在靜磁學裏，設定電流密度分佈 
，則其產生的磁向量勢 
為
；
其中，
是場位置，
是源位置。
將前面推導出的 
在原點 
的泰勒級數帶入磁向量勢方程式，則可得到
![{\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} '}](//wikimedia.org/api/rest_v1/media/math/render/svg/7597d4bd98544b9adb66afb44b05882348f79108)
。
由於在靜磁學裏 
，
![{\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a1d447916cbb803dcb14bce19f3408ee8c9b0432)
；
應用高斯散度定理，由於電流密度分佈 
是局部的，假若積分體積 
足夠大，則位於包含積分體積的曲面 
的電流密度分佈為零：
。
所以，磁單極子項目 
等於零。
磁偶極子項目不等於零。首先，應用高斯散度定理和電流密度分佈的局部性這事實，可以得到
![{\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5bc5e90fc5cc9fd36caaecd411f2d861a760daf2)
。
注意到以下關係式：
![{\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/eb82506e5f533a2dd214a08731551b5c4f723c02)
。
定義磁偶極矩 
為
。
只取至最低階項目，即磁偶極矩項目，則磁向量勢 為
。
![{\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} '}](http://wikimedia.org/api/rest_v1/media/math/render/svg/41d475b895b6103168d1d7301d9a9ea5d9f15abb)
![{\displaystyle \rho (\mathbf {r} ')=q[\delta (\mathbf {r} '-\mathbf {r} '_{1})-\delta (\mathbf {r} '-\mathbf {r} '_{2})]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/9f9df3345abd6273be5e0fff5823a747d094262a)
![{\displaystyle \int _{\mathbb {V'} }q[\delta (\mathbf {r} '-\mathbf {r} '_{1})-\delta (\mathbf {r} '-\mathbf {r} '_{2})]\ \mathrm {d} ^{3}\mathbf {r} '=0}](http://wikimedia.org/api/rest_v1/media/math/render/svg/734cc210275fdc2178f8e51e4bc6d9ba91c5a172)
![{\displaystyle \int _{\mathbb {V'} }\mathbf {r} 'q[\delta (\mathbf {r} '-\mathbf {r} '_{1})-\delta (\mathbf {r} '-\mathbf {r} '_{2})]\ \mathrm {d} ^{3}\mathbf {r} '=q(\mathbf {r} '_{1}-\mathbf {r} '_{2})}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c7696acde3701b54ac0c436ca2d8b4092313f75c)
![{\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} '}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7597d4bd98544b9adb66afb44b05882348f79108)
![{\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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/a1d447916cbb803dcb14bce19f3408ee8c9b0432)
![{\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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5bc5e90fc5cc9fd36caaecd411f2d861a760daf2)
![{\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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/eb82506e5f533a2dd214a08731551b5c4f723c02)
