"Electromagnetics"의 두 판 사이의 차이

2010년 1월 26일 (화) 15:34 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

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.

electromagnetic field

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
\(F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}\)

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

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

@@ 17번째 줄: / 17번째 줄: @@
 <h5>Maxwell's equations</h5>
-<math>\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}</math>
+*  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>
-<math>\nabla \cdot \mathbf{B} = 0</math>
-<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
-<math>\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ </math>
@@ 91번째 줄: / 85번째 줄: @@
 <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">메모</h5>
+* 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>