年泛珠三角及名校物理奥林匹克邀请赛.docx

Pan Pearl River Delta Physics Olympiad 2007年泛珠三角及中华名校物理竞赛Pan Pearl River Delta Physics Olympiad 20072007年泛珠三角及中华名校物理奥林匹克邀请赛Part-1 (Total 7 Problems) 卷-1（共7题）(9:30 am 12:30 pm, 02-26-2007)Q.1 (3 points) 题1（3分）An airplane is initially rising up at speed v0 at an angle q to the horizon. Find the trajectory of the plane such that weightless condition can be achieved in the plane.一架飞机以与水平面成q 角的初速度v0上升。求飞机以什么样的轨迹飞行，能使飞机里的物体处于失重状态。Q.2 (6 points) 题2（6分）As shown, two identical weights are fixed on the two ends of a uniform rigid rod of length L. The upper weight is restricted to move on a smooth horizontal rail and the rod is free to swing along the rail. The masses of the weights and the rod are equal. Find the small angle vibration frequency of the system. 如图所示，两个质量为m的重块分别固定在一根长度为L质量为m的均匀杆两端。上面的重块可以沿光滑的水平轨道滑行，杆可沿轨道方向自由摆动。求整个系统的小角度振动频率。Q.3 (6 points) 题3（6分）(a) A disc shaped medium block of radius R and thickness d ( R) is uniformly magnetized with magnetization perpendicular to the disc plane. Find the magnetic field at point-O on the central axis of the disk and at a distance h from the cavity center.一半径为R，厚度为d ( R)的圆盘形均匀磁化介质，磁化强度为。盘的表面垂直于。求圆盘中心轴上到圆盘中心距离为h的点O的磁场。(b)A long and thin cylindrical medium is uniformly magnetized with magnetization along the cylinder long axis. Find the magnetic field inside and outside the medium. 一细长圆柱型介质沿柱轴方向均匀磁化，磁化强度为。求介质里、外的磁场。Q.4 (5 points) 题4（5分）A large flat dielectric slab of thickness d and dielectric constant e is moving along the x-direction at speed v. Its large surface plane is perpendicular to the y-axis. A magnetic field of strength B is applied along the z-direction. Find the surface bound charge density on the two large surfaces of the slab, and the electric field in the slab.一个厚度为d，介电常数为e 的大平板以速度v 沿X-方向运动。它的表面与Y-轴垂直。Z-方向加有磁场B。求平板两表面上的束缚电荷密度，以及平板中的电场。Q.5 (10 points) 题5（10分）The space between two concentric conductor spherical shells of radii R1 and R3 is filled with two types of media. The dielectric constant and the conductivity of medium-1 and medium-2 are e1, s1 and e2, s2, respectively. The voltage difference between the two shells is V0. (a) In case-A, the media form two concentric shells with the conductor shells, and the radius of the boundary between the two media is R2. Find the following: (i) total current from the inner shell to the outer shell; (ii) total free charge on the two conductor shells and on the boundary between the two media. (b) In case-B, medium-1 fills the upper hemisphere and medium-2 fills the other half. Find the following: (i) total current from the inner shell to the outer shell; (ii) total free charge on the upper and lower halves of the two conductor shells. Case-ACase-B如图所示，两个半径分别为R1和R3的同心导电球壳之间充满了两种介质。球壳之间电势差为V0。介质1和2的介电常数和导电率分别为 e1, s1 和 e2, s2。(a) 两种介质为与导电球壳同心的球壳，其界面为半径为R2的球面。(i) 求两导电球壳间的总电流；(ii) 求两导电球壳以及两介质之间界面上的电荷。(b) 介质-1填充上半部分，介质-2填充下半部分。(i) 求两导电球壳间的总电流；(ii) 求两导电球壳上、下部分的电荷。Q.6 (12 points) 题6（12分）(a) Assume that atmosphere is made of diatom ideal gas in adiabatic equilibrium. Determine air pressure P, temperature T and density r as a function of altitude h, provided that their values at h = 0 are known. (Hint: Set up a differential equation for a thin layer of air at some altitude. , where is a constant.) (6 points)(a) 大气可看成绝热平衡下的双原子理想气体。求空气压强P、温度T和密度r作为高度h的函数，假定它们在h = 0处的值为已知。（提示：对某高度的一薄层气体建立微分方程。，）(6 分)(b) When the partial pressure of water vapor in air exceeds the saturated water vapor pressure (Ps) at a given temperature, the water vapor will condense into droplets which fall down as rain. Ps = 55.35 mmHg at 40C, and Ps = 6.50 mmHg at 5 C. The air/vapor mixture can be considered as diatom ideal gas and the mass of a water molecule is approximately the same as an air molecule. In the humid air at sea level at 40C the water vapor partial pressure is 90 % of Ps. The density of air is 0 = 1.18 kg m-3 at 20 C and 1.0 atm. The humid air then rises adiabatically to an altitude where the temperature is 5 C. Ignore air pressure change due to the reduction of water vapor.(b1) How much rain can one cubic meter of the humid air at sea level generate? (5 points)(b2) Use the results in (a), find the altitude where the temperature is 5 C. (1 point)(b) 当空气中水蒸汽的分压强超过该温度下的饱和水蒸汽压(Ps)时，水蒸汽将凝聚成滴导致下雨。已知40C时Ps = 55.35 mmHg，5 C时 Ps = 6.50 mmHg。空气/水蒸汽的混合物可当作是双原子理想气体，水分子的质量近似等于空气分子的质量。40C时海平面上的潮湿空气中，水蒸汽分压是Ps的90 %。已知20 C时，1个大气压下的空气密度 0 = 1.18 kg m-3。忽略由于水蒸汽的减少导致的气压改变。该潮湿空气绝热上升到某一高度，该处温度为5 C。（b1）一立方米海平面上的潮湿空气能够产生多少雨？(5 分)（b2）用（a）的结果，求温度为5 C处的高度。(1 分)Q7 (8 points) 题7（8分）(i) Find the torque on an electric dipole in a uniform electric field. (1 point)(ii) A medium is uniformed polarized with polarizationby an electric field. Find the torque per volume on the medium exerted by the electric field. (1 point)(iii) An electromagnetic wave is propagating along the z-axis in an isotropic medium. In such medium the relation between the electric displacement and is given by, soandare always pointing in the same direction. Find the torque per volume on the medium exerted by the electromagnetic wave. (1 point)(iv) An electromagnetic wave is propagating along the z-axis in an anisotropic medium. In such medium the electric displacement is, sois not parallel to. Note that and, where c is the speed of light in vacuum. Find the time-averaged (over one period) torque per volume on the medium exerted by the electromagnetic wave. (3 points)(v) Following (iv), find the time-averaged total torque on a section of cylindrical shaped medium of unit cross section area with its long axis along the z-direction from z = 0 to z = d, and the smallest value of d at which the total torque is maximum. (2 points)4(i) 求一个电偶极子在电场中受到的力矩。(1 分)(ii) 某介质在电场中均匀极化，极化强度为。求单位体积介质在该电场中受到的力矩。(1 分)(iii) 在一各向同性的介质中，电磁波 沿z-轴传播 。在该介质中电位移矢量和电场 的关系满足，因此和总是保持同一方向。求单位体积介质在该电磁波中受到的力矩。(1 分)(iv) 在一各向异性的介质中，电磁波沿z-轴传播 。在该介质中电位移矢量为，因此通常和不平行。这里，c是真空中光速。求单位体积介质在该电磁波中受到的一个周期里的平均力矩。 (3分)(v) 根据 (iv), 求长轴平行于z-轴，单位横截面积的圆柱形介质中z = 0 到 z = d部分所受的一个周期的平均力矩，以及使力矩最大所需的d的最小值。（2分）THE END完Pan Pearl River Delta Physics Olympiad 20072007年泛珠三角及中华名校物理奥林匹克邀请赛Part-2 (Total 3 Problems) 卷-2（共3题）(2:30 pm 5:30 pm, 02-26-2007)Q1 Folded Space (6 points) 题1卷起的空间（6分）(a) Consider a one-dimensional standing electromagnetic wave in the form of along the x-direction confined within the space between x = 0 and x = a. The wave must vanish at these two end points. Find the allowed values of kx. (1 point) (b) The String Theory predicts that our space is more than three-dimension, and the additional hidden dimensions are folded up like the dimension y on the surface of a thin cylinder shown in the figure. Suppose the radius of the cylinder is b ( a), and the electromagnetic wave on the surface now takes the form , where y is the coordinate of the folded space around the cylinder. Find the allowed values of ky. (3 points)(c) The photon energy is given by, and hc = 1239 (eV nanometer), where eV stands for electron volt and 1 nanometer is 10-9 meters. The highest energy photons human can make so far is about 1.0 1012 eV. If this is sufficient to create a photon in the folded space, what should be the value of b? (2 points)(a) 一维电磁驻波 在 x-方向限制在x = 0 和 x = a之间。 在两个端点处驻波消失。求 kx的可能值。(1分) (b) 弦理论认为物理空间多于三维，多出的隐藏维空间象细圆柱的表面一样卷了起来，如图中y坐标所示。设圆柱的半径为b ( a), 在圆柱面上电磁波的形式为，其中y 是绕圆柱的折叠空间的坐标。求 ky的可能值。(3分)(c) 光子能量, 其中 hc = 1239 (eV nm)，eV 表示1电子伏特, 1 nm 等于10-9 米。目前人类能产生的最高能量的光子大约为1.0 1012 eV。如果该能量能够产生一个折叠空间的光子，b的值满足什么条件？(2分)Q2 Atomic Force Microscope (AFM) in thermal noise (22 points) 题2 热噪声下的原子力显微镜 (22 分)(i) An AFM is modeled as a uniform rigid rod of length l and mass m1 with a point mass m2 on one end (the tip), and the other end is fixed at point O around which the rod is free to rotate. A spring of force constant K is attached to the tip. Find the resonant frequency w0 of the AFM. (4 points)原子力显微镜能够简化为一个长度为l，质量为m1的均匀硬杆，一端有一个质量为 m2 的质点 (针尖)，另一端固定在点 O ，杆可绕点 O自由转动。一个弹性系数为 K 的弹簧连着针尖。求原子力显微镜的共振频率w0。(4分)(ii) Given an external driving force, derive the differential equation for the small vertical displacement x(t) of the tip from its equilibrium position, and solve it using a trial solution where the amplitude A1 and phase F1 are to be determined. (4 points)给定一个外驱动力, 推导针尖离平衡位置的小位移x(t)的微分方程，并用试探解解它，其中振幅A1 和位相F1 待定。(4 分)(iii) Given two driving forces, find. (4 points)给定两个外驱动力, 求。 (4分)(iv) The driving force comes from thermal noise, which can be described as a sum of many harmonic driving forces in the entire frequency range. Find under the thermal noise driving force. (2 points)驱动力来自于热噪声，它能够写成覆盖所有频率的许多简谐驱动力的和。求热驱动力下的 。(2 分)(v) Consider the electronic band pass filter as shown. Given the input voltage, find the value of inductance L such that the denominator of the absolute value of the output voltage is minimum. (2 points)考虑一个如图所示的电子带通滤波器。输入电压为，求使输出电压绝对值分母最小的电感L 的值。(2 分)(vi) The AFM signal which is proportional to the solution in (iv) is applied as the input signal to the filter. Assuming that only the signal with the frequency wn = w, where w makes the denominator of the output voltage amplitude minimum in (v), can pass through the filter, draw a sketch of the amplitude of the output voltage vs L if Fn = 1 for all n, and describe briefly how the AFM resonant frequency in (i) can be found experimentally. (6 points)将正比于（iv）中的原子力显微镜信号输入到电子滤波器。假设仅有频率wn等于（v）中使输出电压绝对值分母最小的w 的信号能通过该滤波器，假定对所有n，Fn = 1 ，试画出输出电压的大小随L 变化的简图， 并简单描述实验上如何找到（i）中所述原子力显微镜的共振频率。(6 分)Q3 The Lorentz-Lorenz Relation (22 points) 题3 洛伦兹-洛伦兹关系 (22分)The dielectric constant of a dielectric medium is given by the so called Lorentz-Lorenz Relation, where n is a number and K is a material-related constant that depends explicitly on the frequency w of the applied electric field. You are to derive the relation through the steps below.介质的介电常数满足所谓的洛伦兹-洛伦兹关系，其中 K 与所加电场的频率w 以及介质的物质常数有关，n是一数字。下面逐步推出这一关系。(i) An atom can be approximately treated as consisting of a uniform spherical electron density (electron cloud) of radius R with total charge Ze and positive charge nucleus Ze at the center, where e is the charge of a positron. The nucleus mass is much larger than an electron mass me. In a uniform external electric field E0 the electron cloud is displaced slightly from the nucleus while maintaining its spherical shape. Find the displacement. (4 points)一个原子可以近似看成由总电荷为Ze、均匀分布成半径为R的球形电子云，和处于中心的带电Ze的原子核组成，其中e 是正电子的电荷。 核的质量远大于电子质量 me。在均匀外电场E0 中电子云轻微偏离核，但保持球形。求偏离的位移。(4 分)(ii) The atom is placed in an oscillating uniform electric field. Find the induced dipole moment of the atom. (6 points)将原子放在震荡的均匀外场中， 求原子的电偶极矩。(6 分)(iii) In a medium the number of atoms per unit volume is N, find the polarization P of the medium. (2 points)已知介质的单位体积原子数是 N，求介质的极化矢量 P 。(2 分)(iv) Note that in (iii) the electric field is the external field Eext. Consider a small sphere containing many atoms in the large medium. The total field Etotal in the sphere consists of two contributions, namely that from the medium inside the sphere Eself and the external field Eext. Given that the electric fields are uniform inside the sphere, find the relation between Eext and Etotal, and determine n and K in the Lorentz-Lorenz Relation. (7 points)在 (iii) 中的电场是外加电场 Eext。考虑大介质中的一个包含很多原子的小球。球中的总电场 Etotal 来源于两部分，一是球中的介质产生的电场 Eself ，二是外加电场Eext。已知球中的电场是均匀的，求 Eext 和 Etotal的关系，并求洛伦兹-洛伦兹关系中的n 和K。 (7 分)(v) Use the result in (iv), briefly explain the Mirage phenomenon. (3 points)用(iv)的结果简单解释海市蜃楼现象。（3 分）THE END完AnswersPart IQ1.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-1The 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-2Given a small angle deviation q from equilibrium, the potential energy is给定一个角度的小位移q ，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)(a) 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 drop from R1 to R2 is (1 point)介质-1,，从R1 到 R2 的电压为Likewise, in medium-2, and the voltage drop from R2 to R3 is .(1 point)同样，在介质-2, ，从R2 到 R3 的电压为The total voltage drop between R1 and R3 is总电压 V = V1 + V2. So 得The current is电流为. (1 point)The electric displacement in media are, so the charge on the inner, outer, and boundary shells are, and, respectively.介质-1里电位移, 因此在各面上的电荷为, 。(b) Due to symmetry, the electric field is of the form, so, and. (2 point)由对称性可知，电场为，因此电压为，得。The current densities in the two hemispheres are and. The total current is. (2 point)上下半部的电流密度为，。 总电流为 . The total free charge is on the upper half and on the lower half of the inner shell. On the outer shell the charges are negative of the corresponding ones of the inner shell. (1 point) 内球面上半部总自由电荷为，下半部总自由电荷为。外球面的电呵与内球面相反。 Q6.(12 points)a)(1 point) (1 point)Combine these two equations, 合并两式得 (2 point), and.(2 point)b)(1 point)The density at 40C is (1 point)40C的空气密度为The fraction of water vapor at 40C at sea level 40C的水蒸汽分压为, (1 point)The fraction of water at 5C at high altitude 5C的水蒸汽分压为, (1 point)Ra