题目内容
如图所示,在xoy平面内存在B=2T的匀强磁场,OA与OCA为置于竖直平面内的光滑金属导轨,其中OCA满足曲线方程
,C为导轨的最右端,导轨OA与OCA相交处的O点和A点分别接有体积可忽略的定值电阻R1=6Ω和R2=12Ω。现有一长L=1m、质量m=0.1kg的金属棒在竖直向上的外力F作用下,以v=2m/s的速度向上匀速运动,设棒与两导轨接触良好,除电阻R1、R2外其余电阻不计,求:
(1)金属棒在导轨上运动时R2上消耗的最大功率
(2)外力F的最大值
(3)金属棒滑过导轨OCA过程中,整个回路产生的热量。![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241146544572366.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654426438.gif)
(1)金属棒在导轨上运动时R2上消耗的最大功率
(2)外力F的最大值
(3)金属棒滑过导轨OCA过程中,整个回路产生的热量。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241146544572366.gif)
(1)R2上消耗的功率最大值为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654457336.gif)
(2)外力的最大值![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654472289.gif)
(3) 产生的热量为
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654457336.gif)
(2)外力的最大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654472289.gif)
(3) 产生的热量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654488301.gif)
(1)金属棒向上匀速运动的过程中切割磁感线,产生电动势,接入电路的有效长度即为OCA导轨形状所满足的曲线方程,因此接入电路的金属棒长度为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654504466.gif)
所以当棒运动到C点时,感应电动势最大,为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654519425.gif)
电阻R1、R2并联,此时R2上消耗的功率最大,最大值为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654550547.gif)
(2)金属棒相当于电源,外电路中R1、R2并联,其并联阻值为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654566477.gif)
通过金属棒的最大电流为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654582391.gif)
所以最大安培力![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654597385.gif)
因为金属棒受力平衡,所以外力的最大值![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654613398.gif)
(3)金属棒中产生的感应电动势为:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654628427.gif)
显然为正弦交变电动势,所以有效值为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654628433.gif)
该过程经历的时间:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654753372.gif)
所以产生的热量为
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654504466.gif)
所以当棒运动到C点时,感应电动势最大,为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654519425.gif)
电阻R1、R2并联,此时R2上消耗的功率最大,最大值为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654550547.gif)
(2)金属棒相当于电源,外电路中R1、R2并联,其并联阻值为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654566477.gif)
通过金属棒的最大电流为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654582391.gif)
所以最大安培力
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654597385.gif)
因为金属棒受力平衡,所以外力的最大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654613398.gif)
(3)金属棒中产生的感应电动势为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654628427.gif)
显然为正弦交变电动势,所以有效值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654628433.gif)
该过程经历的时间:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654753372.gif)
所以产生的热量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114654769434.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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