# 四極子

（重定向自四極矩

## 數學定義

${\displaystyle Q_{ij}=\sum _{l}q_{l}(3r_{i}r_{j}-\|{\vec {r}}\|^{2}\delta _{ij})}$ .

${\displaystyle Q_{ij}=\int \,\rho (3r_{i}r_{j}-\|{\vec {r}}\|^{2}\delta _{ij})\,d^{3}\mathbf {r} }$

${\displaystyle V_{q}(\mathbf {R} )={\frac {k}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,n_{i}n_{j}\ ,}$

## 电四极子

${\displaystyle V_{q}(\mathbf {R} )={\frac {1}{4\pi \epsilon _{0}}}{\frac {1}{|\mathbf {R} |^{3}}}\sum _{i,j}{\frac {1}{2}}Q_{ij}\,n_{i}n_{j}\ ,}$

## 重力四極子

${\displaystyle V_{q}(\mathbf {R} )=-G{\frac {1}{2}}{\frac {1}{|\mathbf {R} |^{3}}}\sum _{i,j}Q_{ij}\,n_{i}n_{j}\ .}$

${\displaystyle Q_{ij}=M(3x_{i}x_{j}-\delta _{ij})}$

## 參考資料

1. ^ Weisstein, Eric. Electric Quadrupole Moment. Eric Weisstein's World of Physics. Wolfram Research. [May 8, 2012]. （原始内容存档于2019-02-16）.
2. ^ Jackson, John David. Classical Electrodynamics. John Wiley & Sons. 1975. ISBN 0-471-43132-X.
3. ^ Milbert, D. G.; Smith, D. A. Converting GPS Height into NAVD88 Elevation with the GEOID96 Geoid Height Model. National Geodetic Survey, NOAA. [2007-03-07]. （原始内容存档于2007-06-04）.
4. ^ Thorne, Kip S. Multipole Expansions of Gravitational Radiation. Reviews of Modern Physics. April 1980, 52 (2): 299–339. Bibcode:1980RvMP...52..299T. doi:10.1103/RevModPhys.52.299.