问题 计算题

如图所示,abcd是长为2L、宽为L的长方形区域,该区域内存在垂直于纸面向里匀强磁场,磁感应强度的大小为B。在ab边中点M有一粒子源,该粒子源能不断地向区域内发出质量为、电量大小为q的带负电的粒子,粒子速度的大小恒定,沿纸面指向各个方向,不计粒子重力。其中垂直于ab边入射的粒子恰能从ad边中点N射出磁场。求:

(1)粒子入射的速度大小;

(2)bc边有粒子射出的宽度。

答案

电荷在磁场里做匀速圆周运动,洛伦兹力充当向心力,从M点垂直进入磁场的粒子到达N点,运动轨迹如图Ⅰ所示,则:

(1)…①

由几何关系: …②

 …③ 所以:…④

(2)由于粒子的速度大小相同,所以粒子在磁场里做圆周运动的半径均为

粒子打到bc边上的最左端是粒子速度竖直向上时,如轨迹图Ⅱ,设轨迹与bc边的交点与b点的水平距离为x1

由几何关系: …⑤   得:x1= 

由图可知:x1=…⑥

粒子打到bc边上的最右端是粒子的运动轨迹与bc相切时,如轨迹图Ⅲ,设轨迹与bc边的交点与b点的水平距离为x2

由几何关系: …⑦ 得:x2=L…⑧

所以: …⑨ =…⑩

①②④⑤⑥⑦⑧⑩各2分 ③⑨各1分  共计18分

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