2007年泛珠三角及中华名校物理奥林匹克邀请1

Pan Pearl River Delta Physics Olympiad 2007年泛珠三角及中华名校物理竞赛

2007年泛珠三角及中华名校物理奥林匹克邀请赛 Pan Pearl River Delta Physics Olympiad 2007

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 ? 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? 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。 m

Q.3 (6 points) 题3（6分）

m(a) A disc shaped medium block of radius R and thickness d (<< R) is uniformly magnetized with magnetization M

? 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M

?

。盘的表面垂直于M

?

。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 magnetizationM?

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M?。求介质里、外的磁场。

OM?(a)(b)1

Pan Pearl River Delta Physics Olympiad 2007年泛珠三角及中华名校物理竞赛

Q.4 (5 points) 题4（5分）

A large flat dielectric slab of thickness d and dielectric constant ? 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? 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。

y

x z

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 ?1, ?1 and ?2, ?2, 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.

R12 12R1 R 1 R2R33 Case-A Case-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 ?1, ?1 和 ?2, ?2。 (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 ? 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.

?x?dx?1??1x??1, where ??1 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?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。?x?dx?1??1x??1，???1）(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 40?C, 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 40?C the water vapor partial pressure is 90 % of Ps. The density of air is ρ0 = 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40?C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。40?C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 ρ0 = 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)

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Pan Pearl River Delta Physics Olympiad 2007年泛珠三角及中华名校物理竞赛

Q7 (8 points) 题7（8分） (i) Find the torque on an electric dipole ?pin a uniform electric fieldE?. (1 point)

(ii)

A medium is uniformed polarized with polarizationP?by an electric fieldE?. Find the torque per volume on the medium

exerted by the electric field. (1 point) E?(iii)

An electromagnetic wave

?E??yi(kz??t)0(x0?0)eD?is propagating along the z-axis in an isotropic medium. In such medium the relation between the electric displacement

and E??? is given byD??D?andE?0?E, soare 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

E??E(x?ik1z?ik2z?i?t00e??y0e)eis propagating along the z-axis in an anisotropic ?medium. In such medium the electric displacement isD??ik0E0(?xx0e1z???ikyy0e2z)e?i?t, soD?is not parallel toE?. Note that k?1?c?x andk2??c?y, 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)

(i) 求一个电偶极子?p在电场E?中受到的力矩。(1 分)

(ii) 某介质在电场E?中均匀极化，极化强度为P?。求单位体积介质在该电场中受到的力矩。(1 分)

(iii)

E?在一各向同性的介质中，电磁波?Ex??ei(kz??t)???0(0?y0)沿z-轴传播 。在该介质中电位移矢量D和电场 E

???

的关系满足D??0?E，因此D和E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)

??ik(iv)在一各向异性的介质中，电磁波?y?ik

1z2z)e?i?t0e沿z-轴传播 D?E?E0(x0e。在该介质中电位移矢量为??x?ik?ik?E?0E0(?x0e1z??yy0e2z)e?i?t，因此通常D和不平行。这里k1??c?x，k2??c?y，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）

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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 2007 2007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

E(x)?Asin(kxx) 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

E(x,y)?Asin(kxx)cos(kyy), 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 byW?hc2?2kx2?ky, 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

E(x)?Asin(kxx) 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E(x,y)?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WAsin(kxx)cos(kyy)，其

?hc2?2kx2?ky, 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)

xxy4

OF(t)Pan Pearl River Delta Physics Olympiad

(i)

2007年泛珠三角及中华名校物理竞赛

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 ?0 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?0。(4分)

(ii) Given an external driving forceF(t)?F1cos(?1t), 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

x(t)?A1cos(?1t??1) where the amplitude A1 and phase ?1 are to be determined. (4 points)

给定一个外驱动力F(t)?解x(t)?并用试探F1cos(?1t), 推导针尖离平衡位置的小位移x(t)的微分方程，

A1cos(?1t??1)解它，其中振幅A1 和位相?1 待定。(4 分)

(iii) Given two driving forcesF(t)?F1cos(?1t)?F2cos(?2t), findx(t). (4 points)

给定两个外驱动力F(t)?F1cos(?1t)?F2cos(?2t), 求x(t)。 (4分)

(iv) The driving force comes from thermal noise, which can be described as a sum of many harmonic driving

forcesFthermal(t)??Fncos(?nt) in the entire frequency range. Findx(t) under the thermal

nnoise 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

Fther()??s(。求热驱动力下的tnx(t) 。(2 分) mtal?Fconn(v) Consider the electronic band pass filter as shown. Given the input voltageVin(t)?V0ei?t, find

VinCRLVoutthe 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in对值分母最小的电感L 的值。(2 分)

(vi) The AFM signal which is proportional to the solutionx(t) in (iv) is applied as the input signal to the filter.

Assuming that only the signal with the frequency ?n = ?, where ? 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x(t)的原子力显微镜信号输入到电子滤波器。假设仅有频率?n等于（v）中使输

V(t)?V0ei?t，求使输出电压绝

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