# 时变场的位函数

\begin{align*} \newcommand{\dif}{\mathop{}\!\mathrm{d}}\\ \newcommand{\p}{\partial}\\ \newcommand{\bd}{\boldsymbol}\\ \newcommand{\E}{\mathscr{E}}\\ \newcommand{\db}[1]{\dot{\boldsymbol{#1}}} \end{align*}

$\nabla\times\bd{E}=-\frac{\p}{\p t} \nabla\times \bd{A}=-\nabla\times \frac{\p A}{\p t}\\ \therefore \nabla\times(\bd{E}+\frac{\p \bd{A}}{\p t})=0$

$\bd{E}+\frac{\p A}{\p t}=-\nabla\phi\\ \bd{E}=-\nabla \phi-\frac{\p \bd{A}}{\p t}$

$\nabla\cdot \bd{E}=\nabla\cdot (-\nabla\phi-\frac{\p \bd{A}}{\p t})=\frac{\rho}{\varepsilon}$

$\nabla\times\bd{H}=\frac{1}{\mu} \nabla\times \nabla\times \bd{A}=\bd{J}+\varepsilon \frac{\p}{\p t} (-\nabla\phi-\frac{\p \bd{A}}{\p t})$

$\begin{cases} \nabla^2 \phi + \dfrac{\p}{\p t} \nabla \cdot \bd{A}=-\dfrac{\rho}{\varepsilon}\\ \nabla\nabla\cdot \bd{A}-\nabla^2 \bd{A}=\mu \bd{J}-\mu\varepsilon \nabla\dfrac{\p \phi}{\p t}-\mu\varepsilon \dfrac{\p^2 \bd{A}}{\p t^2} \end{cases}$

$\nabla\cdot \bd{A}+\mu\varepsilon \frac{\p \phi}{\p t}=0$

$\nabla^2 \phi + \mu\varepsilon\frac{\p^2 \phi}{\p t^2}=-\frac{\rho}{\varepsilon}$

$\nabla^2 \bd{A}-\mu\varepsilon \frac{\p^2 \bd{A}}{\p t^2}=-\mu \bd{J}$

$\phi ( \bd{r},t)=\frac{1}{r} [ f(t-\frac{r}{v})+g(t+\frac{r}{v}) ]$

$\phi(\bd{r},t)=\frac{1}{4\pi\varepsilon} \int_\tau \frac{\rho(\bd{r}', t-\frac{R}{v})}{R} \dif \tau'$

$\bd{A}(\bd{r},t)=\frac{\mu}{4\pi} \int_\tau \frac{\bd{J}(\bd{r}', t-\frac{R}{v})}{R} \dif \tau'$

# 位函数的复数表示

$\begin{cases} \bd{E}=-\nabla \phi - \frac{\p \bd{A}}{\p t}\\ \bd{B}=\nabla \times \bd{A}\\ \nabla\cdot\bd{A}=-\mu\varepsilon \frac{\p \phi}{\p t} \end{cases}$

$\begin{cases} \db{E}=-\nabla\dot{\phi}-j\omega \db{A}\\ \db{B}=\nabla\times\db{A}\\ \nabla\times\db{A}=-j\omega\mu\varepsilon \dot{\phi} \end{cases}$

$\nabla^2 \db{A}+k^2 \db{A}=-\mu\db{J}\\ \nabla^2 \db{\phi}+k^2 \db{\phi}=-\frac{\dot{\rho}}{\varepsilon}$

$\db{A}=\frac{\mu}{4\pi} \int_\tau \frac{\db{J}e^{-jkR}}{R} \dif \tau'\\ \db{\phi}=\frac{1}{4\pi\varepsilon} \int_\tau \frac{\dot{\rho}e^{-jkR}}{R} \dif \tau'$