题目内容
(16分)在如图所示xoy坐标系第一象限的三角形区域(坐标如图中所标注)内有垂直于纸面向外的匀强磁场,在x 轴下方有沿+y方向的匀强电场,电场强度为E。将一个质量为m、带电量为+q的粒子(重力不计)从P(-a,0)点由静止释放。由于x轴上存在一种特殊物质,使粒子每经过一次x轴后速度大小变为穿过前的
倍。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082413411096571429.png)
(1)欲使粒子能够再次经过x轴,磁场的磁感应强度B0最小是多少?
(2)在磁感应强度等于第(1)问中B0的情况下,求粒子在磁场中的运动时间;
(3)若磁场的磁感应强度变为第(1)问中B0的2倍,求粒子运动的总路程。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134110918413.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082413411096571429.png)
(1)欲使粒子能够再次经过x轴,磁场的磁感应强度B0最小是多少?
(2)在磁感应强度等于第(1)问中B0的情况下,求粒子在磁场中的运动时间;
(3)若磁场的磁感应强度变为第(1)问中B0的2倍,求粒子运动的总路程。
(1)
(2)
(3)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111074577.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134110996703.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111043923.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111074577.png)
⑴设粒子到O点时的速度为v0,由动能定理有
解得
(1分)
粒子经过O点后,速度为v1,
(1分)
如图甲所示,粒子进入磁场后的轨迹圆与磁场边界相切时,磁感应强度最小为B0。设粒子轨道半径为R1,有
(1分)
由
得
(2分)
⑵如图甲,粒子经O1点进入电场区域做匀减速运动,后又加速返回,再次进入磁场时的速率
(1分)
此时粒子做圆周运动的半径
(1分)
其运动轨迹如图甲所示,此后不再进入磁场。由几何
关系可知,![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111433721.png)
则粒子在磁场中运动的时间为
(3分)
⑶若B=2B0,粒子的运动情况如图乙所示,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341115274231.png)
粒子经过O
点第一次进入磁场时的速率仍为v1,在磁场中做圆
周运动的半径记为
,由第⑴问可知,
,
(1分)
粒子从O1点穿过x轴进入电场时速率为
,运动到P1点后返回,则由动能定理
解得
(1分)
当粒子第二次进入磁场时的速率![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341118541095.png)
做圆周运动的半径为
(1分)
粒子从O2点穿过x轴进入电场时速率为
,
运动到P2点后返回,则由动能定理
解得
(1分)
…………
依此类推可知,当粒子第n次进入磁场时,其在磁场中做圆周运动的轨道半径为
,再进入电场中前进的距离
(1分)
因此,粒子运动的总路程为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341122441505.png)
=
=
(1分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111106766.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111152784.png)
粒子经过O点后,速度为v1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111184939.png)
如图甲所示,粒子进入磁场后的轨迹圆与磁场边界相切时,磁感应强度最小为B0。设粒子轨道半径为R1,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111215725.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111262861.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341112931081.png)
⑵如图甲,粒子经O1点进入电场区域做匀减速运动,后又加速返回,再次进入磁场时的速率
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111340852.png)
此时粒子做圆周运动的半径
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111402652.png)
其运动轨迹如图甲所示,此后不再进入磁场。由几何
关系可知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111433721.png)
则粒子在磁场中运动的时间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341115111565.png)
⑶若B=2B0,粒子的运动情况如图乙所示,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341115274231.png)
粒子经过O
点第一次进入磁场时的速率仍为v1,在磁场中做圆
周运动的半径记为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111558339.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111605720.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111652525.png)
粒子从O1点穿过x轴进入电场时速率为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111698948.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111761913.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111808567.png)
当粒子第二次进入磁场时的速率
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341118541095.png)
做圆周运动的半径为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111901534.png)
粒子从O2点穿过x轴进入电场时速率为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341119481008.png)
运动到P2点后返回,则由动能定理
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111995925.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134112088601.png)
…………
依此类推可知,当粒子第n次进入磁场时,其在磁场中做圆周运动的轨道半径为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134112120583.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134112198654.png)
因此,粒子运动的总路程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341122441505.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241341122911200.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824134111074577.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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