"Electromagnetics"의 두 판 사이의 차이
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(피타고라스님이 이 페이지의 이름을 Electromagnetism로 바꾸었습니다.) |
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<h5>potentials</h5> | <h5>potentials</h5> | ||
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− | <h5 style="line-height: 3.428em | + | <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</h5> |
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+ | * also called four vector potential | ||
+ | * 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 |
− | * | + | * gague field describing the photon |
− | * | + | * composed of a scalar electric potential and a three-vector magnetic potential |
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− | <h5 | + | <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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+ | <h5 style="margin: 0px; line-height: 2em;">gauge theoretic understanding</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> | ||
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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 |
− | * | + | * ρ the [http://en.wikipedia.org/wiki/Charge_density charge density] |
− | * | + | * 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> | |
− | < | + | * <br> |
+ | * charge density and current density<br> | ||
− | + | : | |
− | + | <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] | |
+ | : ρ the [http://en.wikipedia.org/wiki/Charge_density charge density] | ||
+ | : j the conventional [http://en.wikipedia.org/wiki/Current_density current density]. | ||
+ | : a labels the [http://en.wikipedia.org/wiki/Spacetime space-time] [http://en.wikipedia.org/wiki/Dimension dimensions] | ||
− | <h5 style=" | + | <h5 style="margin: 0px; line-height: 2em;">메모</h5> |
− | * http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf<br> | + | * [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> |
* Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects<br> | * 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%;"> | + | <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/ | ||
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* 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= | * 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= | * 다음백과사전 http://enc.daum.net/dic100/search.do?q= | ||
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2010년 3월 3일 (수) 17:37 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
\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\)
\(\nabla \cdot \mathbf{B} = 0\)
\(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\)
\(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \)
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
\(E=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi \)
electromagnetic field
- also called four vector potential
- 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
\(F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}\)
\(F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c} & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)\)
gauge theoretic understanding
- 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)
charge density and current density
- this is necessary for Maxwell equations with sources
- ρ the charge density
- j the conventional current density.
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
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
encyclopedia
- 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=