anonymous

Anonymous Mon 24 Aug 2020 13:43:01 No.12042927 Report

Quoted By: >>12042952

You might be retarded. Mu = mu_0 * mu_r. Mu_0 is a fundamental constant and mu_r is material dependent but bounded below by 1. High permeability materials supress magnetic fields. Often called "mu metal"

Anonymous

Anonymous Mon 24 Aug 2020 13:56:32 No.12042952 Report

Quoted By:

>>12042927
That doesn't make sense. Why would you say a ferromagnetic material suppresses magnetic fields? And mu isn't bounded from below. According to Wikipedia it is less than 1 for a diamagnetic material and 0 for a superconductor

Anonymous

Anonymous Mon 24 Aug 2020 15:23:07 No.12043146 Report

Quoted By: >>12043167

>>12042881
This is quite a topic to unpack so let me give it a try and see if you get it all in one go.

Due to the fact that media are generally composed of many charges, electric dipoles, magnetic dipoles (spin/angular momentum of atoms) and so on, it is usefull to make a "macroscopic approximation" that leaves out all of the microscopic intricacies which would make any problem unsolvable due to the high number of degrees of freedom. Therfore we define:

$\vec B = \mu_0 (\vec H + \vec M)$ (1)
Where $\vec B[\math] is the magnetic field in a vacuum, <script type="math/tex">\vec H$ represents a "free" (that is to say "behaves like B would in a vacuum") magnetic field inside of a medium and $\vec M$ is a field that represents magnetisation.

$\vec M$ can further be split up into a spontanious magnetisation (for instance in a permanent magnet) and a induced magnetisation (magnetisation caused by the ensemble atoms reacting to the presence of an outer magnetic field). That is to say:
$\vec M=\vec M_s + \vec M_i$ .

For most materials $\vec M_s=0$ and $\vec M_i = \chi \vec H$ where $\chi=\chi(\vec r, \vec H)$ is the "magnetic susceptibility" which is in the most general case a 3x3 matrix, allowing for a location based, direction based and H field strength dependent response of M to H. For linear, homogenious, and isotropic media (LHI-media)...

linear: $\chi$ independent of $\vec H$ , therefore $|\vec M|\propto |\vec H|$ is linearly proportional
homogenious: $\chi$ independent of $\vec r$
isotropic: $\chi$ diagonal

... the 3x3 $\chi$ matrix can be replaced with a scalar constant (which I will from now on also refer to simply as $\chi$ ).

Anonymous

Anonymous Mon 24 Aug 2020 15:30:06 No.12043167 Report

Quoted By: >>12043361

>>12043146
Given we're talking about LHI-media (which an introductory course generally is) we can rewrite (1) as

$\vec B = \mu_0 (\vec H + \chi \vec H) = \mu_0(1+\chi)\vec H = \mu_0\mu_r \vec H =\mu \vec H$

In vacuum the magnetostatic field equations are:
$\vec n \cdot \vec B=0$
$\vec n \times \vec B=\mu_0\vec j$

whereas in a medium the magnetostatic field equations are given by:
$\vec n \cdot \vec B=0$
$\vec n \times \vec H=\vec j$

Anonymous

Anonymous Mon 24 Aug 2020 16:39:58 No.12043361 Report

Quoted By:

>>12043167
Analogously we can define

$\vec E=\frac{1}{\epsiolon_0}(\vec D - \vec P)$

where $\vec P = \chi_E \epsilon_0 \vec E$ with $\chi_E$ the electric permetivity of the medium and for LHI media we find

$\vec D = (1+\chi_E)\epsilon_0 \vec E = \epsilon_r\epsiolon_0 \vec E = \epsilon \vec E$

(the definitions are a bit different but the fields are fully analogous)

Anonymous

Anonymous Mon 24 Aug 2020 16:48:13 No.12043398 Report

Quoted By:

>>12042881
The short summary is that take maxwell's equations, label charges and currents as either bound or free depending on whether they are caused by dipoles or not dipoles, apply some averaging, and the result is the D and H field pop out. keep in mind D and H are defined in terms of polarization P and magnetization M. also keep in mind that P and M only refer to dipoles, and do not model higher-order terms of the multipole expansion. this is usually sufficient because most natural materials are well modeled by only taking into account dipole effects. however quadrupole and higher order terms can be isolated from the free charge and currents to define higher order material fields analogous to D and H in origin.

Anonymous

Anonymous Mon 24 Aug 2020 16:56:09 No.12043423 Report

Quoted By:

>>12042881
also, "zero mu" really means "zero real mu". this statement is in the context of time harmonic analysis where $\tilde{\mu} = \mu - \frac{\sigma_m}{j \omega}$ . in the case that the real part is close to zero, the imaginary part won't be. this occurs in conductive (lossy) materials.

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