五 23
![](http://www.qiusir.com/wp-content/gallery/physics/gfleareth.png)
最近在讲电场,翰铮同学分享他看学习心得,说有老师用“臭脚丫”类比电荷,臭味是看不见的那种物质...后来在日语班上课,还特地让打篮球的苏同学拖鞋,如果付出闻点味道的代价就能很好理解“field”倒也合算。
后来我觉得用榴莲代表带电物体可能更好,有人对榴莲的味道喜欢的不得了,而有人则唯恐避之不及,相当于同一点正负电荷受力相反...
尽管我本人不曾吃过榴莲,很多年前还写过关于它的心得:
@qiusir:榴莲似乎可以用来形容一种人,粗硬多刺的外表却裹着柔软丰实的心。不少人被其外表所拒,而能触及其内在的也未必受得那气味,但对敢能享食者来说却多有水果之王之美誉.
@qiusir:有的知识也如榴莲,不畏其表不惧味,能敢食之方知其美味.
![](http://www.qiusir.com/wp-content/gallery/physics/GravitationalPotentialEnergy.png)
电场和引力场中的势能问题是另一个难点。在学习这件事上,也很能体现困难源自美德(深究悉讨)...
1、引力势能
![\int\frac{1}{r^2}dr=-\frac{1}{r}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_a61f9021ea331437b52d96150833d297.gif)
![E_p=-\int F\cdot dx](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_5647cd2ccd017672b31ea29e0c6f8706.gif)
![E_p(r)=-\int_{\infty }^{r}-\frac{GMm}{r^2}dr](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_0d9c6537f5bbbd152b16ece693082f8e.gif)
![E_p(r)=-\frac{GMm}{r}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_88cd87e348336f4662e826b17f12bae5.gif)
2、重力势能
以地面为零势面(通常是以保存力为零的位置为零势面,这里是为了表达式简洁和方便,就如通常以地面为参照物一样)
![E_p=-\frac{GMm}{h+R}-(-\frac{GMm}{R})](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_e4f10a04ca56752f3326f53aeab683ad.gif)
![E_p=\frac{GMm}{R}-\frac{GMm}{R+h}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_fd9f349a4aa610aa50bb0d918e3c3a30.gif)
![E_p=\frac{GMm}{R}(1-\frac{1}{1+\frac{h}{R} })](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_e1bd3527dadb8413f6288071954d8a12.gif)
如果物体在地球表面附近,h
![\ll](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_92dc8479f0b468bea555daf5e02865f3.gif)
![\frac{1}{1+\frac{h}{R} } \approx 1-\frac{h}{R}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_215ceddd75eb90be6471a9dd173acfc8.gif)
![E_p\approx\frac{GMm}{R^2}h](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_be8306cd378595cd491a4117138a10f4.gif)
所以有
![E_p\approx mgh](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_9c2adc9e258976b4ae7784e066d65706.gif)
3、电势能
类比万有引力
![F=\frac{GMm}{r^2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_25136050437dad353c920f18105cbf9e.gif)
![F=\frac{kQq}{r^2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_19d99e924dcb6f39a2d5a3584208b033.gif)
不难得出对于异种电荷间的电势能和引力势能具有相同表达式,
![E_p(r)=-\frac{kQq}{r}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_ca133ea3354bde9dad9aa7853796404c.gif)
如果考虑到电荷的正负问题,以及r的方向问题,库仑定律更一般表达
![\mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} = k \frac{{Qq}}{{{r^2}}} \hat{r}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_532ae1750925a90daa50519c8bc6365b.gif)
![\mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} =-G \frac{{Mm}}{{{r^2}}} \hat{r}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_a0925453b80ec19ccf3adc1667b94f2c.gif)
![](http://www.qiusir.com/wp-content/gallery/physics/electric_fieldlines2-005.png)
![](http://www.qiusir.com/wp-content/gallery/physics/electric_fieldlines2-006.png)
电势能的表达式:
![E_p(r)=\frac{kQq}{r}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_918431defa98751164f95cdf4539891c.gif)
由电势的定义式
![\varphi =\frac{E_p}{q}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_905ee3295205d828a1ea502246738123.gif)
![\varphi =\frac{kQ}{r}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_a7d24f8027077912bd6072cc1185fcc3.gif)
如此容易理解,正电荷的电场电势为正,负电荷的电场电势小于零;负电荷在负电荷电场的电势能大于零...对等量异种电荷中垂线为零势能面,除了用电场力方向理解,用
![\varphi =\frac{kQ}{r}+\frac{-kQ}{r}=0](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_0e6c62175c1790dfe056deb43cc22457.gif)
![](http://www.qiusir.com/wp-content/gallery/physics/landfe2003.png)
套用点电荷电势的方程,可以得出关于两个关于y轴对称的点电荷在平面某点的电势:
![\varphi_n=\frac{k_0Q_1}{\sqrt{(x+l)^2+y^2}}+\frac{k_0Q_2}{\sqrt{(x-l)^2+y^2}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_f7ffff9d1245468e9dca250bd81c7c5e.gif)
电场是保守力场...电场线方程和等势面方程互为共轭调和函数(不懂,安排远卓等同学暴力求解去了)...
![\psi_n=\frac{k_0Q_1(x+l)}{\sqrt{(x+l)^2+y^2}}+\frac{k_0Q_2(x-l)}{\sqrt{(x-l)^2+y^2}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_90f1857e4d939d0c58a1415cf1fe97e4.gif)
根据上年的两个方程,用GeoGebra的曲线“序列”就可以绘制上图了...
另,最近这里[?]发现这张图,按照这里物理量表达习惯略微做了修改(日本教材中,U代表势能,而电势用V表示):
![](http://www.qiusir.com/wp-content/gallery/physics/fomulae.png)
关于电势的高低,用地势的高低类比,而等势面和地里的等高线类同...
![](http://www.qiusir.com/wp-content/gallery/physics/41pns2.png)
![](http://www.qiusir.com/wp-content/gallery/physics/ephipn3.png)
![](http://www.qiusir.com/wp-content/gallery/physics/4ycm.png)
![](http://www.qiusir.com/wp-content/gallery/physics/4fchargesf.png)
On this day..
- what if - 2019
- 莫避春阴上马迟,石干 - 2016
- 黑板:物理公式的头脑风暴 - 2011
- 人的正负属性 - 2007
- 反正我是想您了! - 2006
- 赤峰很近! - 2005
- 人认知的平等性 - 2004
5月 26th, 2023 at 18:25
参考:
https://zhuanlan.zhihu.com/p/150129957
https://zhuanlan.zhihu.com/p/148638744
https://max.book118.com/html/2017/0721/123376408.shtm
5月 28th, 2023 at 19:45
https://alienkevin.github.io/electric-fields/
5月 29th, 2024 at 16:25