题目内容
如图15-5-25所示,一半径为R的绝缘圆筒中有沿轴线方向的匀强磁场,磁感应强度为B,一质量为m、带电荷量为q的正粒子(不计重力)以速度为v从筒壁的A孔沿半径方向进入筒内,设粒子和筒壁的碰撞无电荷量和能量的损失,那么要使粒子与筒壁连续碰撞,绕筒壁一周时恰好又从A孔射出,问:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210003631742.jpg)
图15-5-25
(1)磁感应强度B的大小必须满足什么条件?
(2)粒子在筒中运动的时间为多少?![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210003792930.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210003631742.jpg)
图15-5-25
(1)磁感应强度B的大小必须满足什么条件?
(2)粒子在筒中运动的时间为多少?
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210003792930.jpg)
(1)B=
(n=3,4,5,…)
(2)
(n=3,4,5,…)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000394588.gif)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000488679.gif)
(1)粒子射入圆筒后受洛伦兹力作用而偏转,设第一次与B点碰撞,碰后速度方向又指向O点,假设粒子与筒壁撞n-1次,运动轨迹是n段相等的圆弧,再从A孔射出.
设第一段圆弧的圆心为O′,半径为r,如图所示(n=3的情景),
则θ=2π/2n=π/n
由几何关系有:r=Rtan![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000504338.gif)
又r=mv/Bq,故
B=
(n=3,4,5,…).
(2)粒子运动的周期T=![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000519805.gif)
所对的圆心角φ=2(
-θ)=![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000566418.gif)
粒子从A→B用的时间
t′=![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000582845.gif)
粒子运动的总时间
t=nt′=
(n=3,4,5,…).
设第一段圆弧的圆心为O′,半径为r,如图所示(n=3的情景),
则θ=2π/2n=π/n
由几何关系有:r=Rtan
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000504338.gif)
又r=mv/Bq,故
B=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000394588.gif)
(2)粒子运动的周期T=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000519805.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210005351175.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000550233.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000566418.gif)
粒子从A→B用的时间
t′=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000582845.gif)
粒子运动的总时间
t=nt′=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121000488679.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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