"Electromagnetics"의 두 판 사이의 차이

수학노트
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1번째 줄: 1번째 줄:
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<h5>Lorentz force</h5>
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==basic history==
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* Leyden jar : capacitor
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* Volta vs Galvani
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* Humphrey Davy
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* Oesrsted
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* Faraday
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* Maxwell
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* Lodge
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* Marconi
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* Tesla : alternating current
  
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* almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities
 
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* Lorentz force is a rare example of velocity dependent force
 
  
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==gauge invariance==
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*  the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field
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*  the electromagnetism is a gauge field theory with structure group U(1)
  
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<h5>polarization of light</h5>
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* has two possibilites<br>
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==Lorentz force==
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** what does this mean?
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* almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities
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* Lorentz force is a rare example of velocity dependent force
  
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<h5>notations</h5>
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==polarization of light==
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*  has two possibilites
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* vector potential <math>\mathbf{A}(x,y,z,t)=(A_{x},A_{y},A_{z})</math>
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** what does this mean?
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* electrostatic potential <math>\phi(x,y,z,t)</math> (scalar)
 
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* electric field <math>\mathbf{E}(x,y,z,t)</math>
 
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* magnetic field <math>\mathbf{B}(x,y,z,t)</math>
 
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* charge density <math>{\rho} </math> (for point charge, density will be a Dirac delta function)
 
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* current density <math>\mathbf{J}=(J_x,J_y,J_z)</math>
 
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* <math>\mu_0</math>
 
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* <math>\varepsilon_0</math>
 
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* <math>c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}</math>
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">맥스웰 방정식</h5>
 
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*  전기장에 대한 가우스의 법칙<br><math>\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}</math><br>
 
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* 자기장에 대한 가우스의 법칙<br><math>\nabla \cdot \mathbf{B} = 0</math><br>
 
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* 패러데이의 법칙
 
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<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
 
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* 앙페르-패러데이 법칙
 
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<math>\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ </math>
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">electromagnetic field (four vector potential)</h5>
 
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* [http://pythagoras0.springnote.com/pages/12166408 포벡터 포텐셜과 맥스웰 방정식]
 
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*  defined as follows<br><math>A_{\alpha} = \left( - \phi, \mathbf{A} \right)=(-\phi,A_{x},A_{y},A_{z})</math><br><math>\phi</math> is the scalar potential<br><math>A</math>  is the vector potential.<br>
 
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*  in covariant formulation, this is a '''1-form'''<br><math>A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}</math><br>
 
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* gauge field describing the photon
 
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<h5>electromagnetic field strength</h5>
 
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* in covariant formulation, this is a  '''2-form'''
 
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* <math>F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots</math><br>
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">conserved four-current</h5>
 
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* this is necessary for Maxwell equations with sources
 
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* describes the distribution and motion of charged particles
 
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* charge density <math>{\rho} </math> (for point charge, density will be a Dirac delta function)
 
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* current density <math>\mathbf{J}=(J_x,J_y,J_z)</math>
 
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*  charge density and current density<br><math>J^a = \left(c \rho, \mathbf{J} \right)</math><br>
 
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*  four vector is called a conserved current if <math>\partial_{a}J^{a}=0</math><br>
 
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*  in covariant formulation, this is a '''3-form'''<br><math>J=\rho dx\wedge dy \wedge dz - J_{z}dx\wedge dy \wedge dt -J_{x}dy\wedge dz\wedge dt-J_{y}dz\wedge dx\wedge dt</math><br>
 
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<h5>covariant formulation using differential form</h5>
 
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*  electromagnetic field strength<br><math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!</math><br><math>F_{\alpha \beta} = \left( \begin{matrix} 0 &  \frac{E_x}{c} &  \frac{E_y}{c} &  \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c}  & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)</math><br>
 
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*  In [[Gauge theory]], we regard F as 2-form, A as 1-form<br>
 
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* <math>A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}</math><br>
 
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* <math>F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots</math><br>
 
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* <math>J=(-\rho,J_1,J_2,J_3)</math><br>
 
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*  Maxwell's equation can be recast into<br>
 
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** <math>dF=0</math> (<math>\nabla \cdot \mathbf{B} = 0</math>, <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>)<br>
 
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** <math>d*F=J</math> (<math>\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}</math>,  <math>\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ </math>)<br>  <br>
 
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*  See [http://www.math.purdue.edu/%7Edvb/preprints/diffforms.pdf Introduction to differential forms]<br>
 
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* [http://resources.aims.ac.za/archive/2009/solomon.pdf Maxwell's Equations in Terms of Dierential Forms]<br>
 
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* [http://www22.pair.com/csdc/pd2/pd2fre21.htm Maxwell Theory and Differential Forms]<br>
 
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** [http://www22.pair.com/csdc/pdf/maxwell.pdf http://www22.pair.com/csdc/]<br>
 
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** [http://www22.pair.com/csdc/pdf/maxwell.pdf Maxwell Faraday and Maxwell Ampere Equations]<br>
 
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* [http://www.math.sunysb.edu/%7Ebrweber/401s09/coursefiles/ElectromagneticNotes.pdf http://www.math.sunysb.edu/~brweber/401s09/coursefiles/ElectromagneticNotes.pdf]
 
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* http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.7828&rep=rep1&type=pdf
 
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* https://www.nottingham.ac.uk/ggiemr/downloads/GCEM.pdf
 
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<h5 style="margin: 0px; line-height: 2em;">Lagrangian formulation</h5>
 
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*  Lagrangian for a charged particle in an electromagnetic field<br><math>L=T-V</math><br><math>L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}</math><br>
 
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*  action<br><math>S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x</math><br>
 
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*  Euler-Lagrange equations<br><math>p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}</math><br><math>F_{i}=\frac{\partial{L}}{\partial{{q}^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}}}</math><br>
 
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*  equation of motion<br><math>\dot{p}=F</math> Therefore we get<br><math>m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}</math>. This is what we call the Lorentz force law.<br>
 
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* force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math>
 
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* http://dexterstory.tistory.com/888<br>
 
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* [[path integral formulation of quantum mechanics|path integral]]<br>
 
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<h5 style="margin: 0px; line-height: 2em;">Hamiltonian formulation</h5>
 
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*  total energy of a charge particle in an electromagnetic field<br><math>E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi</math><br>
 
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*  replace the momentum with the canonical momentum<br>
 
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**  similar to covariant derivative<br>
 
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<h5 style="margin: 0px; line-height: 2em;">force on a particle</h5>
 
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* force on a particle is same as <math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math>
 
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<h5 style="margin: 0px; line-height: 2em;">gauge transformation</h5>
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==Lagrangian formulation==
 +
* [[Lagrangian formulation of electromagetism]]
  
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* For any scalar field <math>\Lambda(x,y,z,t)</math>, the following transformation does not change any physical quantity<br><math>\mathbf{A} \to \mathbf{A} +\del \Lambda</math><br><math>\phi\to \phi-\frac{\partial\Lambda}{\partial t}</math><br>
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*  unchanged quantities<br><math>\mathbf{B}=\nabla \times \mathbf{A}</math><br><math>\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br>
 
  
−
*  the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field<br>
+
==Hamiltonian formulation==
−
*  the electromagnetism is a gauge field theory with structure group U(1)<br>
+
*  total energy of a charge particle in an electromagnetic field
 +
:<math>E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi</math>
 +
*  replace the momentum with the canonical momentum
 +
**  similar to covariant derivative
  
−
 
 
  
−
 
 
  
−
 
+
==force on a particle==
 +
* force on a particle is same as
 +
:<math>e\mathbf{E}+e\mathbf{v}\times \mathbf{B}</math>
  
−
 
 
  
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<h5 style="margin: 0px; line-height: 2em;">메모</h5>
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==메모==
  
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* [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf]<br>
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* [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf]
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*  Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects<br>
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*  Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects
  
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==related items==
  
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<h5>related items</h5>
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* [[Gauge theory]]
 +
* [[QED]]
  
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* [[Gauge theory]]<br>
 
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* [[QED]]<br>
 
  
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==encyclopedia==
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
 
  
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
185번째 줄: 72번째 줄:
 
* http://en.wikipedia.org/wiki/Four-current
 
* http://en.wikipedia.org/wiki/Four-current
  
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==books==
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* ELECTROMAGNETIC THEORY AND COMPUTATION
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* [[The Early History of Radio from Faraday to Marconi]]
  
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books
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[[분류:math and physics]]
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[[분류:gauge theory]]
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[[분류:migrate]]
  
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ELECTROMAGNETIC THEORY AND COMPUTATION
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==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q377930 Q377930]
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===Spacy 패턴 목록===
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* [{'LOWER': 'classical'}, {'LEMMA': 'electromagnetism'}]
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* [{'LEMMA': 'electrodynamic'}]

2021년 2월 17일 (수) 03:16 기준 최신판

basic history

  • Leyden jar : capacitor
  • Volta vs Galvani
  • Humphrey Davy
  • Oesrsted
  • Faraday
  • Maxwell
  • Lodge
  • Marconi
  • Tesla : alternating current


gauge invariance

  • the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field
  • the electromagnetism is a gauge field theory with structure group U(1)


Lorentz force

  • almost all forces in mechanics are conservative forces, those that are functions only of positions, and certainly not functions of velocities
  • Lorentz force is a rare example of velocity dependent force



polarization of light

  • has two possibilites
    • what does this mean?


Lagrangian formulation


Hamiltonian formulation

  • total energy of a charge particle in an electromagnetic field

\[E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi\]

  • replace the momentum with the canonical momentum
    • similar to covariant derivative


force on a particle

  • force on a particle is same as

\[e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\]


메모


related items


encyclopedia



books

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'classical'}, {'LEMMA': 'electromagnetism'}]
  • [{'LEMMA': 'electrodynamic'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Electromagnetics&oldid=51946"