# 我的电磁学讲义14：动生电动势和感生电动势

## 电动势

\begin{equation*} \mathcal{E}=\frac{\mathrm dA}{\mathrm dq} \end{equation*}

\begin{equation*} A=\int_{-}^{+}q\vec{K}\cdot\mathrm d\vec{l} \end{equation*}

\begin{equation*} \mathcal{E}=\frac{A}{q}=\int_{-}^{+}\vec{K}\cdot\mathrm d\vec{l} \end{equation*}

\begin{equation*} \mathcal{E}=\int_{-}^{+}\vec{K}\cdot\mathrm d\vec{l} = \int_{-}^{+}\vec{E}\cdot\mathrm d\vec{l}=-U_{+-} \end{equation*}

## 动生电动势

\begin{equation*} U_{ab}=EL=vBL \end{equation*}

\begin{equation*} \mathcal{E}=vBL \end{equation*}

\begin{equation*} \mathcal{E}=\int \left (\vec{v}\times \vec{B}\right ) \cdot \mathrm d\vec{l} \end{equation*}

\begin{equation*} \mathcal{E}=\oint \left (\vec{v}\times \vec{B}\right ) \cdot \mathrm d\vec{l} \end{equation*}

\begin{equation*} \mathcal{E}=\int \left (\vec{v}\times \vec{B}\right ) \cdot \mathrm d\vec{l} = \int vB\mathrm dl=\int_0^L B\omega l\mathrm dl=\frac{1}{2}B\omega L^2 \end{equation*}

\begin{equation*} \mathrm dS=\frac{1}{2}L^2\mathrm d\theta = \frac{1}{2}L^2\omega \mathrm dt \end{equation*}

\begin{equation*} \mathrm d\Phi=B\mathrm dS = \frac{1}{2}BL^2\omega \mathrm dt \end{equation*}

\begin{equation*} \mathcal{E}=\Big|\frac{\mathrm d\Phi}{\mathrm dt}\Big|=\frac{1}{2}BL^2\omega \end{equation*}

\begin{equation*} \mathrm d\mathcal{E}=\left (\vec{v}\times \vec{B}\right ) \cdot \mathrm d\vec{x}=-vB\mathrm dx \end{equation*}

\begin{equation*} \mathcal{E}=\int \mathrm d\mathcal{E}=-\int_d^{d+L} vB\mathrm dx=-\frac{\mu_0 Iv}{2\pi}\int_d^{d+L}\frac{\mathrm dx}{x}=\frac{\mu_0 Iv}{2\pi}\ln\frac{d}{d+L} \end{equation*}

## 感生电动势

\begin{equation*} \mathcal{E}=-\frac{\mathrm d\Phi}{\mathrm dt}=-\mu_0 n S\frac{\mathrm dI}{\mathrm dt} \end{equation*}

\begin{equation*} \mathcal{E}=-\frac{\mathrm d\Phi}{\mathrm dt}=\oint \vec{E}_v\cdot\mathrm d\vec{l}=-\int \frac{\partial \vec{B}}{\partial t}\cdot\mathrm d\vec{S} \end{equation*}

## 参考资料

• Fundamentals of Physics, Extended 10th
• Young and Freedman, University Physics, 13th Ed
• 张三慧《电磁学》
• 赵凯华《电磁学》
posted @ 2015-11-23 00:26  瞿立建  阅读(2199)  评论(0编辑  收藏