2007年综合试答案.docVIP

Answers Part I Q1. The plane should follow the parabola 飞机须沿抛物线运动。 (3 points) Q2 (6 points) The center of the rod will not move in the horizontal direction杆中心在水平方向不动。 (2 points) There are two ways to find the torque. 找力距的方法有两种。 Method-1方法-1 The forces acting upon the rod are shown. The torque to the center of the rod is由如图力的分析，可得 . (2 points) Method-2方法-2 Given a small angle deviation ( from equilibrium, the potential energy is给定一个角度的小位移( ，势能为 . (2 points) Finally, 最后得. (2 points) Q3 (6 points) (a) The bound current density on the disk edge is盘边的束缚电流密度为, (1 point) The bound current is束缚电流, (1 point) The B-field is 磁场为 (1 point) (b) The bound current density is, which is on the side wall of the cylinder. (1 point) 柱侧面上的束缚电流密度为 The problem is then the same as a long solenoid. Take a small Ampere loop we get inside; 为求一长线圈的磁场，取一小闭合路径，得介质内 (1 point) Outside介质外 B = 0 (1 point) Q4 (5 points) Each unit charge in the slab experiences the Lorentz force . (1 point) The problem is then the same as a dielectric slab placed between two parallel conductor plates that carry surface charge density. In such case, the electric displacement is. . (2 point) 介质内单位电荷受力。问题变成两电荷面密度为的导电板间充满介质。因此. . Finally, the bound surface charge is. The upper surface carries positive bound charge, and the lower surface carries negative charge. (1 point) 最后得束缚电荷密度，上表面带正电，下表面带负电。 The electric field is, which is along the y-direction (opposite to the Lorentz force). (1 point) 电场为，与Lorentz力方向相反。 Q5. (10 points) Because of the spherical symmetry, the E-field and the current density are all along the radial direction. In steady condition, the electric current I through any spherical interfaces must be equal. Since the area of the sphere is proportional to r2, must be proportional to 1/r2. So let, where K is a constant to be determined, and the expression holds in both media. (1 point) 由对称性可知，电场和电流密度须沿半径方向。稳态时，流过每个包住球心的球面的电流相等，因此与1/r2成正比。设在两介质里，K为待定常数。 In medium-1介质-1,, and the voltage d