Electromagnetic Wave and its E-Field and B-Field
Electromagnetic wave with E-field and B-field component.
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E-field and B-field vectors of an electromagnetic wave
\boldsymbol{E} ~=~ \begin{bmatrix}E_{\text x}(x,y,z,t)\\ E_{\text y}(x,y,z,t) \\ E_{\text z}(x,y,z,t) \end{bmatrix}; ~~~ \boldsymbol{B} ~=~ \begin{bmatrix}B_{\text x}(x,y,z,t)\\ B_{\text y}(x,y,z,t) \\ B_{\text z} (x,y,z,t)\end{bmatrix}
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Wave equation for the E-field
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Wave equation for the B-field
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Laplace operator in Cartesian coordinates
\nabla^2 ~=~ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}
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3d wave equation
\nabla^2 \, \boldsymbol{F} ~=~ \frac{1}{{v_{\text p}}^2} \, \frac{\partial^2 \boldsymbol{F}}{\partial t^2}
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Phase velocity here indicates how fast a wave crest moves from A to B.
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Field constants equal to one divided by phase velocity squared
\mu_0 \, \varepsilon_0 ~=~ \frac{1}{{v_{\text p}}^2}
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Phase velocity is equal to the speed of light
v_{\text p} &~=~ \frac{1}{\sqrt{ \mu_0 \, \varepsilon_0 }} \\\\
&~=~ 3 \cdot 10^8 \, \frac{\mathrm m}{ \mathrm s} \\\\
&~=~ c
&~=~ 3 \cdot 10^8 \, \frac{\mathrm m}{ \mathrm s} \\\\
&~=~ c
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Wave equations for E- and B-field expressed with speed of light
\nabla^2 \, \boldsymbol{E} &~=~ \frac{1}{c^2} \, \frac{\partial^2 \boldsymbol{E}}{\partial t^2} \\\\
\nabla^2 \, \boldsymbol{B} &~=~ \frac{1}{c^2} \, \frac{\partial^2 \boldsymbol{B}}{\partial t^2}
\nabla^2 \, \boldsymbol{B} &~=~ \frac{1}{c^2} \, \frac{\partial^2 \boldsymbol{B}}{\partial t^2}
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Written out wave equation for the E-field
\begin{bmatrix} \frac{\partial^2 E_{\text x}}{\partial x^2} + \frac{\partial^2 E_{\text x}}{\partial y^2} + \frac{\partial^2 E_{\text x}}{\partial z^2} \\ \frac{\partial^2 E_{\text y}}{\partial x^2} + \frac{\partial^2 E_{\text y}}{\partial y^2} + \frac{\partial^2 E_{\text y}}{\partial z^2} \\ \frac{\partial^2 E_{\text z}}{\partial x^2} + \frac{\partial^2 E_{\text z}}{\partial y^2} + \frac{\partial^2 E_{\text z}}{\partial z^2} \end{bmatrix} ~=~ \frac{1}{c^2} \, \begin{bmatrix} \frac{\partial^2 E_{\text x}}{\partial t^2} \\ \frac{\partial^2 E_{\text y}}{\partial t^2} \\ \frac{\partial^2 E_{\text z}}{\partial t^2} \end{bmatrix}
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0First component of the 3d wave equation for the E-field\frac{\partial^2 E_{\text x}}{\partial x^2} + \frac{\partial^2 E_{\text x}}{\partial y^2} + \frac{\partial^2 E_{\text x}}{\partial z^2} ~=~ \frac{1}{c^2} \, \frac{\partial^2 E_{\text x}}{\partial t^2}0
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0Second component of the 3d wave equation for the E-field\frac{\partial^2 E_{\text y}}{\partial x^2} + \frac{\partial^2 E_{\text y}}{\partial y^2} + \frac{\partial^2 E_{\text y}}{\partial z^2} ~=~ \frac{1}{c^2} \, \frac{\partial^2 E_{\text y}}{\partial t^2}0
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0Third component of the 3d wave equation for the E-field\frac{\partial^2 E_{\text z}}{\partial x^2} + \frac{\partial^2 E_{\text z}}{\partial y^2} + \frac{\partial^2 E_{\text z}}{\partial z^2} ~=~ \frac{1}{c^2} \, \frac{\partial^2 E_{\text z}}{\partial t^2}0
