# 鏡像法

## 唯一定理

${\displaystyle \nabla ^{2}\phi =f}$

${\displaystyle \phi =g}$

${\displaystyle \phi (x,\,y,\,z)}$ 是唯一的解答函數。

1. 给出了整个边界的势函数；
2. 给出整个边界的势函数的法向导函数；
3. 给出整个边界部分场的势函数和其他部分的势函数的法向导函数；

## 範例

### 點電荷與無限平面導體

${\displaystyle V(\rho ,\,z)={\frac {1}{4\pi \epsilon _{0}}}\left({\frac {q}{\sqrt {\rho ^{2}+(z-a)^{2}}}}+{\frac {-q}{\sqrt {\rho ^{2}+(z+a)^{2}}}}\right)}$

${\displaystyle \sigma =-\epsilon _{0}{\frac {\partial V}{\partial z}}{\Bigg |}_{z=0}=-\ {\frac {qa}{2\pi (\rho ^{2}+a^{2})^{3/2}}}}$

 ${\displaystyle Q_{t}}$ ${\displaystyle =\int _{0}^{2\pi }\int _{0}^{\infty }\sigma \left(\rho \right)\,\rho \,d\rho \,d\theta }$ ${\displaystyle =-\ {\frac {qa}{2\pi }}\int _{0}^{2\pi }d\theta \int _{0}^{\infty }{\frac {\rho \,d\rho }{\left(\rho ^{2}+a^{2}\right)^{3/2}}}}$ ${\displaystyle =-q}$ 。

### 點電荷與圓球殼導體

${\displaystyle 4\pi \epsilon _{0}V(\mathbf {r} )={\frac {q}{|\mathbf {r} _{1}|}}+{\frac {(-qR/p)}{|\mathbf {r} _{2}|}}={\frac {q}{\sqrt {r^{2}+p^{2}-2\mathbf {r} \cdot \mathbf {p} }}}+{\frac {(-qR/p)}{\sqrt {r^{2}+{\frac {R^{4}}{p^{2}}}-{\frac {2R^{2}}{p^{2}}}\mathbf {r} \cdot \mathbf {p} }}}}$

${\displaystyle V(\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\left[{\frac {q}{\sqrt {r^{2}+p^{2}-2\mathbf {r} \cdot \mathbf {p} }}}-{\frac {q}{\sqrt {{\frac {r^{2}p^{2}}{R^{2}}}+R^{2}-2\mathbf {r} \cdot \mathbf {p} }}}\right]}$

${\displaystyle \sigma (\theta )=\epsilon _{0}{\frac {\partial V}{\partial r}}{\Bigg |}_{r=R}={\frac {-q(R^{2}-p^{2})}{4\pi R(R^{2}+p^{2}-2pR\cos \theta )^{3/2}}}}$

${\displaystyle Q_{t}=\int _{0}^{\pi }d\theta \int _{0}^{2\pi }d\phi \,\,\sigma (\theta )R^{2}\sin \theta =-q}$

${\displaystyle \mathbf {E} =\mathbf {0} }$

${\displaystyle \sigma =q/4\pi R^{2}}$

${\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0}r^{2}}}{\hat {\mathbf {r} }}}$

### 電偶極子與圓球殼導體

${\displaystyle q'={\frac {R\mathbf {p} \cdot \mathbf {M} }{p^{3}}}}$

${\displaystyle \mathbf {M} '=R^{3}\left[-{\frac {\mathbf {M} }{p^{3}}}+{\frac {2\mathbf {p} (\mathbf {p} \cdot \mathbf {M} )}{p^{5}}}\right]}$

## 反演法

${\displaystyle \Phi '(r,\,\theta ,\,\phi )={\frac {R}{r}}\Phi \left({\frac {R^{2}}{r}},\,\theta ,\,\phi \right)}$

## 參考文獻

1. ^ David J. Griffiths. Introduction to Electrodynamics (4th Ed.). Glenview, IL: Pearson. 2013: 121. ISBN 0-321-85656-2.
2. ^ Dick, B. G., Images and the Point Charge-Capacitor Problem, American Journal of Physics, 1973, 41 (11): pp. 1289–1290
3. ^ Tikhonov, A. N.; Samarskii, A. A. Equations of Mathematical Physics. New York: Dover Publications. 1963. ISBN 0-486-66422-8.