题目内容
如图,ABCD是边长为
的正方形。质量为
、电荷量为
的电子以大小为
的初速度沿纸面垂直于BC边射入正方形区域。在正方形内适当区域中有匀强磁场。电子从BC边上的任意点入射,都只能从A点射出磁场。不计重力,求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241136586141858.jpg)
(1)次匀强磁场区域中磁感应强度的方向和大小;
(2)此匀强磁场区域的最小面积。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658504192.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658536204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658551185.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658598204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241136586141858.jpg)
(1)次匀强磁场区域中磁感应强度的方向和大小;
(2)此匀强磁场区域的最小面积。
见解析
(1)设匀强磁场的磁感应强度的大小为B。令圆弧
是自C点垂直于BC入射的电子在磁场中的运行轨道。电子所受到的磁场的作用力
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658645317.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241136586606010.jpg)
应指向圆弧的圆心,因而磁场的方向应垂直于纸面向外。圆弧
的圆心在CB边或其延长线上。依题意,圆心在A、C连线的中垂线上,故B 点即为圆心,圆半径为
按照牛顿定律有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658707479.gif)
联立①②式得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658707484.gif)
(2)由(1)中决定的磁感应强度的方向和大小,可知自
点垂直于
入射电子在A点沿DA方向射出,且自BC边上其它点垂直于入射的电子的运动轨道只能在BAEC区域中。因而,圆弧
是所求的最小磁场区域的一个边界。
为了决定该磁场区域的另一边界,我们来考察射中A点的电子的速度方向与BA的延长线交角为
(不妨设
)的情形。该电子的运动轨迹
如右图所示。
图中,圆
的圆心为O,pq垂直于BC边,由③式知,圆弧
的半径仍为
,在D为原点、DC为x轴,AD为
轴的坐标系中,P点的坐标
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241136589101031.gif)
这意味着,在范围
内,p点形成以D为圆心、
为半径的四分之一圆周
,它是电子做直线运动和圆周运动的分界线,构成所求磁场区域的另一边界。
因此,所求的最小匀强磁场区域时分别以
和
为圆心、
为半径的两个四分之一圆周
和
所围成的,其面积为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113659050808.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658629398.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658645317.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241136586606010.jpg)
应指向圆弧的圆心,因而磁场的方向应垂直于纸面向外。圆弧
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658629398.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658504192.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658707479.gif)
联立①②式得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658707484.gif)
(2)由(1)中决定的磁感应强度的方向和大小,可知自
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658723205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658738241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658629398.gif)
为了决定该磁场区域的另一边界,我们来考察射中A点的电子的速度方向与BA的延长线交角为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658770200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658801481.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658801371.gif)
图中,圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658816251.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658816251.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658504192.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658894193.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658910381.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241136589101031.gif)
这意味着,在范围
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658926481.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658504192.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658957390.gif)
因此,所求的最小匀强磁场区域时分别以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658972206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658988210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658504192.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658629398.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113658957390.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824113659050808.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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