题目内容
如图甲所示足够长的平行光滑金属导轨ab、cd倾斜放置,两导轨之间的距离为L=0.5m,导轨平面与水平面间的夹角为θ=30°,导轨上端a、c之间连接有一阻值为R1=4Ω的电阻,下端b、d之间接有一阻值为R2=4Ω的小灯泡。有理想边界的匀强磁场垂直于导轨平面向上,虚线ef为磁场的上边界,ij为磁场的下边界,此区域内的感应强度B,随时间t变化的规律如图乙所示,现将一质量为m=0.2kg的金属棒MN,从距离磁场上边界ef的一定距离处,从t=0时刻开始由静止释放,金属棒MN从开始运动到经过磁场的下边界ij的过程中,小灯泡的亮度始终不变。金属棒MN在两轨道间的电阻r=1Ω,其余部分的电阻忽略不计,ef、ij边界均垂直于两导轨。重力加速度g=10m/s2。求:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082422231103113562.png)
(1)小灯泡的实际功率;
(2)金属棒MN穿出磁场前的最大速率;
(3)整个过程中小灯泡产生的热量。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/2014082422231103113562.png)
(1)小灯泡的实际功率;
(2)金属棒MN穿出磁场前的最大速率;
(3)整个过程中小灯泡产生的热量。
(1)1W (2) 3m/s (3)1.08J
试题分析:(1)由于小灯泡的亮度始终不变,说明金属棒MN进入磁场后作匀速直线运动,速度达到最大,由平衡条件得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222312310749.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222314213404.png)
小灯泡的电功率
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222315571714.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222316444514.png)
(2)由闭合电路的欧姆定律得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222317177521.png)
其中,总电阻
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222317911583.png)
由法拉第电磁感应定律得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222318971527.png)
由以上各式代入数据解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222320329569.png)
(3)金属棒进入磁场前,由牛顿第二定律得:
加速度a=gsin30°=5m/s2
进入磁场前所用的时间:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222321561496.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222322887520.png)
设磁场区域的长度为x。在0-t1时间内,由法拉第电磁感应定律得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408242223236051292.png)
金属棒MN进入磁场前,总电阻
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222324400609.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222325399361.png)
又感应电动势
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222326210577.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222327458539.png)
在磁场中运动的时间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222328924791.png)
整个过程中小灯泡产生的热量
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222330125977.png)
代入数据解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824222331358571.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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