# 本構關係

${\displaystyle \mathbf {D} (\mathbf {r} ,t)\ {\stackrel {def}{=}}\ \epsilon _{0}\mathbf {E} (\mathbf {r} ,t)+\mathbf {P} (\mathbf {r} ,t)}$
${\displaystyle \mathbf {H} (\mathbf {r} ,t)\ {\stackrel {def}{=}}\ {\frac {1}{\mu _{0}}}\mathbf {B} (\mathbf {r} ,t)-\mathbf {M} (\mathbf {r} ,t)}$

${\displaystyle \mathbf {D} =\mathbf {D} (\mathbf {E} ,\mathbf {B} )}$
${\displaystyle \mathbf {H} =\mathbf {H} (\mathbf {E} ,\mathbf {B} )}$

## 自由空間案例

${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} }$
${\displaystyle \mathbf {H} =\mathbf {B} /\mu _{0}}$

## 線性物質案例

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

(法拉第电磁感应定律)
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$ ${\displaystyle \oint _{\mathbb {L} }\ \mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {\mathrm {d} \Phi _{\mathbf {B} }}{\mathrm {d} t}}}$

(含馬克士威加法)
${\displaystyle \nabla \times (\mathbf {B} /\mu )=\mathbf {J} _{f}+\varepsilon {\frac {\partial \mathbf {E} }{\partial t}}\ }$ ${\displaystyle \oint _{\mathbb {L} }\ (\mathbf {B} /\mu )\cdot \mathrm {d} {\boldsymbol {\ell }}=I_{f}+{\frac {\mathrm {d} \Phi _{\varepsilon \mathbf {E} }}{\mathrm {d} t}}}$

${\displaystyle \Phi _{\varepsilon \mathbf {E} }=\iint _{\mathbb {S} }\!\!\!\!\!\!\!\!\!\!\!\!\;\subset \!\supset \varepsilon \mathbf {E} \cdot \mathrm {d} \mathbf {s} }$

## 一般案例

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

• 色散吸收${\displaystyle \varepsilon }$${\displaystyle \mu }$頻率的函數。因果論不允許物質具有非色散性，例如，克拉莫-克若尼關係式。場與場之間的相位可能不同相，這導致${\displaystyle \varepsilon }$${\displaystyle \mu }$為複值，也導致電磁波被物質吸收。[2]:330-335
• 非線性${\displaystyle \varepsilon }$${\displaystyle \mu }$都是電場與磁場的函數。例如，克爾效應[3]波克斯效應Pockels effect）。
• 各向異性：例如，雙折射二向色性dichroism）。${\displaystyle \varepsilon }$${\displaystyle \mu }$都是二階張量[4]
${\displaystyle D_{i}=\sum _{j}\epsilon _{ij}E_{j}}$
${\displaystyle B_{i}=\sum _{j}\mu _{ij}H_{j}}$
• 雙耦合各向同性Bi-isotropy）或雙耦合各向異性Bi-anisotropy）：在雙耦合各向同性物質裏，${\displaystyle \mathbf {D} }$場與${\displaystyle \mathbf {H} }$場分別各向同性地耦合於${\displaystyle \mathbf {E} }$場與${\displaystyle \mathbf {B} }$[4]
${\displaystyle \mathbf {D} =\epsilon \mathbf {E} +\xi \mathbf {H} }$
${\displaystyle \mathbf {B} =\mu \mathbf {H} +\zeta \mathbf {E} }$

• 在不同位置和時間，${\displaystyle \mathbf {P} }$場與${\displaystyle \mathbf {M} }$場分別跟${\displaystyle \mathbf {E} }$場、${\displaystyle \mathbf {B} }$場有關：這可能是因為「空間不勻性」。例如，一個磁鐵的域結構異質結構液晶，或最常出現的狀況是多種材料占有不同空間區域。這也可能是因為隨時間而改變的物質或磁滯現象。對於這種狀況，${\displaystyle \mathbf {P} }$場與${\displaystyle \mathbf {M} }$場計算為[5][2]:14
${\displaystyle \mathbf {P} (\mathbf {r} ,t)=\varepsilon _{0}\int d^{3}\mathbf {r} 'dt'\;\chi _{\mathrm {e} }(\mathbf {r} ,\mathbf {r} ',t,t';\mathbf {E} )\,\mathbf {E} (\mathbf {r} ',t')}$
${\displaystyle \mathbf {M} (\mathbf {r} ,t)={\frac {1}{\mu _{0}}}\int d^{3}\mathbf {r} 'dt'\;\chi _{\mathrm {m} }(\mathbf {r} ,\mathbf {r} ',t,t';\mathbf {B} )\,\mathbf {B} (\mathbf {r} ',t')}$

## 參考文獻

1. ^ Andrew Zangwill. Modern Electrodynamics. Cambridge University Press. 2013. ISBN 978-0-521-89697-9.
2. Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc., 1999, ISBN 978-0-471-30932-1
3. ^ Weinberger, P. John Kerr and his Effects Found in 1877 and 1878 (PDF). Philosophical Magazine Letters. 2008, 88 (12): 897–907. Bibcode:2008PMagL..88..897W. doi:10.1080/09500830802526604. （原始内容 (PDF)存档于2011-07-18）.
4. 通常，物質都具有雙耦合各向異性。TG Mackay and A Lakhtakia. Electromagnetic Anisotropy and Bianisotropy: A Field Guide. World Scientific. 2010: pp. 7–11.
5. ^ Halevi, Peter, Spatial dispersion in solids and plasmas, Amsterdam: North-Holland, 1992, ISBN 978-0444874054
6. ^ Aspnes, David E., "Local-field effects and effective-medium theory: A microscopic perspective," Am. J. Phys. 50, p. 704-709 (1982).
7. ^ O. C. Zienkiewicz, Robert Leroy Taylor, J. Z. Zhu, Perumal Nithiarasu. The Finite Element Method Sixth. Oxford UK: Butterworth-Heinemann. 2005: 550 ff. ISBN 0750663219.
8. ^ N. Bakhvalov and G. Panasenko, Homogenization: Averaging Processes in Periodic Media (Kluwer: Dordrecht, 1989); V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer: Berlin, 1994).
9. ^ Vitaliy Lomakin, Steinberg BZ, Heyman E, & Felsen LB. Multiresolution Homogenization of Field and Network Formulations for Multiscale Laminate Dielectric Slabs (PDF). IEEE Transactions on Antennas and Propagation. 2003, 51 (10): 2761 ff. Bibcode:2003ITAP...51.2761L. doi:10.1109/TAP.2003.816356. （原始内容 (PDF)存档于2012-05-14）.
10. ^ Edward D. Palik & Ghosh G, Handbook of Optical Constants of Solids, London UK: Academic Press: pp. 1114, 1998, ISBN 0125444222