Question

如圖所示，勻強電場方向豎直向上，勻強磁場方向水平且垂直紙面向里，有兩個帶電小球a和b，a恰能在垂直于磁場方向的豎直平面內做半徑r=0.8 m的勻速圓周運動，b恰能以v=2 m/s的水平速度在垂直于磁場方向的豎直平面內向右做勻速直線運動．小球a、b質量m_a=10 g，m_b=40 g，電荷量q_a=1×10^－2C，q_b= 2×10^－2C，g=10m/s²。求：
（1）小球a和b分別帶什么電？電場強度E與磁感應強度B的大小？
（2）小球a做勻速周周運動繞行方向是順時針還是逆時針？速度大小v_a是多大？
（3）設小球b的運動軌跡與小球a的運動軌跡的最低點相切，當小球a運動到最低點即切點時小球b也同時運動到切點，a、b相碰后合為一體，設為c，在相碰結束的瞬間，c的加速度a_c=？

Answer 1

（1）小球a在互相垂直的勻強電場和勻強磁場中做勻速圓周運動，電場力和重力的合力為零，電場力方向向上，所以小球a帶正電，且有
m_ag=q_aE··································································①
E="10 N" / C································································②
小球b做勻速直線運動，合力為零，帶正電，且有
m_bg=q_bvB+q_bE·····························································③
B=5T·····································································④
（2）小球a做勻速圓周運動繞行方向是逆時針方向。
由
得······························································⑤
v_a="4 m" / s·································································⑥
（3）a、b相碰前速度方向相同，設碰后的共同速度為v_c，則
m_a v_a+m_b v=(m_a+m_b) v_c······················································⑦
v_c="2.4 m" / s································································⑧
由牛頓第二定律得
m_ca_c=q_cE+q_c v_cB – m_eg······················································⑨
m_c=m_a+m_b
q_c=(q_a+q_b)
解得a_c="3.2 m" / s²··························································⑩
評分標準：本題滿分18分。其中判斷a、b帶電性質各1分，a繞行方向1分，①式2分，②式1分，③式2分，④式1分，⑤式2分，⑥式1分，⑦式2分，⑧式1分，⑨式2分，⑩式1分。

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