"Electromagnetics"의 두 판 사이의 차이

2010년 5월 13일 (목) 20:30 판

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

using vector calculus notation
\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\)
\(\nabla \cdot \mathbf{B} = 0\)
\(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\)
\(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \)

vector potential
from \(\nabla \cdot \mathbf{B} = 0\), we can find a vector potential such that \(\mathbf{B}=\nabla \times \mathbf{A}\)
scalar potential
\(\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi \)

defined as follows
\(A_{\alpha} = \left( - \phi/c, \mathbf{A} \right)=(\phi,A_{x},A_{y},A_{z})\)
\(\phi\) is the scalar potential
\(A\) is the vector potential.
gague field describing the photon

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)
For any scalar field \(\Lambda(x,y,z,t)\),
\(\mathbf{A} \to \mathbf{A} +\del \Lambda\)
\(\phi\to \phi-\frac{\partial\Lambda}{\partial t}\)

electromagnetic field strength
\(F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}\)
\(F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c} & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)\)

\[J^a = \left(c \rho, \mathbf{j} \right)\] where

j the conventional current density.

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

@@ 41번째 줄: / 41번째 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>
-*  scalar potential<br><math>E=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br>
+*  scalar potential<br><math>\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br>
@@ 60번째 줄: / 60번째 줄: @@
 *  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>
-*  For any scalar field <math>\Lambda(x,y,z,t)</math>,<br><math>\mathbf{A} \to \mathbf{A} +\del \Lambda</math><br><math>\phi\to \phi-\frac{1}{c}\frac{\partial\Lambda}{\partial t}</math><br>  <br>
+*  For any scalar field <math>\Lambda(x,y,z,t)</math>,<br><math>\mathbf{A} \to \mathbf{A} +\del \Lambda</math><br><math>\phi\to \phi-\frac{\partial\Lambda}{\partial t}</math><br>  <br>