Question

如圖所示，單匝線圈ABCD在外力作用下以速度v向右勻速進入勻強磁場，第二次又以速度2v勻速進入同一勻強磁場．求：

（1）第二次與第一次線圈中最大電流之比；
（2）第二次與第一次外力做功的最大功率之比；
（3）第二次與第一次線圈中產生熱量之比．

Answer 1

（1）2∶1 （2）4∶1（3）2∶1  

（1）線圈以速度v勻速進入磁場，當CD邊在磁場中時，線框中感應電動勢其中L為CD邊的長度．                                            
此時線框中的感應電流為，其中R為線框的總電阻．   
同理，線圈以速度2v勻速進入磁場時，線框中的感應電流最大值為
所以第二次與第一次線圈中最大電流之比為2∶1                     （4分）
（2）線圈以速度v勻速進入磁場，當CD邊在磁場中時，CD邊受安培力最大，最大值為 ．由于線圈做勻速運動，所以此時外力也最大，且外力大小等于安培力大小，此時外力的功率為．                 
同理，線圈以速度2v勻速進入磁場時，外力的最大功率為．
所以第二次與第一次外力做功的最大功率之比為4∶1                   （4分）
（3）線圈以速度v勻速進入磁場，線框中的感應電流為
設AD邊長為l，則線框經過時間完全進入磁場，此后線框中不再有感應電流．所以第一次線框中產生的熱量為         
同理，線圈以速度2v勻速進入磁場時，線框中產生的熱量為                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
所以第二次與第一次線圈中產生熱量之比為2∶1                       （4分）