电偶极子辐射
1令原点处的电偶极子为
\begin{equation}
\boldsymbol{\mathbf{p}} (t) = p_0 \cos\left(\omega t\right) \hat{\boldsymbol{\mathbf{z}}}
\end{equation}
使用洛伦兹规范，在 $r \gg \lambda$ 的近似下
\begin{equation}
\varphi(r, \theta, t) = -\frac{p_0\omega}{4\pi\epsilon_0 c} \left(\frac{\cos\theta}{r} \right) \sin[\omega(t - r/c)]
\end{equation}
\begin{equation}
\boldsymbol{\mathbf{A}} (r, \theta, t) = -\frac{\mu_0 p_0 \omega}{4\pi r} \sin[\omega(t - r/c)] \hat{\boldsymbol{\mathbf{z}}}
\end{equation}
进而得
\begin{equation}
\boldsymbol{\mathbf{E}} = -\frac{\mu_0 p_0\omega^2}{4\pi} \left(\frac{\sin\theta}{r} \right) \cos[\omega(t - r/c)] \hat{\boldsymbol{\mathbf{\theta}}}
\end{equation}
\begin{equation}
\boldsymbol{\mathbf{B}} = -\frac{\mu_0 p_0\omega^2}{4\pi c} \left(\frac{\sin\theta}{r} \right) \cos[\omega(t - r/c)] \hat{\boldsymbol{\mathbf{\phi}}}
\end{equation}
其中用到了近似（$d$ 是偶极子的长度，$k = 2\pi/\lambda$）
\begin{equation}
d \ll \frac{1}{k} \ll r
\end{equation}
辐射功率
\begin{equation}
\boldsymbol{\mathbf{s}} ( \boldsymbol{\mathbf{r}} , t) = \frac{1}{\mu_0} \boldsymbol{\mathbf{E}} \boldsymbol\times \boldsymbol{\mathbf{B}} = \frac{\mu_0p_0^2\omega^4}{16\pi^2c} \frac{\sin^2\theta}{r^2} \cos^2[\omega(t - r/c)] \hat{\boldsymbol{\mathbf{r}}}
\end{equation}
时间平均值为
\begin{equation}
\left\langle \boldsymbol{\mathbf{s}} \right\rangle = \frac{1}{\mu_0} \boldsymbol{\mathbf{E}} \boldsymbol\times \boldsymbol{\mathbf{B}}
= \frac{\mu_0p_0^2\omega^4}{32\pi^2c} \frac{\sin^2\theta}{r^2} \hat{\boldsymbol{\mathbf{r}}}
\end{equation}
总辐射功率（时间平均）为
\begin{equation}
\left\langle P \right\rangle = \oint \left\langle \boldsymbol{\mathbf{s}} \right\rangle \boldsymbol\cdot \,\mathrm{d}{ \boldsymbol{\mathbf{a}} } = \frac{\mu_0 p_0^2 \omega^4}{12\pi c}
\end{equation}
1. ^ 参考 [12]．