密立根通过观测油滴的运动规律证明了电荷的量子性,因此获得了1923年的诺贝尔奖。图13是密立根油滴实验的原理示意图,两个水平放置、相距为d的足够大金属极板,上极板中央有一小孔。通过小孔喷入一些小油滴,由于碰撞或摩擦,部分油滴带上了电荷。有两个质量均为
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3Em%3C%2Fmi%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
、位于同一竖直线上的球形小油滴A和B,在时间t内都匀速下落了距离
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3Eh%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
。此时给两极板加上电压U(上极板接正极),A继续以原速度下落,B经过一段时间后向上匀速运动。B在匀速运动时间t内上升了距离
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3Eh%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3Eh%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo%3E%E2%89%A0%3C%2Fmo%3E%3Cmsub%3E%3Cmi%3Eh%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,随后与A合并,形成一个球形新油滴,继续在两极板间运动直至匀速。已知球形油滴受到的空气阻力大小为
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Ef%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmsup%3E%3Cmi%3Em%3C%2Fmi%3E%3Cmrow%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmi%3Ev%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,其中k为比例系数,m为油滴质量,v为油滴运动速率。不计空气浮力,重力加速度为g。求:
![](http://tikupic.21cnjy.com/2022/06/30/1a/f7/1af7acb23cb583c15d2d8fff0a0dc7bc.png)