查看“︁學院物理/連續電荷分布與高斯定律”︁的源代码

==高斯定律==
<math>\Phi=\vec E\cdot\vec A=EA\cos\theta=\int\vec E\cdot d\vec A=\frac{kq}{r^2}\cdot4\pi r^2=\frac{\quad q\;\;4\pi r^2}{4\pi\epsilon\,r^2\quad\;\;}=\frac q\epsilon</math>
<div>封閉面的總電通量 = <div style=display:inline-block;text-align:center;vertical-align:middle><span style="border-bottom:1px solid;">封閉面內的淨電荷</span><br><math>\epsilon</math></div></div>
==例題==
===一===
<div style=float:right>
<div style=height:100px;position:relative;width:200px>
<div style="border:solid;border-width:0 0 4px 4px;height:50px;left:30px;position:absolute;top:0;width:20px;"></div>
<div style="background-color:#eee;border:4px solid;height:6px;left:50px;position:absolute;top:44px;width:60px;"></div>
<div style=left:0;position:absolute;top:38px><math>\vec E\gets</math></div>
<div style=left:30px;position:absolute;top:14px><math>\begin{align}\gets x\to\\\\a\qquad l\end{align}</math></div>
</div>
圖23.2
</div>
<math>\begin{align}E&=\int dE=k\int\frac{dq}{r^2}=k\lambda\int_a^{l+a}\frac{dx}{x^2}=k\lambda\left[-\frac 1 x\right]^{l+a}_a\\&\qquad\qquad\qquad dq=\lambda dx\\&=k\frac Q l\left(-\frac1{l+a}+\frac1 a\right)=\frac{kQ}{a\left(l+a\right)}\end{align}</math>

===二===
<div style=float:right;position:relative;width:200px>
<div style="border:4px solid;border-radius:100px;height:100px;width:100px;"></div>
<div style="border:solid;border-width:0 0 2px 2px;height:54px;left:52px;position:absolute;top:0;width:140px;"></div>
<div style=left:40px;position:absolute;top:20px><math>a</math></div>
<div style=left:120px;position:absolute;top:52px>'''''x'''''</div>
圖23.3


</div>
<math>E_x=\int dE\,\cos\theta=\int\frac{kdq}{r^2}\frac x r=\int\frac{kxdq}{r^3}=\frac{kxQ}{\sqrt{x^2+a^2}^3}</math>
===三===
<div style=clear:right;float:right;position:relative;width:200px><b><i>
<div style="background-color:#eee;border:4px solid;border-radius:100px;height:100px;width:100px;"></div>
<div style="border:2px dashed;border-radius:60px;height:60px;left:22px;position:absolute;top:22px;width:60px;"></div>
<div style="border:solid;border-width:0 0 2px 2px;height:53px;left:53px;position:absolute;top:0;width:120px;"></div>
<div style="border:dashed;border-width:0 0 2px;height:30px;left:23px;position:absolute;top:23px;width:30px;"></div>
<div style=left:57px;position:absolute;top:2px>R</div>
<div style=left:120px;position:absolute;top:54px>x</div>
<div style=left:40px;position:absolute;top:54px>r</div></i></b>
圖23.4</div>
<math>\begin{align}E_x=\int\frac{kx\sigma rdrd\theta}{\sqrt{x^2+r^2}^3}=kx2\pi\sigma \int_0^R\frac{rdr}{\sqrt{x^2+r^2}^3}=kx\pi\sigma\int_0^R\frac{d\left(x^2+r^2\right)}{\sqrt{x^2+r^2}^3}\\dq=\sigma dA=\sigma rdrd\theta=2\pi\sigma rdr\qquad\qquad\qquad\qquad\qquad\qquad\quad\\=kx\pi\sigma\int_0^R\left(x^2+r^2\right)^{-\frac3 2}d\left(x^2+r^2\right)=-2kx\pi\sigma\left[\frac1\sqrt{x^2+r^2}\right]_0^R\\=-2kx\pi\sigma\left(\frac1\sqrt{x^2+R^2}-\frac1\sqrt{x^2}\right)=2kx\pi\sigma\left(\frac1x-\frac1\sqrt{x^2+R^2}\right)\\=2k\pi\sigma\left(1-\frac x\sqrt{x^2+R^2}\right)\end{align}</math>

===四===
<div style=float:right;position:relative;width:200px>
<div style="background-color:#eee;border:20px solid;border-color:#eaeaea #ccc;height:80px;width:80px;"></div>
<div style=left:44px;position:absolute;top:26px><math>\begin{align}\to\\\to&\vec E\\\to\end{align}</math></div>
圖23.9


</div>
<math>\Phi_E=0</math>
===六===
<div style=clear:right;float:right;position:relative;width:200px>
<div style="background-image:radial-gradient(at 20% 20%,#fff 4%,#eee 10% 40%,#ddd);border:4px solid;border-radius:100px;height:100px;width:100px;"></div>
<div style="border:1px dashed;border-radius:50%;height:60px;left:23px;position:absolute;top:23px;width:60px;"></div>
<div style="border:dashed;border-width:0 0 0 1px;height:30px;left:54px;position:absolute;top:54px;width:30px;"></div>
<div style=left:2px;position:absolute;top:42px><math>\gets a\to\longleftarrow r\longrightarrow</math></div>
<div style=left:44px;position:absolute;top:58px>'''''r'''''</div>
圖23.14</div>
<math>\begin{align}E\left(r>a\right)=\frac{kQ}{r^2}\\\Phi_E=E\oint dA=\frac Q\epsilon=E4\pi r^2\\E\left(r<a\right)=\frac{kQr}{a^3}\\E4\pi r^2=\frac{Q_{in}}\epsilon=Q\frac{r^3}{a^3}\big/\epsilon=\frac{Qr^3}{\epsilon a^3}\\E=\frac{kQ_{in}}{r^2}\qquad Q_{in}=Q\frac{\frac4 3\pi r^3}{\frac4 3\pi a^3}=Q\frac{r^3}{a^3}\end{align}</math>

===七===
<div style=float:right;position:relative;width:200px>
<div style="border:1px dashed;border-radius:100px;height:100px;width:100px;"></div>
<div style="background:#eee;border: 2px solid;border-radius:10px;box-sizing:border-box;height:20px;left:41px;line-height:1;position:absolute;top:41px;width:20px;">&nbsp;+</div>
<div style=margin-top:10px;position:relative>
<div style="background-image:linear-gradient(#fafafa,#eee 40% 80%,#ddd);border:2px dashed;height:56px;margin-left:12px;width:74px;"></div>
<div style="border:solid;border-width:0 0 4px;height:28px;left:0;position:absolute;top:0;width:102px;"></div>
</div>
<div style=left:12px;position:absolute;top:16px><math>\begin{matrix}\nwarrow\quad\;\nearrow\\\longleftarrow\quad\longrightarrow\\\swarrow\vec E\searrow\\\\\longleftarrow l\longrightarrow\\\qquad\quad\;\,r\end{matrix}</math></div>
圖23.16</div>
<math>E2\pi rl=\frac{\lambda l}\epsilon\qquad E=\frac\lambda{2\pi\epsilon r}=\frac{2k\lambda}r</math><div style=clear:both></div>
===八===
<div style=float:right;position:relative;width:200px>
<div style="background-color:#efefef;border:4px solid;box-sizing:border-box;height:100px;letter-spacing:10px;overflow:hidden;padding-left:6px;white-space:nowrap;width:100px;">
+++++

+++++
+++++</div>
圖23.17</div>
<math>2EA=\frac{\sigma A}\epsilon\qquad E=\frac\sigma{2\epsilon}</math>