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@@ -7,7 +7,6 @@ - [1.3 叠加原理](#13-叠加原理) - [1.4 高斯定理](#14-高斯定理) - [1.5 电势能](#15-电势能) - - [](#) - [2 电势](#2-电势) - [2.1 环路定理](#21-环路定理) - [2.2 电势](#22-电势) @@ -185,10 +184,48 @@ $$ W = \iiint \frac{\varepsilon_0 E^2}{2} dx\ dy\ dz $$ -## +**例7简化版。** 求一个均匀带电球壳所具有的电势能,半径为 $R$,总带电量为 $Q$ + +先求电势做法: + +$$ +\phi = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{r} \\ +W = \phi q = \frac{1}{2} \cdot \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q^2}{r} = \frac{Q^2}{8 \pi\varepsilon_0 r} +$$ + +直接微分做法: + +$$ +W = \int_r^\infty \frac{1}{2} \varepsilon_0 (\frac{Q}{4\pi \varepsilon_0 x^2})^2 4\pi x^2\ dx = \frac{Q^2}{8 \pi\varepsilon_0 r} +$$ + +**例7.** 求一个均匀带电球体所具有的电势能,半径为 $R$,总带电量为 $Q$ + +先将这个求按层微分,将每一层移动到无穷远所做的功即为电势能。 + +$$ +\int_0^R\frac{Q4\pi x^2\ dx}{\frac{4}{3}\pi r^3}\cdot\frac{Qx^3}{R^3}\cdot\frac{1}{4\pi\varepsilon_0}\cdot\frac{1}{x} = \frac{3}{5}\cdot\frac{1}{4\pi\varepsilon_0}\cdot\frac{Q^2}{R} +$$ + +**例8.** 相距为 $d$ 的两个点电荷,在空间中某一点产生的电场强度分别为 $E_1$ 和 $E_2$,则总电场强度可以写成 $\int \frac{\varepsilon_0}{2}(E_1+E_2)^2$,展开后为 $\int \frac{\varepsilon_0}{2}E_1^2 + \int \frac{\varepsilon_0}{2}E_2^2 + \int \frac{\varepsilon_0}{2}E_1\cdot E_2$,通过计算说明三项分别是点电荷 $1$ 的自能[^6],点电荷 $2$ 的自能,$1$ 与 $2$ 的互能,你可以做合理的假设。 + +![例题8图](image-1.png) + +两个自能:易证,略去。 + +即为证 + +$$ +\iiint \frac{1}{2}\varepsilon_0 E_1E_2\ dV = \frac{1}{4\pi\varepsilon_0} \frac{q_1q_2}{b} \\ +\varepsilon_0 E_1E_2 = \varepsilon_0\frac{1}{4\pi\varepsilon_0}\cdot \frac{q_1}{r^2} \cdot \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_2}{r^2+d^2-2rdcos\theta} \cdot \frac{2r^2-2rdcos\theta}{2r\sqrt{r^2+d^2-2rdcos\theta}} +$$ + +化简三重积分即可得左边等于右边 ## 2 电势 +我们可以将其与重力势能进行类比,可以定义一个物体在某个位置的重力势为 $\frac{W_G}{m} = gh$,则重力势能即为 $W_G = mgh$ + ### 2.1 环路定理 ### 2.2 电势 @@ -214,3 +251,7 @@ $$ [^3]: 需要特别注意,这个转换公式必须谨慎使用,因为公式两边的单位并不具有相同的量纲 [^4]: 场的定义为某种物理量在**空间**中的分布情况 + +[^5]: 保守力的定义为一个力满足其功只取决于初始和终了位置,而与物体路径无关 + +[^6]: 自能指物体中电荷之间的相互作用产生的电势能,故点电荷没有自能