题目内容
(16分)如图甲所示,两水平放置的平行金属板A、B的板长
,板间距离d=0.10m,在金属板右侧有一范围足够大,方向垂直于纸面向里的匀强磁场,磁感应强度B=1.0×10-2T,其左边界为y轴.在t=0时刻,两金属板间加如图乙所示的正弦交变电压.现从t=0开始,位于极板左侧的粒子源沿x轴向右以1000个/秒的数量连续均匀发射带正电的粒子,粒子均以
的初速度沿x轴进入电场,经电场后部分粒子射入磁场.已知带电粒子的比荷
,粒子通过电场区域的极短时间内,极板间的电压可以看作不变,不计粒子重力,不考虑极板边缘及粒子间相互影响.试求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241614454339890.png)
(1)t=0时刻进入的粒子,经边界y轴射入磁场和射出磁场时两点间的距离;
(2)每秒钟有多少个粒子进入磁场;
(3)何时刻由粒子源进入的带电粒子在磁场中运动时间最长,求最长时间tm(π≈3).
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161444700583.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161444934853.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161445215973.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241614454339890.png)
(1)t=0时刻进入的粒子,经边界y轴射入磁场和射出磁场时两点间的距离;
(2)每秒钟有多少个粒子进入磁场;
(3)何时刻由粒子源进入的带电粒子在磁场中运动时间最长,求最长时间tm(π≈3).
(1)1.38m(2)每秒钟有500个粒子进入磁场(3)tm="3.2" ×10-5s
试题分析:(1)t=0时,UAB=0,带电粒子在极板间不偏转,水平射入磁场,
由:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161445714762.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161446120718.png)
射入和射出磁场时,两点间的距离为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241614463381097.png)
(2)假设两极板电压为u时,由O点射入的粒子刚好从板的边缘进入磁场,则有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161446588723.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161446853592.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161446869584.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161447118475.png)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241614474461227.png)
由乙图可知,在每个周期内,只有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161447727551.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161448085647.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161448351551.png)
(3)只有当A板电势高于B板电势,且沿B板右侧边缘进入磁场的粒子,由于在电场中的偏转角最大,使得其在磁场中做圆周运动对应的圆心角最大,故在磁场中运动的时间最长,如图所示.即:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161448631730.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161448897566.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241614490377756.png)
或:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161449271743.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161449536603.png)
这些时刻入射的粒子在磁场中运动的时间最长.
由带电粒子在电场中做类平抛运动的特点可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161449786990.png)
设粒子在磁场中做匀速圆周运动的周期T,所求的最长 时间为tm,则有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161450051773.png)
由几何关系得粒子在磁场中运动的圆心角为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161450379426.png)
则:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824161450722577.png)
点评:本题难度较大,如果粒子以垂直磁场边界射入,必垂直磁场边界射出,处理带电粒子在匀强磁场中的偏转问题,首先要先画出大概的运动轨迹,先找圆心后求半径,处理边界磁场问题时,应巧妙应用对称性是解题的技巧
![](http://thumb2018.1010pic.com/images/loading.gif)
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