北大博雅试题及答案
01、空间直角坐标系中，六个完全相同、均匀带电的正方形绝缘平板构成一个正方体，其中心O位于坐标原点，各棱方向与坐标轴平等，记与z轴平行的棱的中点为A，正方体与x轴的交点为B，则A、B、C三点的电场强度 D
A、全部为0
B、全部不为0
C、有两个满足至少在两个方向上的分量不为0
D、有一个满足恰好在两个方向上的分量不为0
![](https://wkrtcs.bdimg.com/rtcs/image?w=140.00&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","r":"r_40","w":"140.00","h":"101.00","dataType":"png","c":"word/media/image1.png","imgOriW":321,"imgOriH":231})
解：答案：D
根据对称性，O点的电场为零。A点的电场Z分量为零，但X、Y分量不等于零。B点的电场只有X分量。
02、如图，用轻绳悬挂一带电小球A，绳长为L，小球质量为m，现将一无穷远处的相同小球B移到图示位置，原小球偏转角度为θ，求移动小球过程中外力做的功
![](https://wkrtcs.bdimg.com/rtcs/image?w=86.00&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","r":"r_55","w":"86.00","h":"113.00","dataType":"png","c":"word/media/image2.png","imgOriW":197,"imgOriH":259})
解：答案：![](https://wkrtcs.bdimg.com/rtcs/image?w=170&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"170","h":"21","c":"9b9266f96f76aabeb946e0472968dc9f.png"})
如图位置处于平衡 ![](https://wkrtcs.bdimg.com/rtcs/image?w=226&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"226","h":"71","c":"9bf34d467b9e25788fa9298951b4f4b5.png"})
电势能的增量 ![](https://wkrtcs.bdimg.com/rtcs/image?w=394&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"394","h":"67","c":"00d8b90bbabfb9ef88a9e6c3b9f0e362.png"})
重力势能的增量 ![](https://wkrtcs.bdimg.com/rtcs/image?w=188&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"188","h":"23","c":"bf41ff6927bf8e985993b1a67e2fddb4.png"})
则，外力做功 ![](https://wkrtcs.bdimg.com/rtcs/image?w=291&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"291","h":"23","c":"a4034ebce57aa93fe68decebf159b91d.png"})
03、如图，有一等腰三棱镜，底角为θ，从侧面沿平行于底边的方向射入一光束，其折射后在底面发生全反射并从另一个侧面射出。已知三棱镜的材料的折射率为![](https://wkrtcs.bdimg.com/rtcs/image?w=26&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"26","h":"29","c":"d21848cdd835abcb491be1f151e9b6c6.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=154.00&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","r":"r_55","w":"154.00","h":"114.00","dataType":"jpeg","c":"word/media/image9.jpeg","imgOriW":353,"imgOriH":261})
解：第一次折射。入射角![](https://wkrtcs.bdimg.com/rtcs/image?w=91&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"91","h":"28","c":"ed0fa02425a1e7020715f7266f9a7424.png"})
折射定律 ![](https://wkrtcs.bdimg.com/rtcs/image?w=82&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"82","h":"49","c":"b6ccde2a7f506e36f3f2e000759e1f93.png"})
解得 ![](https://wkrtcs.bdimg.com/rtcs/image?w=139&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"139","h":"54","c":"54abe50629ff7d195db8c2a330b59903.png"})
光线在底面发生全反射的临界角C与折射率n的关系 ![](https://wkrtcs.bdimg.com/rtcs/image?w=74&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"74","h":"43","c":"4ed69a86ccd200c1b02be009d12116f7.png"})
解得 ![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"61","h":"25","c":"f3adea3721338dcf5fe70a899d695126.png"})
发生全反射的条件是![](https://wkrtcs.bdimg.com/rtcs/image?w=77&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"77","h":"23","c":"c1f12e4c744ffdd9fb89382b4bada5e3.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=182&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"182","h":"54","c":"58ed68ce3d27c3bcad1614260f671646.png"})
04、长度分别为L1和L2的两根不可伸长的细绳悬挂着质量分别为m1和m2的两个小球，处于静止状态，如图所示，中间小球突然受到一水平方向的冲击力，瞬间获得水平向右的速度v，求此时两绳中的拉力。
![](https://wkrtcs.bdimg.com/rtcs/image?w=83.00&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","r":"r_55","w":"83.00","h":"165.00","dataType":"png","c":"word/media/image17.png","imgOriW":177,"imgOriH":351})
解：m1相对于悬点O做圆周运动，根据牛顿定律，有 ![](https://wkrtcs.bdimg.com/rtcs/image?w=173&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"173","h":"47","c":"ce1d65b29bc1f721fe61e70ddfd69b83.png"})
m2相对于m1做圆周运动，此时m1的向心力加速度为![](https://wkrtcs.bdimg.com/rtcs/image?w=24&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"24","h":"47","c":"e8b257eeb4f34d6cc982058f0e4dc7e7.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=201&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"201","h":"47","c":"e01a997f3e0e9c82091eb8371a23606d.png"})
由上二式可得
![](https://wkrtcs.bdimg.com/rtcs/image?w=314&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"314","h":"47","c":"784073fd348dc0a0eab3e2ecd34cd1f7.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=213&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"213","h":"47","c":"5ad0a16ac62350eb804ccbb682d2d680.png"})
05、一平行板电容器，面积为S，板间距为d，与电动势为U的恒定电源串联，现将一厚度为d，面积为S、相对介电常数为ε的电介质插入极板之间，求该过程中外力做的功。
解：插入电介质前电容器的电容为![](https://wkrtcs.bdimg.com/rtcs/image?w=88&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"88","h":"43","c":"c95b6ebb357ce44fa692286e78965fcc.png"})
这个过程中外力做的功是 ![](https://wkrtcs.bdimg.com/rtcs/image?w=292&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"292","h":"48","c":"16b8461f68b94425be2430933b317f40.png"})
06、如图，真空中有四块完全相同且彼此相近的大金属板A、B、C、D平行放置，表面涂黑（可看成黑体），最外侧两块板的热力学温度各维持T1和T4，且T1>T4。当达到热稳定时，求B板的温度。
![](https://wkrtcs.bdimg.com/rtcs/image?w=96.00&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","r":"r_3","w":"96.00","h":"120.00","dataType":"png","c":"word/media/image27.png","imgOriW":220,"imgOriH":275})
解：温度为T2的B板左、右侧的单位时间内单位面积上净获得的辐射热量分别为
![](https://wkrtcs.bdimg.com/rtcs/image?w=146&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"146","h":"33","c":"2cba3d2dad67b99af4dc523dc56a360e.png"})
达到热稳定时，有 ![](https://wkrtcs.bdimg.com/rtcs/image?w=118&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"118","h":"28","c":"679e01cbca271802f91bd8b7b24866e0.png"})
由上得 ![](https://wkrtcs.bdimg.com/rtcs/image?w=109&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"109","h":"31","c":"63d62ac8e54f38be62f239d4794e059a.png"})
温度T3的C板左、右侧的单位时间内单位面积上净获得的辐射热量分别为
![](https://wkrtcs.bdimg.com/rtcs/image?w=146&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"146","h":"33","c":"3516611eff6672a2d17243bc74024214.png"})
达到热稳定时，有 ![](https://wkrtcs.bdimg.com/rtcs/image?w=118&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"118","h":"28","c":"07c7fdac68d21608e40687f15ecf0339.png"})
由上得 ![](https://wkrtcs.bdimg.com/rtcs/image?w=109&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"109","h":"31","c":"fcc958e14c49e605b090c67bf0d73f32.png"})
由上可得 ![](https://wkrtcs.bdimg.com/rtcs/image?w=124&md5sum=c2930c3b43a29b7658737847dad97cc8&sign=4d91f54bb6&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=292&ipr={"t":"img","w":"124","h":"68","c":"5ae12f3b99740462cecb518a51b4df1f.png"})
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