坡印亭矢量本文重定向自 坡印廷向量

定义

${\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} }$

说明

${\displaystyle {\frac {\partial u}{\partial t}}=-\mathbf {\nabla } \cdot \mathbf {S} -\mathbf {J_{\mathrm {f} }} \cdot \mathbf {E} }$

${\displaystyle u={\frac {1}{2}}\!\left(\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} \right)}$

• E 是电场强度；
• D 是电位移矢量；
• B 是磁感应强度；
• H 是磁场强度。:258-260

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }$${\displaystyle \mathbf {H} ={\frac {1}{\mu }}\mathbf {B} }$

增添场旋度的不变性

${\displaystyle \mathbf {S} '=\mathbf {S} +\nabla \times \mathbf {F} \Rightarrow \nabla \cdot \mathbf {S} '=\nabla \cdot \mathbf {S} }$

${\displaystyle \nabla \cdot \left(\mathbf {E} \times \mathbf {H} \right)=\mathbf {H} \cdot \nabla \times \mathbf {E} -\mathbf {E} \cdot \nabla \times \mathbf {H} }$

${\displaystyle \nabla \times \mathbf {H} =\mathbf {J_{\mathrm {f} }} +{\frac {\partial \mathbf {D} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {H} ={\frac {\partial \mathbf {D} }{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {S} =-\mathbf {E} \cdot {\frac {\partial \mathbf {D} }{\partial t}}}$

${\displaystyle \mathbf {S'} =-V{\frac {\partial \mathbf {D} }{\partial t}}}$

微观领域的形式

${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} }$

${\displaystyle {\frac {\partial u}{\partial t}}+\nabla \cdot \mathbf {S} =-\mathbf {J} \cdot \mathbf {E} }$

${\displaystyle u={\frac {1}{2}}\!\left(\varepsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)}$

时间平均坡印亭矢量

{\displaystyle {\begin{aligned}\mathbf {S} &=\mathbf {E} \times \mathbf {H} \\&=\operatorname {Re} \!\left(\mathbf {E_{\mathrm {a} }} \right)\times \operatorname {Re} \!\left(\mathbf {H_{\mathrm {a} }} \right)\\&=\operatorname {Re} \!\left({\underline {\mathbf {E_{m}} }}e^{j\omega t}\right)\times \operatorname {Re} \!\left({\underline {\mathbf {H_{m}} }}e^{j\omega t}\right)\\&={\frac {1}{2}}\!\left({\underline {\mathbf {E_{m}} }}e^{j\omega t}+{\underline {\mathbf {E_{m}^{*}} }}e^{-j\omega t}\right)\times {\frac {1}{2}}\!\left({\underline {\mathbf {H_{m}} }}e^{j\omega t}+{\underline {\mathbf {H_{m}^{*}} }}e^{-j\omega t}\right)\\&={\frac {1}{4}}\!\left({\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}^{*}} }}+{\underline {\mathbf {E_{m}^{*}} }}\times {\underline {\mathbf {H_{m}} }}+{\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}} }}e^{2j\omega t}+{\underline {\mathbf {E_{m}^{*}} }}\times {\underline {\mathbf {H_{m}^{*}} }}e^{-2j\omega t}\right)\\&={\frac {1}{4}}\!\left[{\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}^{*}} }}+\left({\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}^{*}} }}\right)^{*}+{\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}} }}e^{2j\omega t}+\left({\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}} }}e^{2j\omega t}\right)^{*}\right]\\&={\frac {1}{2}}\operatorname {Re} \!\left({\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}^{*}} }}\right)+{\frac {1}{2}}\operatorname {Re} \!\left({\underline {\mathbf {E_{m}} }}\times \mathbf {H_{m}} e^{2j\omega t}\right)\end{aligned}}}

${\displaystyle \langle \mathbf {S} \rangle ={\frac {1}{T}}\int _{0}^{T}\mathbf {S} (t)\mathrm {d} t={\frac {1}{T}}\int _{0}^{T}\!\left[{\frac {1}{2}}\operatorname {Re} \!\left({\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}^{*}} }}\right)+{\frac {1}{2}}\operatorname {Re} \!\left({\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}} }}e^{2j\omega t}\right)\right]\mathrm {d} t}$

${\displaystyle \operatorname {Re} \!\left(e^{2j\omega t}\right)=\cos(2\omega t)}$

${\displaystyle \langle \mathbf {S} \rangle ={\frac {1}{2}}\operatorname {Re} \!\left({\underline {\mathbf {E_{m}} }}\times {\underline {\mathbf {H_{m}^{*}} }}\right)={\frac {1}{2}}\operatorname {Re} \!\left({\underline {\mathbf {E_{m}} }}e^{j\omega t}\times {\underline {\mathbf {H_{m}^{*}} }}e^{-j\omega t}\right)={\frac {1}{2}}\operatorname {Re} \!\left(\mathbf {E_{\mathrm {a} }} \times \mathbf {H_{\mathrm {a} }^{*}} \right)}$

例子与应用

平面波

${\displaystyle \langle S\rangle ={\frac {1}{2\mu _{0}\mathrm {c} }}E_{\mathrm {m} }^{2}={\frac {\varepsilon _{0}\mathrm {c} }{2}}E_{\mathrm {m} }^{2}}$

数学推导

${\displaystyle B_{\mathrm {m} }={\frac {1}{\mathrm {c} }}E_{\mathrm {m} }}$

${\displaystyle E(\mathbf {r} ,t)=E_{\mathrm {m} }\cos(\omega t-\mathbf {k} \cdot \mathbf {r} )}$
${\displaystyle B(\mathbf {r} ,t)=B_{\mathrm {m} }\cos(\omega t-\mathbf {k} \cdot \mathbf {r} )}$

${\displaystyle S(\mathbf {r} ,t)={\frac {1}{\mu _{0}}}E_{\mathrm {m} }B_{\mathrm {m} }\cos ^{2}(\omega t-\mathbf {k} \cdot \mathbf {r} )={\frac {1}{\mu _{0}c}}E_{\mathrm {m} }^{2}\cos ^{2}(\omega t-\mathbf {k} \cdot \mathbf {r} )=\varepsilon _{0}\mathrm {c} E_{\mathrm {m} }^{2}\cos ^{2}(\omega t-\mathbf {k} \cdot \mathbf {r} )}$

${\displaystyle \langle S\rangle ={\frac {1}{2\mu _{0}\mathrm {c} }}E_{\mathrm {m} }^{2}={\frac {\varepsilon _{0}\mathrm {c} }{2}}E_{\mathrm {m} }^{2}}$

辐射压

${\displaystyle P_{\mathrm {rad} }={\frac {\langle S\rangle }{\mathrm {c} }}}$

书目

• Harrington, Roger F. Time-Harmonic Electromagnetic Fields. McGraw-Hill. 1961.
• Hayt, William. Engineering Electromagnetics 4th. McGraw-Hill. 1981. ISBN 0-07-027395-2.
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. Foundations of Electromagnetic Theory 4th. Addison-Wesley. 1993. ISBN 0-201-52624-7.