题目内容
如图所示,矩形区域Ⅰ和Ⅱ内分别存在方向垂直于纸面向外和向里的匀强磁场(AA′、BB′、CC′、DD′为磁场边界,四者相互平行),磁感应强度大小均为B,矩形区域的长度足够长,两磁场宽度及BB′与CC′之间的距离均相同。某种带正电的粒子从AA′上的O1处以大小不同的速度沿与O1A成α=30°角进入磁场(如图所示,不计粒子所受重力),当粒子的速度小于某一值时,粒子在区域Ⅰ内的运动时间均为t0;当速度为v0时,粒子在区域Ⅰ内的运动时间为
。求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241628350903773.png)
⑴粒子的比荷
;
⑵磁场区域Ⅰ和Ⅱ的宽度d;
⑶速度为v0的粒子从O1到DD′所用的时间。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162834934419.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241628350903773.png)
⑴粒子的比荷
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162835480451.png)
⑵磁场区域Ⅰ和Ⅱ的宽度d;
⑶速度为v0的粒子从O1到DD′所用的时间。
(1)
(2)
(3)t=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162835979854.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241628363381223.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162836525890.png)
试题分析:解:(1)若速度小于某一值时粒子不能从BB′ 离开区域Ⅰ,只能从AA′边离开区域Ⅰ。则无论粒子速度大小,在区域Ⅰ中运动的时间相同。轨迹如图所示(图中只画了一个粒子的轨迹)。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241628366818885.jpg)
则粒子在区域Ⅰ内做圆周运动的圆心角为φ1=300o
由Bqv=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162836696581.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162837055655.png)
得:粒子做圆周运动的周期 T =
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162837211682.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241628373981048.png)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162835979854.png)
(2)速度为v0时粒子在区域I内的运动时间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162837773419.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162837913711.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162838069804.png)
得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162838241773.png)
所以其圆心在BB′上,穿出BB′ 时速度方向与BB′ 垂直,其轨迹如图所示,设轨道半径为R
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162838709835.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241628389121049.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241628363381223.png)
(3)区域I、Ⅱ宽度相同,则粒子在区域I、Ⅱ中运动时间均为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162837773419.png)
穿过中间无磁场区域的时间为t′ =
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162839770491.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162839941680.png)
则粒子从O1到DD′所用的时间t=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824162836525890.png)
点评:带电粒子在匀强磁场中的运动是整个高中的重点,也是高考的必考的内容,粒子的运动过程的分析是解题的关键.
![](http://thumb2018.1010pic.com/images/loading.gif)
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