题目内容
如图—26所示,在xoy坐标系中分布着四个有界场区,在第三象限的AC左下方存在垂直纸面向里的匀强磁场B1=0.5T,AC是直线y=-x—0.425(单位:m)在第三象限的部分,另一沿y轴负向的匀强电场左下边界也为线段AC的一部分,右边界为y轴,上边界是满足
(单位:m)的抛物线的一部分,电场强度E=2.5N/C。在第二象限有一半径为r=0.1m的圆形磁场区域,磁感应强度B2=1T,方向垂直纸面向里,该区域同时与x轴、y轴相切,切点分别为D、F,在第一象限的整个空间存在垂直纸面向外的匀强磁场,磁感应强度B3=1T,另有一厚度不计的挡板PQ垂直纸面放置,其下端坐标P(0.1m,0.1m),上端Q在y轴上,且∠PQF=30°现有大量m=1×10-6kg,q=-2×10-4C的粒子(重力不计)同时从A点沿x轴负向以v0射入,且v0取0<v0<20m/s之=4-4d
之间的一系列连续值,并假设任一速度的粒子数占入射粒子总数的比例相
同。
(1)求所有粒子从第三象限穿越x轴时的速度;
(2)设从A点发出的粒子总数为N,求最终打在挡板PQ右侧的粒子数N'.![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241229432206325.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943188553.gif)
之间的一系列连续值,并假设任一速度的粒子数占入射粒子总数的比例相
同。
(1)求所有粒子从第三象限穿越x轴时的速度;
(2)设从A点发出的粒子总数为N,求最终打在挡板PQ右侧的粒子数N'.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241229432206325.jpg)
(1)设某速度为
的粒子从A点入射后到达AC上的G点,因
与AC成450角,其对应圆心角为900,即恰好经过四分之一圆周,故到达G点时速度仍为
,方向沿Y轴正向. ----------------------------------------------------------------------(2分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082412294328214649.jpg)
粒子在电场中沿Y轴正向加速运动,设G点坐标为G(x,y),刚好穿出电场时坐标为(x,
),粒子穿出电场时速度为
,在电场中运动的过程中,由动能定理得:
------------------------------------------ (2分)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943391467.gif)
又
-----------------------------------------------(2分)
代入数据解得
,可见粒子穿出电场时速度大小与x无关。---------(3分)
因
,由![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943454525.gif)
代入数据得:
----------------------------------------------------------------------(1分)
由数学知识可知,k点坐标为k(-0.2m,-0.225m),故从A点射出的所有粒子均从AK之间以20m/s的速度沿Y轴正向射出电场,在到达X轴之前粒子作匀速直线运动,故所有粒子从第三象限穿越X轴时的速度大小均为20m/s的速度沿Y轴正向。---------------------------------------------------(1分)
(2)因为
,故离子束射入
时,离子束宽度刚好与2
相等,设粒子在
中运动轨道半径为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943547213.gif)
,解得
=
---------------------------------------------------(1分)
考察从任一点J进入
的粒子,设从H穿出
磁场,四边形
为菱形,又因为
水平,而
,故H应与F重合,即所有粒子经过
后全部从F点离开
进入
磁场。对
趋于
的粒子,圆心角∠
,故射入
时速度趋于Y轴负向;对
趋于0的粒子,圆心角∠
,故射入
时速度趋于Y轴正向,即进入
的所有粒子速度与Y轴正向夹角在0~1800之间。-- ----------------------(4分)
由于
=
,所以
,由几何关系知:
无限靠近Y轴负向射入的粒子轨迹如图所示,最终打在PQ板的右侧
; ---(1分)
与Y轴负向成600角的粒子刚好经过P点到达Q点;-------- ----------- -(1分)
因此与Y轴正向在0~1200之间从F点射出的粒子要么打在PQ板的左侧,要么打不到板上而穿越Y轴离开
。由于是“大量”粒子,忽略打在P或Q的临界情况,所以最终打在挡板PQ右侧的粒子数
------------------------------ -----------------------------------------(3分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943235195.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943235195.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943235195.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082412294328214649.jpg)
粒子在电场中沿Y轴正向加速运动,设G点坐标为G(x,y),刚好穿出电场时坐标为(x,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943298197.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943360200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943360770.gif)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943376558.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943391467.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943407629.gif)
代入数据解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943438454.gif)
因
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943454442.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943454525.gif)
代入数据得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943469436.gif)
由数学知识可知,k点坐标为k(-0.2m,-0.225m),故从A点射出的所有粒子均从AK之间以20m/s的速度沿Y轴正向射出电场,在到达X轴之前粒子作匀速直线运动,故所有粒子从第三象限穿越X轴时的速度大小均为20m/s的速度沿Y轴正向。---------------------------------------------------(1分)
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943485408.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943500215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943516185.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943500215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943547213.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943563595.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943547213.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943485408.gif)
考察从任一点J进入
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943500215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943500215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943672271.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943688240.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943703330.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943500215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943500215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943750215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943750199.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943766383.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943781470.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943797220.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943812204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943828428.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943750215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943750215.gif)
由于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943750215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943500215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943890293.gif)
无限靠近Y轴负向射入的粒子轨迹如图所示,最终打在PQ板的右侧
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943906215.gif)
与Y轴负向成600角的粒子刚好经过P点到达Q点;-------- ----------- -(1分)
因此与Y轴正向在0~1200之间从F点射出的粒子要么打在PQ板的左侧,要么打不到板上而穿越Y轴离开
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943750215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824122943953593.gif)
略
![](http://thumb2018.1010pic.com/images/loading.gif)
练习册系列答案
相关题目