91影视

Write the expression for Maxwell鈥檚 equation.

$\frac{鈭�G^{渭谓}}{鈭�x^{谓}}=0$

....................(1)

$$\begin{gathered} \mathrm{If}\mathrm{the}\mathrm{渭}-0\mathrm{then},\mathrm{equation}\left(1\right)\mathrm{becomes}, \\ \frac{鈭�{\mathrm{G}}^{\mathrm{慰谓}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=\frac{鈭�{\mathrm{G}}^{00}}{鈭�{\mathrm{x}}^0}+\frac{鈭�{\mathrm{G}}^{01}}{鈭�{\mathrm{x}}^1}+\frac{鈭�{\mathrm{G}}^{02}}{鈭�{\mathrm{x}}^2}+\frac{鈭�{\mathrm{G}}^{03}}{鈭�{\mathrm{x}}^3} \\ \frac{鈭�{\mathrm{G}}^{\mathrm{慰谓}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=\frac{鈭�{\mathrm{B}}_{\mathrm{x}}}{鈭�\mathrm{x}}+\frac{鈭�{\mathrm{B}}_{\mathrm{y}}}{鈭�\mathrm{y}}+\frac{鈭�{\mathrm{B}}_{\mathrm{Z}}}{鈭�\mathrm{z}} \\ \frac{鈭�{\mathrm{B}}_{\mathrm{x}}}{鈭�\mathrm{x}}+\frac{鈭�{\mathrm{B}}_{\mathrm{y}}}{鈭�\mathrm{y}}+\frac{鈭�{\mathrm{B}}_{\mathrm{Z}}}{鈭�\mathrm{z}}=鈭�.\mathrm{B} \\ 鈭�.\mathrm{B}=0 \\ \mathrm{If}\mathrm{渭}=1\mathrm{then}\mathrm{equation}\left(1\right)\mathrm{becomes}, \\ \frac{鈭�{\mathrm{G}}^{1\mathrm{谓}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=\frac{鈭�{\mathrm{G}}^{10}}{鈭�{\mathrm{x}}^0}+\frac{鈭�{\mathrm{G}}^{11}}{鈭�{\mathrm{x}}^1}+\frac{鈭�{\mathrm{G}}^{12}}{鈭�{\mathrm{x}}^2}+\frac{鈭�{\mathrm{G}}^{13}}{鈭�{\mathrm{x}}^3} \\ \frac{鈭�{\mathrm{G}}^{1\mathrm{谓}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=-\frac{1}{\mathrm{c}}\frac{鈭�{\mathrm{B}}_{\mathrm{x}}}{鈭�\mathrm{t}}-\frac{1}{\mathrm{c}}\frac{鈭�{\mathrm{E}}_{\mathrm{z}}}{鈭�\mathrm{t}}+\frac{1}{\mathrm{c}}\frac{鈭�{\mathrm{E}}_{\mathrm{y}}}{鈭�\mathrm{z}} \\ -\frac{1}{\mathrm{c}}\frac{鈭�{\mathrm{B}}_{\mathrm{x}}}{鈭�\mathrm{t}}-\frac{1}{\mathrm{c}}\frac{鈭�{\mathrm{E}}_{\mathrm{z}}}{鈭�\mathrm{t}}+\frac{1}{\mathrm{c}}\frac{鈭�{\mathrm{E}}_{\mathrm{y}}}{鈭�\mathrm{z}}=-\frac{1}{\mathrm{c}}\left(\frac{鈭�\mathrm{B}}{鈭�\mathrm{t}}+鈭�脳\mathrm{E}\right) \\ 鈭�脳\mathrm{E}=-\frac{鈭�\mathrm{B}}{鈭�\mathrm{t}} \end{gathered}$$

$\mathrm{Take}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{spatial}\mathrm{components}.$

$$\begin{gathered} \mathrm{If}\mathrm{渭}=1,\mathrm{v}=2\mathrm{and}\mathrm{位}=3,\mathrm{then},\mathrm{the}\mathrm{equation}\left(2\right)\mathrm{becomes}, \\ \frac{鈭�{\mathrm{F}}_{12}}{鈭�{\mathrm{x}}^3}+\frac{鈭�{\mathrm{F}}_{23}}{鈭�{\mathrm{x}}^1}+\frac{鈭�{\mathrm{F}}_{31}}{鈭�{\mathrm{x}}^2}=\frac{鈭�{\mathrm{B}}_{\mathrm{z}}}{鈭�{\mathrm{x}}^2}+\frac{鈭�{\mathrm{F}}_{\mathrm{x}}}{鈭�\mathrm{z}}+\frac{鈭�{\mathrm{F}}_{\mathrm{y}}}{鈭�\mathrm{y}} \\ 鈭�.\mathrm{B}=0 \\ \mathrm{If}\mathrm{渭}=0,\mathrm{v}=1\mathrm{and}\mathrm{位}=2,\mathrm{the}\mathrm{z}鈥�\mathrm{component}\mathrm{from}\mathrm{equation}\left(2\right)\mathrm{becomes}, \\ \frac{鈭�{\mathrm{F}}_{01}}{鈭�{\mathrm{x}}^2}+\frac{鈭�{\mathrm{F}}_{12}}{鈭�{\mathrm{x}}^1}+\frac{鈭�{\mathrm{F}}_{30}}{鈭�{\mathrm{x}}^2}=\frac{鈭�\left({\mathrm{E}}_{\mathrm{x}}/\mathrm{c}\right)}{鈭�\mathrm{y}}+\frac{{\mathrm{B}}_{\mathrm{z}}}{鈭�\left(\mathrm{ct}\right)}+\frac{鈭�\left({\mathrm{E}}_{\mathrm{x}}/\mathrm{c}\right)}{鈭�\mathrm{x}} \\ 鈭�脳\mathrm{E}=-\frac{鈭�\mathrm{B}}{鈭�\mathrm{t}} \\ \mathrm{If}\mathrm{渭}=0,\mathrm{v}=2\mathrm{and}\mathrm{位}=3,\mathrm{the}\mathrm{x}鈥�\mathrm{and}\mathrm{y}\mathrm{component}\mathrm{from}\mathrm{equation}\left(2\right)\mathrm{becomes}, \\ \frac{鈭�{\mathrm{F}}_{02}}{鈭�{\mathrm{x}}^3}+\frac{鈭�{\mathrm{F}}_{23}}{鈭�{\mathrm{x}}^0}+\frac{鈭�{\mathrm{F}}_{30}}{鈭�{\mathrm{x}}^2}=\frac{鈭�\left({\mathrm{E}}_{\mathrm{y}}/\mathrm{c}\right)}{鈭�{\mathrm{x}}^3}+\frac{鈭�{\mathrm{B}}_{\mathrm{x}}}{鈭�\left(\mathrm{ct}\right)}+\frac{鈭�\left({\mathrm{E}}_{\mathrm{z}}/\mathrm{c}\right)}{鈭�{\mathrm{x}}^2} \\ 鈭�脳\mathrm{E}=-\frac{鈭�\mathrm{B}}{鈭�\mathrm{t}} \\ \mathrm{So},\mathrm{It}\mathrm{can}\mathrm{be}\mathrm{seen}\mathrm{thet}\mathrm{the}\mathrm{function}\frac{鈭�{\mathrm{G}}^{\mathrm{渭谓}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=0\mathrm{can}\mathrm{be}\mathrm{expressed}\mathrm{in}\mathrm{terms}\mathrm{of}\mathrm{field} \\ \mathrm{tensor}\mathrm{as}\frac{鈭�{\mathrm{F}}_{\mathrm{渭谓}}}{鈭�{\mathrm{x}}^{\mathrm{位}}}+\frac{鈭�{\mathrm{F}}_{\mathrm{谓位}}}{鈭�{\mathrm{x}}^{\mathrm{渭}}}+\frac{鈭�{\mathrm{F}}_{\mathrm{位渭}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=0 \\ \mathrm{Therefore},\mathrm{the}\mathrm{second}\mathrm{equation}\frac{鈭�{\mathrm{G}}^{\mathrm{渭谓}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=0\mathrm{can}\mathrm{be}\mathrm{expressed}\mathrm{in}\mathrm{terms}\mathrm{of}\mathrm{field} \\ \mathrm{tensor}\mathrm{as}\frac{鈭�{\mathrm{F}}_{\mathrm{渭谓}}}{鈭�{\mathrm{x}}^{\mathrm{位}}}+\frac{鈭�{\mathrm{F}}_{\mathrm{谓位}}}{鈭�{\mathrm{x}}^{\mathrm{渭}}}+\frac{鈭�{\mathrm{F}}_{\mathrm{位渭}}}{鈭�{\mathrm{x}}^{\mathrm{谓}}}=0 \end{gathered}$$