"Electromagnetics"의 두 판 사이의 차이

수학노트
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(사용자 3명의 중간 판 65개는 보이지 않습니다)
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 nly 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}</math>
 
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* magnetic field <math>\mathbf{B}</math>
 
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* <math>{\rho} </math>
 
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* <math>\mathbf{J}</math>
 
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* <math>\mu_0</math>
 
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* <math>\varepsilon_0</math>
 
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<h5>Maxwell's equations</h5>
 
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* using vector calculus notation<br><math>\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}</math><br><math>\nabla \cdot \mathbf{B} = 0</math><br><math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math><br><math>\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ </math><br>
 
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<h5>potentials</h5>
 
  
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*  vector potential<br> from <math>\nabla \cdot \mathbf{B} = 0</math>, we can find a vector potential such that <math>\mathbf{B}=\nabla \times \mathbf{A}</math><br>
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==Lagrangian formulation==
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* scalar potential<br><math>\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br>
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* [[Lagrangian formulation of electromagetism]]
  
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==Hamiltonian formulation==
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*  total energy of a charge particle in an electromagnetic field
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:<math>E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi</math>
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*  replace the momentum with the canonical momentum
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**  similar to covariant derivative
  
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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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*  defined as follows<br><math>A_{\alpha} = \left( - \phi/c, \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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* gague field describing the photon
 
  
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==force on a particle==
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* force on a particle is same as
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:<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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==메모==
  
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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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* [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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*  
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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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*  the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field<br>
 
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*  the electromagnetism is a gauge field theory with structure group U(1)<br>
 
  
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==related items==
  
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* [[Gauge theory]]
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* [[QED]]
  
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<h5>Covariant formulation</h5>
 
  
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*  electromagnetic field strength<br><math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}</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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==encyclopedia==
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<h5 style="margin: 0px; line-height: 2em;">charge density and current density</h5>
 
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* this is necessary for Maxwell equations with sources
 
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* ρ the [http://en.wikipedia.org/wiki/Charge_density charge density]
 
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* j the conventional [http://en.wikipedia.org/wiki/Current_density current density].
 
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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%;">four-current</h5>
 
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* charge density and current density<br>
 
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:
 
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<math>J^a = \left(c \rho, \mathbf{j} \right)</math> where
 
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: c is the [http://en.wikipedia.org/wiki/Speed_of_light speed of light]
 
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: ρ the [http://en.wikipedia.org/wiki/Charge_density charge density]
 
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: j the conventional [http://en.wikipedia.org/wiki/Current_density current density].
 
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: a labels the [http://en.wikipedia.org/wiki/Spacetime space-time] [http://en.wikipedia.org/wiki/Dimension dimensions]
 
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<h5 style="margin: 0px; line-height: 2em;">메모</h5>
 
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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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*  Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects<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%;">encyclopedia</h5>
 
  
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/Classical_electromagnetism
 
* http://en.wikipedia.org/wiki/Classical_electromagnetism
 
* [http://en.wikipedia.org/wiki/Maxwell%27s_equations http://en.wikipedia.org/wiki/Maxwell's_equations]
 
* [http://en.wikipedia.org/wiki/Maxwell%27s_equations http://en.wikipedia.org/wiki/Maxwell's_equations]
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* [http://en.wikipedia.org/wiki/Maxwell%27s_equations#Differential_geometric_formulations http://en.wikipedia.org/wiki/Maxwell's_equations#Differential_geometric_formulations]
 
* http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
 
* http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
 
* http://en.wikipedia.org/wiki/electrical_current
 
* http://en.wikipedia.org/wiki/electrical_current
 
* 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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[[분류:math and physics]]
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[[분류:gauge theory]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* 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"