坡印廷向量

（重定向自坡印亭矢量

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。[9]:258-260

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

增添場旋度的不變性

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

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

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

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

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

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

微观领域的形式

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

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

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

时间平均坡印廷矢量

{\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}}}

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

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

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

例子與應用

平面波

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

數學推導

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

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

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

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

輻射壓

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

參考文獻

1. ^ Julius Adams Stratton. Chap.II Stress and Energy. Electromagnetic Theory First. New York: McGraw-Hill. 1941: 132. ”first derived by Poynting in 1884 and again in the same year by Heaviside.”
2. ^ Janusz Turowski; Marek Turowski. Engineering Electrodynamics: Electric Machine, Transformer, and Power Equipment Design. CRC Press. 6 February 2014. ISBN 978-1-4665-8932-2.
3. ^ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0471927129
4. ^ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
5. ^ Poynting, J. H. On the Transfer of Energy in the Electromagnetic Field. Philosophical Transactions of the Royal Society of London. 1884, 175: 343–361. doi:10.1098/rstl.1884.0016.
6. Kinsler, P.; Favaro, A.; McCall M.W. Four Poynting theorems. Eur. J. Phys. 2009, 30 (5): 983. Bibcode:2009EJPh...30..983K. arXiv:0908.1721. doi:10.1088/0143-0807/30/5/007.
7. ^ Pfeifer, R.N.C.; Nieminen, T.A.; Heckenberg N. R.; Rubinsztein-Dunlop H. Momentum of an electromagnetic wave in dielectric media. Rev. Mod. Phys. 2007, 79 (4): 1197 [2015-05-30]. Bibcode:2007RvMP...79.1197P. doi:10.1103/RevModPhys.79.1197. （原始内容存档于2017-01-28）.
8. ^ Umov, N. A. Ein Theorem über die Wechselwirkungen in Endlichen Entfernungen. Zeitschrift für Mathematik und Physik. 1874, XIX: 97.
9. John David Jackson. Classical electrodynamics Third. New York: Wiley. 1998 [2015-05-21]. ISBN 0-471-30932-X. （原始内容存档于2009-08-04）.
10. ^ Wolfgang K. H. Panofsky; Melba Phillips. Classical Electricity and Magnetism: Second Edition. Courier Corporation. 12 July 2012. ISBN 978-0-486-13225-9.
11. ^ Andrew Chubykalo; Augusto Espinoza; Rumen Tzonchev. Experimental test of the compatibility of the definitions of the electromagnetic energy density and the Poynting vector (PDF). The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics. October 2004, 31 (1): 113–120.
12. ^ Henry Frank Tiersten. Linear Piezoelectric Plate Vibrations: Elements of the Linear Theory of Piezoelectricity and the Vibrations Piezoelectric Plates. Springer. 11 November 2013. ISBN 978-1-4899-6453-3.
13. ^ Bondar, H.; Bastien, F. Quelques remarques sur la transmission de l’énergie électromagnétique en champ proche. Annales de la Fondation Louis de Broglie. 2008, 33 (3-4): 283–305 [2015-05-21]. （原始内容存档于2016-03-04）.
14. ^ Hecht, Eugene, Optics 4th, United States of America: Addison Wesley, 2002, ISBN 0-8053-8566-5 （英语）
15. Richter, F.; Florian, M.; Henneberger, K. Poynting's theorem and energy conservation in the propagation of light in bounded media. Europhys. Lett. 2008, 81 (6): 67005. Bibcode:2008EL.....8167005R. arXiv:0710.0515. doi:10.1209/0295-5075/81/67005.
16. ^ Harrington (1981, p. 61)
17. ^ Hayt (1993, p. 402)
18. ^ Reitz (1993, p. 454)
19. ^ Feynman Lectures on Physics, Sections 17-4 and Volume 2, Chapter 17, section 4 and the end of Chapter 27, Section 6.

書目

• 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.