Electromagnetics
http://bomber0.myid.net/ (토론)님의 2010년 5월 13일 (목) 18:32 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 transformation
- 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://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=