题目内容
如图甲所示的坐标系中,第四限象内存在垂直于纸面向里的有界匀强磁场,
方向的宽度OA=20
cm,
方向无限制,磁感应强度B0=1×10-4T。现有一比荷为q/m=2×1011C/kg的正离子以某一速度从O点射入磁场,α=60°,离子通过磁场后刚好从A点射出。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306424806537.jpg)
小题1:求离子进入磁场B0的速度的大小;
小题2:离子进入磁场B0后,某时刻再加一个同方向的匀强磁场,使离子做完整的圆周运动,求所加磁场磁感应强度的最小值;
小题3:离子进入磁场B0后,再加一个如图乙所示的变化磁场(正方向与B0方向相同,不考虑磁场变化所产生的电场),求离子从O点到A点的总时间。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642386266.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642417337.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642448310.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306424806537.jpg)
小题1:求离子进入磁场B0的速度的大小;
小题2:离子进入磁场B0后,某时刻再加一个同方向的匀强磁场,使离子做完整的圆周运动,求所加磁场磁感应强度的最小值;
小题3:离子进入磁场B0后,再加一个如图乙所示的变化磁场(正方向与B0方向相同,不考虑磁场变化所产生的电场),求离子从O点到A点的总时间。
小题1:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642495548.png)
小题2:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642526615.png)
小题3:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306425731559.png)
(1)如图所示,由几何关系得离子在磁场中运动时的轨道半径
(2分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306426204414.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306426674045.jpg)
离子在磁场中做匀速圆周运动,洛仑兹力提供向前心力
求得:
m/s(2分)
(2)由
知,B越小,r越大。设离子在磁场中最大半径为R,由几何关系得:
m(2分) 由牛顿运动定律得![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642854717.png)
求得
T(2分)则外加磁场
T(1分)
(3)离子在原磁场中运动周期
s(1分)离子在磁场中运动第一次遇到外加磁场的过程中轨迹对应的圆心角
(2分)
此时施加附加磁场时离子在磁场中能做的圆周运动的最大半径为r2由几何关系知:
(m)(1分)
离子在有附加磁场时运动半径为r3 .则
,求得
m
因
,所以离子能做完整的圆周运动(2分)
离子在外加磁场后时,
s 对照外加磁场的规律可知,每隔
s离子在周期性外加磁场时,离子恰可做一次完整的匀速圆周运动,共做三次,最后在A点离开磁场。
离子从O点进入到A点射出的总时间为
s…(3分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642604583.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306426204414.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306426674045.jpg)
离子在磁场中做匀速圆周运动,洛仑兹力提供向前心力
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642698764.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642495548.png)
(2)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642760681.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642792453.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642854717.png)
求得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642870554.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642526615.png)
(3)离子在原磁场中运动周期
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306429321064.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130642963958.png)
此时施加附加磁场时离子在磁场中能做的圆周运动的最大半径为r2由几何关系知:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306429791143.png)
离子在有附加磁场时运动半径为r3 .则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130643026843.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130643057764.png)
因
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130643104447.png)
离子在外加磁场后时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306431351168.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824130643182636.png)
离子从O点进入到A点射出的总时间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241306425731559.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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