# 多極展開

（重定向自多極展開式

## 電勢的多極展開式

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

### 笛卡兒多極展開

${\displaystyle f(\mathbf {r} ')=f(\mathbf {O} )+\mathbf {r} '\cdot \nabla 'f(\mathbf {O} )+{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}r'_{\alpha }r'_{\beta }{\frac {\partial ^{2}f(\mathbf {O} )}{\partial r'_{\alpha }\partial r'_{\beta }}}+\dots }$

${\displaystyle {\frac {\partial f(\mathbf {r} ')}{\partial r'_{\alpha }}}={\frac {(r_{\alpha }-r'_{\alpha })}{|\mathbf {r} -\mathbf {r} '|^{3}}}}$
${\displaystyle {\frac {\partial ^{2}f(\mathbf {r} ')}{\partial r'_{\alpha }\partial r'_{\beta }}}={\frac {3(r_{\alpha }-r'_{\alpha })(r_{\beta }-r'_{\beta })}{|\mathbf {r} -\mathbf {r} '|^{5}}}-{\frac {\delta _{\alpha \beta }}{|\mathbf {r} -\mathbf {r} '|^{3}}}}$

{\displaystyle {\begin{aligned}{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}&={\frac {1}{r}}+{\frac {\mathbf {r} \cdot \mathbf {r} '}{r^{3}}}+{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}\left({\frac {3r_{\alpha }r_{\beta }}{r^{5}}}-{\frac {\delta _{\alpha \beta }}{r^{3}}}\right)r'_{\alpha }r'_{\beta }+\dots \\&={\frac {1}{r}}+{\frac {\mathbf {r} \cdot \mathbf {r} '}{r^{3}}}+{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}\left({\frac {3r_{\alpha }r_{\beta }r'_{\alpha }r'_{\beta }-r^{2}r'_{\alpha }r'_{\beta }\delta _{\alpha \beta }}{r^{5}}}\right)+\dots \\&={\frac {1}{r}}+{\frac {\mathbf {r} \cdot \mathbf {r} '}{r^{3}}}+{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}\left({\frac {3r_{\alpha }r_{\beta }r'_{\alpha }r'_{\beta }-r_{\alpha }r_{\beta }r^{\prime 2}\delta _{\alpha \beta }}{r^{5}}}\right)+\dots \\\end{aligned}}}

${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int _{\mathbb {V'} }\left[{\frac {1}{r}}+{\frac {\mathbf {r} \cdot \mathbf {r} '}{r^{3}}}+{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}{\frac {r_{\alpha }r_{\beta }(3r'_{\alpha }r'_{\beta }-r^{\prime 2}\delta _{\alpha \beta })}{r^{5}}}+\dots \right]\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '}$

${\displaystyle q{\stackrel {def}{=}}\ \int _{\mathbb {V'} }\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '}$
${\displaystyle \mathbf {p} {\stackrel {def}{=}}\ \int _{\mathbb {V'} }\mathbf {r} '\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '}$
${\displaystyle Q_{\alpha \beta }{\stackrel {def}{=}}\ \int _{\mathbb {V'} }(3r'_{\alpha }r'_{\beta }-r^{\prime 2}\delta _{\alpha \beta })\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '}$

${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\left({\frac {q}{r}}+{\frac {\mathbf {p} \cdot \mathbf {r} }{r^{3}}}+{\frac {1}{2r^{5}}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}Q_{\alpha \beta }r_{\alpha }r_{\beta }+\dots \right)}$

### 球多極展開

${\displaystyle {\frac {1}{|\mathbf {r} -\mathbf {r} '|}}=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\frac {4\pi }{2\ell +1}}\ {\frac {r^{\prime \ell }}{r^{\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ')Y_{\ell m}(\theta ,\phi ),\qquad r'

${\displaystyle \Phi (\mathbf {r} )={\frac {1}{\epsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\frac {Y_{\ell m}(\theta ,\phi )}{((2\ell +1)r^{\ell +1}}}\int _{\mathbb {V'} }Y_{\ell m}^{*}(\theta ',\phi ')r^{\prime \ell }\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '}$

${\displaystyle q_{\ell m}{\stackrel {def}{=}}\ \int _{\mathbb {V'} }Y_{\ell m}^{*}(\theta ',\phi ')r^{\prime \ell }\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '}$

${\displaystyle \Phi (\mathbf {r} )={\frac {1}{\epsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\frac {q_{\ell m}Y_{\ell m}(\theta ,\phi )}{(2\ell +1)r^{\ell +1}}}}$

{\displaystyle {\begin{aligned}q_{00}&={\frac {1}{\sqrt {4\pi }}}\int _{\mathbb {V'} }\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '&&={\frac {1}{\sqrt {4\pi }}}\ q\\q_{11}&=-{\sqrt {\frac {3}{8\pi }}}\int _{\mathbb {V'} }r'\sin {\theta '}\ e^{-i\phi '}\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '&&=-{\sqrt {\frac {3}{8\pi }}}\ (p_{x}-ip_{y})\\q_{10}&={\sqrt {\frac {3}{4\pi }}}\int _{\mathbb {V'} }r'\cos {\theta }\ \rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '&&=-{\sqrt {\frac {3}{4\pi }}}\ p_{z}\\q_{22}&={\sqrt {\frac {15}{32\pi }}}\int _{\mathbb {V'} }r^{\prime 2}\sin ^{2}{\theta '}\ e^{-2i\phi '}\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '&&={\sqrt {\frac {15}{288\pi }}}\ (Q_{11}-2iQ_{12}-Q_{22})\\q_{21}&=-{\sqrt {\frac {15}{8\pi }}}\int _{\mathbb {V'} }r^{\prime 2}\sin {\theta '}\cos {\theta '}\ e^{-i\phi '}\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '&&=-{\sqrt {\frac {15}{72\pi }}}\ (Q_{13}-iQ_{33})\\q_{20}&={\sqrt {\frac {5}{16\pi }}}\int _{\mathbb {V'} }r^{\prime 2}(3\cos ^{2}{\theta '}-1)\rho (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '&&={\sqrt {\frac {5}{16\pi }}}\ Q_{33}\end{aligned}}}

## 電能的多極展開式

${\displaystyle U=\int _{\mathbb {V} }\rho (\mathbf {r} )\Phi (\mathbf {r} )\ \mathrm {d} ^{3}\mathbf {r} }$

{\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&=\Phi (\mathbf {O} )+\mathbf {r} \cdot \nabla \Phi (\mathbf {O} )+{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}r_{\alpha }r_{\beta }{\frac {\partial ^{2}\Phi (\mathbf {O} )}{\partial r_{\alpha }\partial r_{\beta }}}+\dots \\&=\Phi (\mathbf {O} )-\mathbf {r} \cdot \mathbf {E} (\mathbf {O} )-{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}r_{\alpha }r_{\beta }{\frac {\partial E_{\beta }(\mathbf {O} )}{\partial r_{\alpha }}}+\dots \\\end{aligned}}}

${\displaystyle \Phi (\mathbf {r} )=\Phi (\mathbf {O} )-\mathbf {r} \cdot \mathbf {E} (\mathbf {O} )-{\frac {1}{6}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}(3r_{\alpha }r_{\beta }-r^{2}\delta _{\alpha \beta }){\frac {\partial E_{\beta }(\mathbf {O} )}{\partial r_{\alpha }}}+\dots }$

${\displaystyle U=q\Phi (\mathbf {O} )-\mathbf {p} \cdot \mathbf {E} (\mathbf {O} )-{\frac {1}{6}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}Q_{\alpha \beta }{\frac {\partial E_{\beta }(\mathbf {O} )}{\partial r_{\alpha }}}+\dots }$

## 磁向量勢的多極展開式

${\displaystyle \mathbf {A} (\mathbf {r} )\ {\stackrel {def}{=}}\ {\frac {\mu _{0}}{4\pi }}\int _{\mathbb {V} '}{\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$

${\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{\mathbb {V'} }\left[{\frac {1}{r}}+{\frac {\mathbf {r} \cdot \mathbf {r} '}{r^{3}}}+{\frac {1}{2}}\sum _{\alpha =1}^{3}\sum _{\beta =1}^{3}{\frac {r_{\alpha }r_{\beta }(3r'_{\alpha }r'_{\beta }-r^{\prime 2}\delta _{\alpha \beta })}{r^{5}}}+\dots \right]\mathbf {J} (\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '}$

{\displaystyle {\begin{aligned}\int _{\mathbb {V} '}J_{\alpha }(\mathbf {r} ')\,d^{3}\mathbf {r} '&=\int _{\mathbb {V} '}[\mathbf {J} (\mathbf {r} ')\cdot \nabla ']r'_{\alpha }\,d^{3}\mathbf {r} '=\int _{\mathbb {V} '}\nabla '\cdot [r'_{\alpha }\mathbf {J} (\mathbf {r} ')]-r'_{\alpha }\nabla '\cdot \mathbf {J} (\mathbf {r} ')\,d^{3}\mathbf {r} '\\&=\int _{\mathbb {V} '}\nabla '\cdot [r'_{\alpha }\mathbf {J} (\mathbf {r} ')]\,d^{3}\mathbf {r} '\\\end{aligned}}}

${\displaystyle \int _{\mathbb {V} '}J_{\alpha }(\mathbf {r} ')\,d^{3}\mathbf {r} '=\int _{\mathbb {S} '}r'_{\alpha }\mathbf {J} (\mathbf {r} ')\cdot \,d\mathbf {S} '=0}$

{\displaystyle {\begin{aligned}\int _{\mathbb {V} '}\nabla '\cdot [r'_{\alpha }r'_{\beta }J(\mathbf {r} ')]\,d^{3}\mathbf {r} '&=\int _{\mathbb {V} '}r'_{\beta }[J(\mathbf {r} ')\cdot \nabla ']r'_{\alpha }+r'_{\alpha }[J(\mathbf {r} ')\cdot \nabla ']r'_{\beta }+r'_{\alpha }r'_{\beta }\nabla '\cdot J(\mathbf {r} ')\,d^{3}\mathbf {r} '\\&=\int _{\mathbb {V} '}r'_{\beta }J_{\alpha }(\mathbf {r} ')+r'_{\alpha }J_{\beta }(\mathbf {r} ')\,d^{3}\mathbf {r} '\\&=0\\\end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {r} \cdot \int _{\mathbb {V'} }\mathbf {r} 'J_{\alpha }(\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '&={\frac {1}{2}}\sum _{\beta =1}^{3}r_{\beta }\int _{\mathbb {V'} }r'_{\beta }J_{\alpha }(\mathbf {r} ')-r'_{\alpha }J_{\beta }(\mathbf {r} ')\ \mathrm {d} ^{3}\mathbf {r} '\\&=-\ {\frac {1}{2}}\left[\mathbf {r} \times \int _{\mathbb {V'} }\mathbf {r} '\times \mathbf {J} (\mathbf {r} )\ \mathrm {d} ^{3}\mathbf {r} '\right]_{\alpha }\\\end{aligned}}}

${\displaystyle \mathbf {m} {\stackrel {def}{=}}\ {\frac {1}{2}}\int _{\mathbb {V} '}\mathbf {r} '\times \mathbf {J} (\mathbf {r} ')\,d^{3}\mathbf {r} '}$

${\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}{\frac {\mathbf {m} \times \mathbf {r} }{r^{3}}}}$

## 參考文獻

1. Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 111, 145–151, 1999, ISBN 978-0-471-30932-1
2. ^ Ross D. Adamson. The Fast Multipole Method. January 21, 1999 [December 10, 2010].