"Electromagnetics"의 두 판 사이의 차이
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12번째 줄: | 12번째 줄: | ||
* has two possibilites<br> | * has two possibilites<br> | ||
** what does this mean? | ** what does this mean? | ||
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37번째 줄: | 39번째 줄: | ||
* 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> | * 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> | ||
* scalar potential<br><math>E=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br> | * scalar potential<br><math>E=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br> | ||
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">four-current</h5> | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">four-current</h5> | ||
− | * charge density and current density | + | * charge density and current density<br> |
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+ | <math>J^a = \left(c \rho, \mathbf{j} \right)</math> | ||
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where | where | ||
55번째 줄: | 61번째 줄: | ||
− | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;"> | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">electromagnetic field</h5> |
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− | < | + | * alfour<br> this is what we call the electromagnetic field<br><math>A_{\alpha} = \left( - \phi/c, \mathbf{A} \right)</math><br> φ is the scalar potential and <math>A</math> is the vector potential.<br> |
* an example of four-vector | * an example of four-vector | ||
71번째 줄: | 71번째 줄: | ||
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− | + | <h5>Covariant formulation</h5> | |
− | <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> | + | * 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> |
85번째 줄: | 85번째 줄: | ||
<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">Weyl's gauge theoretic formulation</h5> | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">Weyl's gauge theoretic formulation</h5> | ||
− | * | + | * the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field<br> |
+ | * the electromagnetism is a gauge field theory with structure group U(1)<br> | ||
2010년 2월 3일 (수) 07:29 판
Lorentz force
- almost all forces in mechanics are conservative forces, those that are functions nly 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?
Maxwell's equations
- using vector calculus notation
\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\)
\(\nabla \cdot \mathbf{B} = 0\)
\(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\)
\(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \)
charge density and current density
- this is necessary for Maxwell equations with sources
- ρ the charge density
- j the conventional current density.
potentials
- vector potential
from \(\nabla \cdot \mathbf{B} = 0\), we can find a vector potential such that \(\mathbf{B}=\nabla \times \mathbf{A}\) - scalar potential
\(E=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi \)
four-current
- charge density and current density
\[J^a = \left(c \rho, \mathbf{j} \right)\]
where
- c is the speed of light
- ρ the charge density
- j the conventional current density.
- a labels the space-time dimensions
electromagnetic field
- alfour
this is what we call the electromagnetic field
\(A_{\alpha} = \left( - \phi/c, \mathbf{A} \right)\)
φ is the scalar potential and \(A\) is the vector potential.
- an example of four-vector
- gague field describing the photon
- composed of a scalar electric potential and a three-vector magnetic potential
Covariant formulation
- electromagnetic field strength
\(F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}\)
\(F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c} & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)\)
Weyl's gauge theoretic formulation
- 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)
메모
- http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf
- Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects
참고할만한 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Classical_electromagnetism
- http://en.wikipedia.org/wiki/Maxwell's_equations
- http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
- http://en.wikipedia.org/wiki/electrical_current
- http://en.wikipedia.org/wiki/Four-current
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://viswiki.com/en/
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- 다음백과사전 http://enc.daum.net/dic100/search.do?q=