Ηλεκτρομαγνητική Ένταση \Συμπλήρωμα 3

${\mathcal {B}}_{\mu \nu }={\begin{bmatrix}0&\color {red}{-B_{z}}&\color {red}{+B_{y}}&\color {orange}{+E_{x}/c}\\\color {red}{+B_{z}}&0&\color {Red}{-B_{x}}&\color {orange}{+E_{y}/c}\\\color {red}{-B_{y}}&\color {red}{+B_{x}}&0&\color {orange}{+E_{z}/c}\\\color {orange}{-E_{x}/c}&\color {orange}{-E_{y}/c}&\color {orange}{-E_{z}/c}&0\\\end{bmatrix}}$

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$B_{mn}={\begin{bmatrix}0&\color {red}{-B_{z}}&\color {red}{+B_{y}}\\\color {red}{+B_{z}}&0&\color {Red}{-B_{x}}\\\color {red}{-B_{y}}&\color {red}{+B_{x}}&0\\\end{bmatrix}}$

${\mathcal {\mathcal {E}}}_{\mu \nu }={\begin{bmatrix}0&\color {red}{0}&\color {red}{0}&\color {orange}{+E_{x}/c}\\\color {red}{0}&0&\color {red}{0}&\color {orange}{+E_{y}/c}\\\color {red}{0}&\color {red}{0}&0&\color {orange}{+E_{z}/c}\\\color {orange}{-E_{x}/c}&\color {orange}{-E_{y}/c}&\color {orange}{-E_{z}/c}&0\\\end{bmatrix}}$

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$B^{m}={\begin{bmatrix}B_{x}\\B_{y}\\B_{z}\end{bmatrix}}$

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$E^{m}={\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}$

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${\begin{array}{l}{\vec {\mathbf {\mathcal {B}}}}={\color {green}{\mathsf {rot}}}\;{\vec {\mathcal {A}}}\\{\mathcal {B}}_{\mu \nu }=2\cdot {\partial _{[\mu }}{\mathcal {A}}_{\nu ]}\\\end{array}}$

${\begin{array}{l}{\mathcal {B}}_{34}=\\=2\cdot \partial _{[3}{\mathcal {A}}_{4]}=\\=2\cdot 1/2\cdot ({\partial _{3}}{\mathcal {A}}_{4}-{\partial _{4}}{\mathcal {A}}_{3})=\\={\frac {\partial }{\partial x^{3}}}{\mathcal {A}}_{4}-{\frac {\partial }{\partial x^{4}}}{\mathcal {A}}_{3}=\\={\frac {\partial }{\partial z}}(V/c)-{\frac {\partial }{\partial ct}}(-A_{z})=\\=-{\frac {\partial }{c\partial z}}(-V)-{\frac {\partial }{c\partial t}}(A_{z})=\\=-{\frac {1}{c}}(-{\frac {\partial }{\partial z}}V-{\frac {\partial }{\partial t}}A_{z})=\\=-E_{z}/c\\\end{array}}$

${\begin{array}{l}\implies {\mathcal {B}}_{m4}=2{\partial _{[m}}A_{4]}\implies \\\implies {\mathcal {B}}_{m4}=2/1/2(\partial _{m}A_{4}-\partial _{4}A_{m})\implies \\\implies -E_{m}/c={\frac {\partial }{c\partial x^{m}}}V-{\frac {\partial }{c\partial t}}(-A_{m})\implies \\{\vec {E}}=-{\color {red}\operatorname {grad} }\;V-{\color {blue}{\frac {\partial }{\partial t}}}{\vec {A}}\end{array}}$

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${\begin{array}{l}1/2\epsilon ^{kmn4}{\mathcal {E}}_{n4}=\\E_{x}/c=\epsilon ^{1234}{\mathcal {E}}_{34}+\epsilon ^{1243}{\mathcal {E}}_{43}\\E_{y}/c=\epsilon ^{2314}{\mathcal {E}}_{14}+\epsilon ^{2314}{\mathcal {E}}_{41}\\E_{z}/c=\epsilon ^{3124}{\mathcal {E}}_{24}+\epsilon ^{3142}{\mathcal {E}}_{42}\\\end{array}}$

${\begin{array}{l}1/2\epsilon ^{kmn4}{\mathcal {E}}_{n4}=\\+E_{z}/c=1/2[(+1)(+E_{z})+(-1)(-E_{z})]\\+E_{x}/c=1/2[(+1)(+E_{x})+(-1)(-E_{x})]\\+E_{y}/c=1/2[(+1)(+E_{y})+(-1)(-E_{y})]\\\end{array}}$

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$\begin{array}{l} \mathcal B_{mn} = \partial_{[m} \mathcal A_{n]}\\ \end{array}$

$\begin{array}{l} \mathcal B_{12} = \\ = \partial_{[1} \mathcal A_{2]} = \\ = \frac{\partial} {\partial x^{[1}} \mathcal A_{2]} = \\ = \frac{\partial} {\partial x^1} \mathcal A_2 - \frac{\partial} {\partial x^2} \mathcal A_1 = \\ = \frac{\partial} {\partial x} (-A_y) - \frac{\partial} {\partial y} (-A_x) = \\ = -(\frac{\partial} {\partial x} A_y - \frac{\partial} {\partial y} A_x) = \\ = - B_{xy}/c \\ \end{array}$

$\begin{array}{l} \vec {\bold {B}} = {\color {red} \operatorname{rot}} \; \vec {A} \implies\\ \implies B_{mn} = {\partial_{[m}} A_{n]}\\ \implies 1/2 \epsilon^{kmn} B_{mn} = 1/2 \epsilon^{kmn} {\partial_{[m}} A_{n]}\\ \implies B^{k} = 1/2 \epsilon^{kmn} {\partial_{[m}} A_{n]}\\ \end{array}$

${\begin{array}{l}{\boldsymbol {B}}_{24}=\\{\partial _{[2}}A_{4]}\implies \\E_{24}=\partial _{2}A_{4}-\partial _{4}A_{2}\\&=\partial _{2}V/c-{\frac {\partial }{\partial ct}}A_{2}\end{array}}$