"미분형식과 맥스웰 방정식"의 두 판 사이의 차이

2012년 4월 19일 (목) 09:16 판

electromagnetic field strength
\(F=\left( \begin{array}{cccc} 0 & -\frac{E_x}{c} & -\frac{E_y}{c} & -\frac{E_z}{c} \\ \frac{E_x}{c} & 0 & B_z & -B_y \\ \frac{E_y}{c} & -B_z & 0 & B_x \\ \frac{E_z}{c} & B_y & -B_x & 0 \end{array} \right)\)
다음과 같은 미분형식으로 이해할 수 있음
\(F=\frac{1}{2}F_{\alpha \beta}dx^{\alpha}\wedge dx^{\beta}\)
\(F=E_1 d x_1\wedge d t+B_3 d x_1\wedge d x_2+E_2 d x_2\wedge d t+B_1 d x_2\wedge d x_3+E_3 d x_3\wedge d t+B_2 d x_3\wedge d x_1\)
맥스웰방정식은 미분형식 F 에 대한 exterior derivative가 만족시키는 방정식으로 이해할 수 있다
이차미분형식으로서 로렌츠 불변이다

\(F'=E'_1 d x'_1\wedge d t'+B'_3 d x'_1\wedge d x'_2+E'_2 d x'_2\wedge d t'+B'_1 d x'_2\wedge d x'_3+E'_3 d x'_3\wedge d t'+B'_2 d x'_3\wedge d x'_1=F\)

\((A_{\alpha})= \left( - \phi, \mathbf{A} \right)=(-\phi,A_{x},A_{y},A_{z})\)
\(\phi\) 스칼라 포텐셜
\(\mathbf{A}\) 벡터 포텐셜
1-미분형식으로서, \(A=-\phi dt+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}\)
\(\mathbf{F}=dA\) 로부터 \(\mathrm{d}\, {\bold{F}}=0\) 를 얻는다

\(\star dx dy =-dzdt\)

\(\star dy dz =-dxdt\)

\(\star dz dx =-dydt\)

\(\star dx dt =dydz\)

\(\star dy dt =dzdx\)

\(\star dz dt=dxdy\)

this is necessary for Maxwell equations with sources
describes the distribution and motion of charged particles
전하 밀도\({\rho} \) (for point charge, density will be a Dirac delta function)
current 밀도\(\mathbf{J}=(J_x,J_y,J_z)\)
current 4-vector
\(J^a = \left(c \rho, \mathbf{J} \right)\)
four vector is called a conserved current if \(\partial_{a}J^{a}=0\)
in covariant formulation,
- 1-form \(J=-\rho dt +J_{x}dx+J_{y}dy+J_{z}dz\)
- dual 3-form \(\star J=\rho dx\wedge dy \wedge dz -J_{x}dy\wedge dz\wedge dt-J_{y}dz\wedge dx\wedge dt- J_{z}dx\wedge dy \wedge dt \)

맥스웰 방정식 을 미분형식의 언어를 통하여 다음과 같이 쓸 수 있다
\(\mathrm{d}\, {\bold{F}}=0\) (\(\nabla \cdot \mathbf{B} = 0\), \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\))
\(\mathrm{d}\, {*\bold{F}}=\bold{J}\) (\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\), \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \))

@@ 56번째 줄: / 56번째 줄: @@
 * this is necessary for Maxwell equations with sources
 * describes the distribution and motion of charged particles
-* charge density <math>{\rho} </math> (for point charge, density will be a Dirac delta function)
+* 전하 밀도<math>{\rho} </math> (for point charge, density will be a Dirac delta function)
-* current density <math>\mathbf{J}=(J_x,J_y,J_z)</math>
+* current 밀도<math>\mathbf{J}=(J_x,J_y,J_z)</math>
-*  charge density and current density<br><math>J^a = \left(c \rho, \mathbf{J} \right)</math><br>
+*  current 4-vector<br><math>J^a = \left(c \rho, \mathbf{J} \right)</math><br>
 *  four vector is called a conserved current if <math>\partial_{a}J^{a}=0</math><br>
 *  in covariant formulation,<br>
 **  1-form <math>J=-\rho dt +J_{x}dx+J_{y}dy+J_{z}dz</math><br>
-**  dual 3-form <math>\star J=\rho dx\wedge dy \wedge dz - J_{z}dx\wedge dy \wedge dt -J_{x}dy\wedge dz\wedge dt-J_{y}dz\wedge dx\wedge dt</math><br>
+**  dual 3-form <math>\star J=\rho dx\wedge dy \wedge dz -J_{x}dy\wedge dz\wedge dt-J_{y}dz\wedge dx\wedge dt- J_{z}dx\wedge dy \wedge dt </math><br>