题目内容
如图所示,在平面直角坐标系xoy的0≤x≤2L、0≤y≤L区域内存在沿y轴正向的匀强电场,一质量为m,电荷量为q,不计重力,带正电的粒子以速度v0从坐标原点O沿x轴正向射入电场后,恰好从M(2L,L)点离开电场,粒子离开电场后将有机会进入一个磁感应强度大小为
、方向垂直于纸面向外的矩形磁场区域,并最终从x轴上的N(4L,0)点与x轴正向成45°角离开第一象限,题中只有m、v0、q、L为已知量,求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241903287591855.jpg)
(1)匀强电场的电场强度E
(2)粒子在第一象限内运动的时间
(3)如果粒子离开M点后有机会进入的是垂直纸面向里的矩形磁场,磁感应强度大小仍然为
,粒子运动一段时间后仍然能从x轴上的N点与x轴正向成45°角离开第一象限,则该矩形区域的最小面积S
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190328525300.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241903287591855.jpg)
(1)匀强电场的电场强度E
(2)粒子在第一象限内运动的时间
(3)如果粒子离开M点后有机会进入的是垂直纸面向里的矩形磁场,磁感应强度大小仍然为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190328525300.png)
(1)
(2)
(3)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190329758339.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190329040317.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190329430342.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190329758339.png)
试题分析: 解:(1)由带电粒子在电场中做类平抛运动有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241903300383875.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190330241290.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190330522251.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190330803250.png)
由①②③有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190329040317.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190331318274.png)
(2)粒子在电场中运动y方向上有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190331614267.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190331848264.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190332129243.png)
粒子在磁场中圆周运动有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190332441312.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190332815302.png)
由几何关系,粒子离开电场后直接进入磁场四分之一圆周之后离开磁场做匀速直线运动,最后运动到N点.
粒子在磁场中运动时间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190333236278.png)
粒子匀速直线运动时间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190333533295.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190333798279.png)
由②⑤⑥⑦有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190329430342.png)
粒子运动轨迹如图所示,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241903344385574.jpg)
矩形边长为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190334469357.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824190334796370.png)
点评:本题考查了带电粒子在匀强电场中的类平抛运动,在磁场中的匀速圆周运动,对数学的几何能力要求较高,关键画出粒子的轨迹图,结合牛顿第二定律以及向心力等知识进行求解.
![](http://thumb2018.1010pic.com/images/loading.gif)
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