给予在源位置
r
′
=
(
r
′
,
θ
′
,
ϕ
′
)
{\displaystyle \mathbf {r} '=(r',\theta ',\phi ')}
的电荷分布,计算在场位置
r
=
(
r
,
θ
,
ϕ
)
{\displaystyle \mathbf {r} =(r,\theta ,\phi )}
产生的电势。
源位置为
r
′
{\displaystyle \mathbf {r} ^{\prime }}
的点电荷
q
{\displaystyle q}
,其电势
Φ
(
r
)
{\displaystyle \Phi (\mathbf {r} )}
在场位置
r
{\displaystyle \mathbf {r} }
为
Φ
(
r
)
=
q
4
π
ε
0
|
r
−
r
′
|
=
q
4
π
ε
0
1
r
2
+
r
′
2
−
2
r
′
r
cos
γ
{\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r^{\prime }} |}}={\frac {q}{4\pi \varepsilon _{0}}}{\frac {1}{\sqrt {r^{2}+r^{\prime 2}-2r^{\prime }r\cos \gamma }}}}
;
其中,
ε
0
{\displaystyle \varepsilon _{0}}
是电常数 ,
γ
{\displaystyle \gamma }
是
r
{\displaystyle \mathbf {r} }
与
r
′
{\displaystyle \mathbf {r} ^{\prime }}
之间的夹角。
假设
r
′
<
r
{\displaystyle r'<r}
,场位置比源位置离原点更远,则此距离倒数函数
1
/
|
r
−
r
′
|
{\displaystyle 1/|\mathbf {r} -\mathbf {r^{\prime }} |}
以
r
′
/
r
{\displaystyle r^{\prime }/r}
的幂 和勒壤得多项式 展开为[ 1] :
Φ
(
r
)
=
q
4
π
ε
0
r
∑
ℓ
=
0
∞
(
r
′
r
)
ℓ
P
ℓ
(
cos
γ
)
{\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}r}}\sum _{\ell =0}^{\infty }\left({\frac {r^{\prime }}{r}}\right)^{\ell }P_{\ell }(\cos \gamma )}
。
应用球馀弦定律 (spherical law of cosine ),
cos
γ
{\displaystyle \cos \gamma }
表示为
cos
γ
=
cos
θ
cos
θ
′
+
sin
θ
sin
θ
′
cos
(
ϕ
−
ϕ
′
)
{\displaystyle \cos \gamma =\cos \theta \cos \theta ^{\prime }+\sin \theta \sin \theta ^{\prime }\cos(\phi -\phi ^{\prime })}
。
这结果也可以直接用向量代数 直接计算出来。
应用球谐函数加法定理 ,
P
ℓ
(
cos
γ
)
{\displaystyle P_{\ell }(\cos \gamma )}
又表示为[ 2]
P
ℓ
(
cos
γ
)
=
4
π
2
ℓ
+
1
∑
m
=
−
ℓ
ℓ
Y
ℓ
m
(
θ
,
ϕ
)
Y
ℓ
m
∗
(
θ
′
,
ϕ
′
)
{\displaystyle P_{\ell }(\cos \gamma )={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ^{\prime },\phi ^{\prime })}
;
其中,
Y
ℓ
m
{\displaystyle Y_{\ell m}}
是球谐函数 。
将这方程式代入电势的方程式,可以得到
Φ
(
r
)
=
q
4
π
ε
0
r
∑
ℓ
=
0
∞
(
r
′
r
)
ℓ
(
4
π
2
ℓ
+
1
)
∑
m
=
−
ℓ
ℓ
Y
ℓ
m
(
θ
,
ϕ
)
Y
ℓ
m
∗
(
θ
′
,
ϕ
′
)
{\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}r}}\sum _{\ell =0}^{\infty }\left({\frac {r^{\prime }}{r}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ^{\prime },\phi ^{\prime })}
。
点电荷的“球多极矩” 定义为
q
ℓ
m
=
d
e
f
q
r
′
ℓ
Y
ℓ
m
∗
(
θ
′
,
ϕ
′
)
{\displaystyle q_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ qr^{\prime \ell }Y_{\ell m}^{*}(\theta ^{\prime },\phi ^{\prime })}
。
则电势的方程式又可写为
Φ
(
r
)
=
1
ε
0
∑
ℓ
=
0
∞
∑
m
=
−
ℓ
ℓ
q
ℓ
m
Y
ℓ
m
(
θ
,
ϕ
)
(
2
ℓ
+
1
)
r
ℓ
+
1
{\displaystyle \Phi (\mathbf {r} )={\frac {1}{\varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\frac {q_{\ell m}Y_{\ell m}(\theta ,\phi )}{(2\ell +1)r^{\ell +1}}}}
。
假设
r
<
r
′
{\displaystyle r<r'}
,场位置比源位置离原点更近,则此距离倒数函数
1
/
|
r
−
r
′
|
{\displaystyle 1/|\mathbf {r} -\mathbf {r^{\prime }} |}
可以以
r
/
r
′
{\displaystyle r/r^{\prime }}
的幂 和勒壤得多项式 展开:
Φ
(
r
)
=
q
4
π
ε
0
r
′
∑
ℓ
=
0
∞
(
r
r
′
)
ℓ
(
4
π
2
ℓ
+
1
)
∑
m
=
−
ℓ
ℓ
Y
ℓ
m
(
θ
,
ϕ
)
Y
ℓ
m
∗
(
θ
′
,
ϕ
′
)
{\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}r^{\prime }}}\sum _{\ell =0}^{\infty }\left({\frac {r}{r^{\prime }}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ^{\prime },\phi ^{\prime })}
。
点电荷的“内部球多极矩”(前述的球多极矩称为外部球多极矩)定义为
I
ℓ
m
=
d
e
f
q
(
r
′
)
ℓ
+
1
Y
ℓ
m
∗
(
θ
′
,
ϕ
′
)
{\displaystyle I_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {q}{\left(r^{\prime }\right)^{\ell +1}}}Y_{\ell m}^{*}(\theta ^{\prime },\phi ^{\prime })}
。
则电势的方程式写为
Φ
(
r
)
=
1
ε
0
∑
ℓ
=
0
∞
∑
m
=
−
ℓ
ℓ
I
ℓ
m
r
ℓ
Y
ℓ
m
(
θ
,
ϕ
)
2
ℓ
+
1
{\displaystyle \Phi (\mathbf {r} )={\frac {1}{\varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }{\frac {I_{\ell m}r^{\ell }Y_{\ell m}(\theta ,\phi )}{2\ell +1}}}
。
^ Griffiths, David J., Introduction to Electrodynamics (3rd ed.), Prentice Hall: pp. 146–148, 1998, ISBN 0-13-805326-X
^ 2.0 2.1 2.2 Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 107–111, 1999, ISBN 978-0-471-30932-1