# 电偶极矩

${\displaystyle \mathbf {p} =q\,\mathbf {d} }$

## 簡單電偶極子案例

${\displaystyle \mathbf {p} (\mathbf {r} )=\int _{\mathbb {V} '}\rho (\mathbf {r} ')\,(\mathbf {r} '-\mathbf {r} )\ d^{3}\mathbf {r} '}$

${\displaystyle \rho (\mathbf {r} ')=\sum _{i=1}^{N}\,q_{i}\delta (\mathbf {r} '-\mathbf {r} _{i}')}$

${\displaystyle \mathbf {p} (\mathbf {r} )=\sum _{i=1}^{N}\,q_{i}\int _{\mathbb {V} '}\delta (\mathbf {r} '-\mathbf {r} _{i}')\,(\mathbf {r} '-\mathbf {r} )\ d^{3}\mathbf {r} '=\sum _{i=1}^{N}\,q_{i}(\mathbf {r} _{i}'-\mathbf {r} )}$

${\displaystyle \mathbf {p} (\mathbf {r} )=q(\mathbf {r} _{+}'-\mathbf {r} )-q(\mathbf {r} _{-}'-\mathbf {r} )=q(\mathbf {r} _{+}'-\mathbf {r} _{-}')=q\mathbf {d} }$

${\displaystyle \mathbf {p} (\mathbf {r} )=\sum _{i=1}^{N}\mathbf {p} _{i}}$

## 電偶極子產生的電勢與電場

${\displaystyle \phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}r_{+}}}-{\frac {q}{4\pi \varepsilon _{0}r_{-}}}}$

{\displaystyle {\begin{aligned}{\frac {1}{r_{\pm }}}&=\left(r^{2}+{\frac {d^{2}}{4}}\mp rd\cos {\theta }\right)^{-1/2}={\frac {1}{r}}\left(1+{\frac {d^{2}}{4r^{2}}}\mp {\frac {d\cos {\theta }}{r}}\right)^{-1/2}\\&\approx {\frac {1}{r}}\left(1\pm {\frac {d\cos {\theta }}{2r}}\right)\\\end{aligned}}}

${\displaystyle \phi (\mathbf {r} )\approx {\frac {qd\cos {\theta }}{4\pi \varepsilon _{0}r^{2}}}}$

${\displaystyle \mathbf {p} =q\mathbf {r} _{+}-q\mathbf {r} _{-}=q\mathbf {d} }$

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\ {\frac {\mathbf {p} \cdot {\hat {\mathbf {r} }}}{r^{2}}}}$

${\displaystyle E_{r}=-\ {\frac {\partial \phi (\mathbf {r} )}{\partial r}}={\frac {p\cos {\theta }}{2\pi \varepsilon _{0}r^{3}}}}$
${\displaystyle E_{\theta }=-\ {\frac {1}{r}}\ {\frac {\partial \phi (\mathbf {r} )}{\partial \theta }}={\frac {p\sin {\theta }}{4\pi \varepsilon _{0}r^{3}}}}$
${\displaystyle E_{\varphi }=-\ {\frac {1}{r\sin {\theta }}}{\frac {\partial \phi (\mathbf {r} )}{\partial \varphi }}=0}$

${\displaystyle \mathbf {E} ={\frac {p(2\cos {\theta }\ {\hat {\mathbf {r} }}+\sin {\theta }\ {\hat {\boldsymbol {\theta }}})}{4\pi \varepsilon _{0}r^{3}}}={\frac {3(\mathbf {p} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {p} }{4\pi \varepsilon _{0}r^{3}}}}$

{\displaystyle {\begin{aligned}\mathbf {E} =-\nabla \Phi &={\frac {1}{4\pi \epsilon _{0}r^{3}}}\left(3(\mathbf {p} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {p} \right)-{\frac {\mathbf {p} }{3\epsilon _{0}}}\delta ^{3}(\mathbf {r} )\\&={\frac {p}{4\pi \epsilon _{0}r^{3}}}(2\cos \theta {\hat {\mathbf {r} }}+\sin \theta {\hat {\boldsymbol {\theta }}})-{\frac {\mathbf {p} }{3\epsilon _{0}}}\delta ^{3}(\mathbf {r} )\end{aligned}}}

## 電偶極矩密度與電極化強度

${\displaystyle \mathbf {p} =\int _{\mathbb {V} '}{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')\ d^{3}\mathbf {r} '}$

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}{\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{3}}}\ d^{3}\mathbf {r} '}$

## 介電質內部的自由電荷與束縛電荷

${\displaystyle \rho _{bound}=-\nabla \cdot \mathbf {P} }$

${\displaystyle \rho _{total}=\rho _{free}+\rho _{bound}}$

${\displaystyle \sigma _{bound}=\mathbf {P} \cdot {\hat {\mathbf {n} }}_{\mathrm {out} }}$

${\displaystyle \nabla \cdot \mathbf {E} =\rho _{total}/\epsilon _{0}}$

${\displaystyle \nabla \cdot \mathbf {P} =-\rho _{bound}}$

${\displaystyle \mathbf {D} \ {\stackrel {\mathrm {def} }{=}}\ \epsilon _{0}\mathbf {E} +\mathbf {P} }$

${\displaystyle \nabla \cdot \mathbf {D} =\rho _{free}}$

### 介電質產生的電勢

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}\left[{\frac {\rho _{free}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}+{\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{3}}}+\sum _{i,j=1}^{3}{\frac {{\mathfrak {Q}}_{ij}(\mathbf {r} ')(x_{i}-x_{i}')(x_{j}-x_{j}')}{2|\mathbf {r} -\mathbf {r} '|^{5}}}\dots \right]\ d^{3}\mathbf {r} '}$

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}\left[{\frac {\rho _{free}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}+{\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{3}}}\right]\ d^{3}\mathbf {r} '}$

${\displaystyle \nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)={\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}}$

{\displaystyle {\begin{aligned}\int _{\mathbb {V} '}{\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{3}}}\ d^{3}\mathbf {r} '&=\int _{\mathbb {V} '}{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')\cdot \nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)\ d^{3}\mathbf {r} '\\&=\int _{\mathbb {V} '}\nabla '\cdot \left({\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)\ d^{3}\mathbf {r} '-\int _{\mathbb {V} '}{\frac {\nabla '\cdot {\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\ d^{3}\mathbf {r} '\\\end{aligned}}}

${\displaystyle \int _{\mathbb {V} '}\nabla '\cdot \left({\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)\ d^{3}\mathbf {r} '=\oint _{\mathbb {S} '}\left({\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)\cdot \ d\mathbf {a} '}$

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}\left[{\frac {\rho _{free}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}-\ {\frac {\nabla '\cdot {\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right]\ d^{3}\mathbf {r} '}$

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}{\frac {\rho _{total}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\ d^{3}\mathbf {r} '}$

${\displaystyle \rho _{total}=\rho _{free}+\nabla \cdot {\boldsymbol {\mathfrak {p}}}(\mathbf {r} )}$

${\displaystyle \rho _{bound}=-\nabla \cdot {\boldsymbol {\mathfrak {p}}}}$

${\displaystyle \nabla \cdot \mathbf {P} =-\rho _{bound}}$

### 面束縛電荷密度

{\displaystyle {\begin{aligned}\phi (\mathbf {r} )&={\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}{\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{3}}}\ d^{3}\mathbf {r} '\\&={\frac {1}{4\pi \varepsilon _{0}}}\oint _{\mathbb {S} '}\left({\frac {{\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)\cdot \ d\mathbf {a} '-{\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}{\frac {\nabla '\cdot {\boldsymbol {\mathfrak {p}}}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\ d^{3}\mathbf {r} '\\\end{aligned}}}

${\displaystyle \sigma _{bound}={\boldsymbol {\mathfrak {p}}}\cdot {\hat {\mathbf {n} }}}$

${\displaystyle \phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\oint _{\mathbb {S} '}{\frac {\sigma _{bound}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\ da'+{\frac {1}{4\pi \varepsilon _{0}}}\int _{\mathbb {V} '}{\frac {\rho _{bound}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\ d^{3}\mathbf {r} '}$

### 範例：處於均勻外電場的介電質球

${\displaystyle \phi (r,\theta )=\sum _{l=0}^{\infty }(A_{l}\ r^{l}+B_{l}\ r^{-(l+1)})P_{l}(\cos {\theta })}$

${\displaystyle \phi _{in}(r,\theta )=\sum _{l=0}^{\infty }A_{l}\ r^{l}P_{l}(\cos {\theta })}$

${\displaystyle \phi _{out}(r,\theta )=-E_{\infty }r\cos {\theta }+\sum _{l=0}^{\infty }B_{l}r^{-(l+1)}P_{l}(\cos {\theta })}$

${\displaystyle \phi _{in}(R,\theta )=\phi _{out}(R,\theta )}$
${\displaystyle \epsilon _{r}\left.{\frac {\partial \phi _{in}(r,\theta )}{\partial r}}\right|_{r=R}=\left.{\frac {\partial \phi _{out}(r,\theta )}{\partial r}}\right|_{r=R}}$

${\displaystyle A_{1}R=-E_{\infty }R+B_{1}R^{-2}}$
${\displaystyle A_{l}R^{l}=B_{l}R^{-(l+1)},\qquad \qquad l\neq 1}$

${\displaystyle \epsilon _{r}A_{1}=-E_{\infty }-2B_{1}R^{-3}}$
${\displaystyle \epsilon _{r}lA_{l}R^{(l-1)}=-(l+1)B_{l}R^{-(l+2)},\qquad \qquad l\neq 1}$

${\displaystyle A_{1}=-\ {\frac {3E_{\infty }}{\epsilon _{r}+2}}}$
${\displaystyle B_{1}={\frac {(\epsilon _{r}-1)R^{3}E_{\infty }}{\epsilon _{r}+2}}}$

${\displaystyle A_{l}=B_{l}=0,\qquad \qquad l\neq 1}$

${\displaystyle \phi _{out}(r,\theta )=-E_{\infty }r\cos {\theta }+{\frac {(\epsilon _{r}-1)R^{3}E_{\infty }\cos {\theta }}{(\epsilon _{r}+2)r^{2}}}}$

${\displaystyle \phi _{in}(r,\theta )=-{\frac {3}{\epsilon _{r}+2}}E_{\infty }r\cos {\theta }}$

${\displaystyle \mathbf {E} _{in}=-\nabla \phi _{in}(r,\theta )={\frac {3}{\epsilon _{r}+2}}\mathbf {E} _{\infty }=\left(1-\ {\frac {\epsilon _{r}-1}{\epsilon _{r}+2}}\right)\mathbf {E} _{\infty }}$

${\displaystyle \mathbf {E} _{p}=\mathbf {E} _{in}-\mathbf {E} _{\infty }=-\ \left({\frac {\epsilon _{r}-1}{\epsilon _{r}+2}}\right)\mathbf {E} _{\infty }=-{\frac {\boldsymbol {\mathfrak {p}}}{3\epsilon _{0}}}}$

${\displaystyle {\boldsymbol {\mathfrak {p}}}=\epsilon _{0}(\epsilon _{r}-1)\mathbf {E} _{in}}$

${\displaystyle \rho _{bound}=-\nabla \cdot {\boldsymbol {\mathfrak {p}}}=0}$

${\displaystyle \sigma _{bound}={3}\varepsilon _{0}{\frac {\epsilon _{r}-1}{\epsilon _{r}+2}}E_{\infty }\cos {\theta }={\boldsymbol {\mathfrak {p}}}\cdot {\hat {\mathbf {r} }}}$

## 註釋

1. ^ 粒子物理學裏，有三種重要的離散對稱性：電荷共軛對稱性是粒子與其反粒子的對稱性，又稱「正反共軛對稱性」。宇稱對稱性是關於粒子位置 ${\displaystyle \mathbf {r} }$${\displaystyle -\mathbf {r} }$ 的對稱性，時間反演對稱性是時間 ${\displaystyle t}$${\displaystyle -t}$ 的對稱性。
2. ^ 時間反演變換將 ${\displaystyle t}$ 改變為 ${\displaystyle -t}$ 。一個載流迴圈的磁偶極矩 ${\displaystyle {\boldsymbol {\mu }}}$ 是其所載電流 ${\displaystyle I}$ 乘於迴圈面積 ${\displaystyle \mathbf {a} }$ ，以方程式表示為 ${\displaystyle {\boldsymbol {\mu }}=I\mathbf {a} ={\frac {\mathrm {d} q}{\mathrm {d} t}}\mathbf {a} }$ 。注意到電流是電荷量對於時間的導數，所以，時間反演會逆反磁偶極矩的方向。電偶磁矩的兩個參數，電荷量和位移向量都跟時間反演無關，所以，時間反演不會改變電偶極矩的方向。
3. ^ 空間反演（宇稱）變換是粒子位置坐標對於參考系原點的反射。電偶極矩是極向量polar vector），而磁偶極矩是軸向量axial vector），所以，空間反演（宇稱）會逆反電偶極矩的方向，不會改變磁偶極矩的方向。

## 參考文獻

1. ^ Christopher J. Cramer. Essentials of computational chemistry 2. Wiley: 307. 2004.
2. ^ Jackson, John David. Classical Electrodynamic 3rd. USA: John Wiley & Sons, Inc. 1999: 107–111145–150, 184–188. ISBN 978-0-471-30932-1.
3. ^ Jack Vanderlinde. §7.1 The electric field due to a polarized dielectric. Classical Electromagnetic Theory. Springer. 2004. ISBN 1-4020-2699-4.
4. ^ George E Owen. Introduction to Electromagnetic Theory republication of the 1963 Allyn & Bacon. Courier Dover Publications. 2003: 80. ISBN 0-486-42830-3.
5. ^ Daniel A. Jelski, Thomas F. George. Computational studies of new materials. World Scientific. 1999: 222ff. ISBN 981-02-3325-6.
6. ^ Andrew Gray. The theory and practice of absolute measurements in electricity and magnetism. Macmillan & Co. 1888: 126–127.，根據這本書內的方程式，引述自威廉·湯姆森的論文。
7. ^ Barr, S.M. A REVIEW OF CP VIOLATION IN ATOMS. International Journal of Modern Physics A. 1993-01-20, 08 (02): 209–236 [2021-01-30]. ISSN 0217-751X. doi:10.1142/S0217751X93000096. （原始内容存档于2021-12-07） （英语）.
8. ^ Peccei, R. D.; Quinn, Helen R. CP Conservation in the Presence of Pseudoparticles. Physical Review Letters. 1977-06-20, 38 (25): 1440–1443. ISSN 0031-9007. doi:10.1103/PhysRevLett.38.1440 （英语）.
9. ^ Abel, S.; Khalil, S.; Lebedev, O. EDM constraints in supersymmetric theories. Nuclear Physics B. 2001-07, 606 (1-2): 151–182 [2021-01-30]. doi:10.1016/S0550-3213(01)00233-4. （原始内容存档于2019-06-26） （英语）.
10. ^ E. Aleksandrov; et al, A new Precision Measurement of the Neutron Electric Dipole Moment (EDM), Neutron Physics Laboratory of Neutron Physics Division of Neutron Research Department at PETERSBURG NUCLEAR PHYSICS INSTITUTE, [2011-01-25], （原始内容存档于2011-08-23）
11. ^ Baker, C. A.; Doyle, D. D.; Geltenbort, P.; Green, K.; van der Grinten, M. G. D.; Harris, P. G.; Iaydjiev, P.; Ivanov, S. N.; May, D. J. R. Improved Experimental Limit on the Electric Dipole Moment of the Neutron. Physical Review Letters. 2006-09-27, 97 (13): 131801. ISSN 0031-9007. doi:10.1103/PhysRevLett.97.131801 （英语）.
12. ^ Hudson, J. J.; Kara, D. M.; Smallman, I. J.; Sauer, B. E.; Tarbutt, M. R.; Hinds, E. A. Improved measurement of the shape of the electron. Nature. 2011-05, 473 (7348): 493–496 [2021-01-30]. ISSN 0028-0836. doi:10.1038/nature10104. （原始内容存档于2020-05-18） （英语）.
13. ^ Griffith, W. C.; Swallows, M. D.; Loftus, T. H.; Romalis, M. V.; Heckel, B. R.; Fortson, E. N. Improved Limit on the Permanent Electric Dipole Moment of Hg 199. Physical Review Letters. 2009-03-10, 102 (10): 101601. ISSN 0031-9007. doi:10.1103/PhysRevLett.102.101601 （英语）.
14. ^ Dar, Shahida. The Neutron EDM in the SM : A Review. arXiv High Energy Physics - Phenomenology e-prints. 2000-08-01: arXiv:hep–ph/0008248.