题目内容
如图所示,电阻忽略不计的、两根平行的光滑金属导轨竖直放置,其上端接一阻值R=3Ω的定值电阻.在水平虚线L1、L2间有一与导轨所在平面垂直的匀强磁场B,磁场区域的高度为d=0.5m.导体杆a的质量ma=0.2kg、电阻Ra=6Ω;导体杆b的质量mb=0.6kg、电阻Rb=3Ω,它们分别从图中M、N处同时由静止开始在导轨上无摩擦向下滑动,且都能匀速穿过磁场区域,当a杆刚穿出磁场时b杆正好进入磁场.设重力加速度为g=10m/s2.(不计a、b之间的作用)求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210272263300.png)
(1)在整个过程中,a、b两杆完全穿出磁场区克服安培力分
别做的功;
(2)设a、b杆在磁场中的运动速率分别为
,则![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027336437.png)
的值为多少?
(3)M点和N点距水平虚线L1的高度.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210272263300.png)
(1)在整个过程中,a、b两杆完全穿出磁场区克服安培力分
别做的功;
(2)设a、b杆在磁场中的运动速率分别为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027273505.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027336437.png)
的值为多少?
(3)M点和N点距水平虚线L1的高度.
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210274141197.png)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027445986.png)
(3)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027507918.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210273981170.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210274141197.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027445986.png)
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027507918.png)
解:(1)因 a、b杆在磁场中做匀速运动,设 a杆克服安培力做功
,由动能定理可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210275543174.png)
…………① (1分)
解得:
② (1分)
同理设b杆克服安培力做功
,由动能定理得:
…………③ (1分)
解得:
④ (1分)
(2)当
杆以速度
在磁场中匀速运动时,回路的
总电阻为
⑤ (1分)
杆中产生感应电动势为:
…⑥
杆产生感应电流为:
……⑦
由
杆做匀速运动有:
……⑧
由⑤⑥⑦⑧可得:
……⑨ (2分)
同理b杆以速度
在磁场中匀速运动时,回路总电阻为
…⑩ (1分)
b杆中产生感应电动势为:
…
b杆产生感应电流为:
……
由b杆做匀速运动有:
…
由⑩![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028381225.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028490239.png)
可得:
…
(1分)
由⑨
可得:
……
(1分)
(3)设a杆在磁场中运动历时为t,则有:
…………
(1分)
…………
(1分)
结合
式
解得:
………
(1分)
依
得:
…
(1分)
联立![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028786235.png)
式解得:
(1分)
所以:
(1分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027523433.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210275543174.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210275701016.png)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210273981170.png)
同理设b杆克服安培力做功
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027663447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210276941018.png)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210274141197.png)
(2)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027741277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027772342.png)
总电阻为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210277881039.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027741277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027850591.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027741277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027913615.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027741277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028162656.png)
由⑤⑥⑦⑧可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028225895.png)
同理b杆以速度
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028287350.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241210283031016.png)
b杆中产生感应电动势为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028365597.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028381225.png)
b杆产生感应电流为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028443634.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028490239.png)
由b杆做匀速运动有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028537670.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028568242.png)
由⑩
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028381225.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028490239.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028568242.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028693905.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028724574.jpg)
由⑨
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028724574.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027445986.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028786235.png)
(3)设a杆在磁场中运动历时为t,则有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028818492.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028864232.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028880572.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028911240.png)
结合
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028786235.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029145589.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029161740.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029239242.png)
依
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029317650.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029348860.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029410247.png)
联立
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121028786235.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029239242.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029613912.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029722246.png)
所以:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121027507918.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121029800246.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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