# 2007年泛珠三角及中华名校物理奥林匹克邀请赛试题

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） (9:3

0 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) (a)

? A disc shaped medium block of radius R and thickness d (<< R) is uniformly magnetized with magnetization M
the disc plane.

m

perpendicular to Find the magnetic field at point-O on the central axis of the disk and at a distance h from the cavity center.

?

?

?

(b)

A long and thin cylindrical medium is uniformly magnetized with magnetization M field inside and outside the medium.

along the cylinder long axis. Find the magnetic

?

O
? M

(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)

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

(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)

x z

Q.5 (10 points)

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.

R2

1
R1

2

1
R1 R3

R3
Case-A Case-B

2

(b)

Q.6 (12 points)

(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 d x ?
?

1

? ?1

x

? ?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 d x ?
?

1

? ?1

x

? ?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)

2

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) (ii) Find the torque on an electric dipole p in a uniform electric field E . (1 point)

?

?

?

?

A medium is uniformed polarized with polarization P by an electric field E . Find the torque per volume on the medium exerted by the electric field. (1 point) An electromagnetic wave

(iii)

? ? ? i ( kz ? ? t ) E ? E 0 ( x0 ? y0 )e is propagating along the z-axis in an isotropic medium. In such
? ?

medium the relation between the electric displacement D and E is given by D ? ? 0 ? E , so D and E are 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

?

?

?

?

? ? ik1 z ? ik z ? i? t ? ? ? y y0 e 2 )e medium. In such medium the electric displacement is D ? ? 0 E 0 ( ? x x 0 e , so D
parallel to E . Note that k 1 ?

? ? ik z ? ik z ? i? t E ? E 0 ( x0 e 1 ? y 0 e 2 )e
?
c

is propagating along the z-axis in an anisotropic is not

?

? x and k 2 ?

?
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) 求一个电偶极子 p 在电场 E 中受到的力矩。(1 分)

(i) (ii) (iii)

?

? ?

?

?

?

?

? ? ? i ( kz ? ? t ) E ? E 0 ( x0 ? y0 )e 沿 z-轴传播
?

?

?

。在该介质中电位移矢量 D 和电场 E

(iv)

? ? ik z ? ik z ? i? t E ? E 0 ( x0 e 1 ? y 0 e 2 )e 在一各向异性的介质中，电磁波
? ? ik z ? ik z ? i? t ? D ? ? 0 E 0 (? x x 0 e 1 ? ? y y 0 e 2 ) e ，因此通常 D
?

?
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。 分） （2

3

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) (a) (b) 题 1 卷起的空间（6 分） (v)

n

x ( t ) 。(2 分)
Consider the electronic band pass filter as shown. Given the input voltage V in ( t )

? V0 e

i? t

, find

Vin C R L

Vout

Consider a one-dimensional standing electromagnetic wave in the form of E ( x ) ? A sin ( k x x ) 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) 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 ) ? A sin ( k x x ) co s( k y y ) , where y is the coordinate of the folded space around the cylinder. Find the allowed values of ky. (3 points) The photon energy is given by W ?

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 V in ( t ) 对值分母最小的电感 L 的值。(2 分)

? V0 e

i? t

，求使输出电压绝

(vi) The AFM signal which is proportional to the solution x ( 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）中使输出电压绝对值分母最小 的? 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)

(c)

hc 2?

k x ? k y , and hc = 1239 (eV × nanometer), where eV stands for electron volt and 1
2 2

nanometer is 10-9 meters. The highest energy photons human can make so far is about 1.0 × 10 12 eV. If this is sufficient to create a photon in the folded space, what should be the value of b? (2 points) (a) (b) 一维电磁驻波 E ( x ) ? A sin ( k x x ) 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) 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 ) ? A sin( k x x ) cos( k y y ) ，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 W ?

hc 2?

k x ? k y , 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
2 2

x

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 ?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)

x

y

O

F(t)

(ii)

Given an external driving force F ( t ) ? F1 co s(? 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 ) ? A1 co s( ? 1 t ? ? 1 ) where the amplitude A1 and phase ?1 are to be determined. (4 points) 给 定 一 个 外 驱 动 力 F ( t ) ? F1 co s(? 1t ) , 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

x ( t ) ? A c o ? (1 t? ? 1 解它，其中振幅 A1 和位相?1 待定。(4 分) s ) 1
(iii) Given two driving forces F ( t ) ? F1 co s( ? 1t ) ? F 2 co s( ? 2 t ) , find x ( t ) . (4 points) 给定两个外驱动力 F ( t ) ? F1 co s( ? 1t ) ? F 2 co s( ? 2 t ) , 求 x ( t ) 。 (4 分) (iv) The driving force comes from thermal noise, which can be described as a sum of many harmonic driving forces Fth erm a l ( t ) ? ? F n co s( ? n t ) in the entire frequency range. Find x ( t ) under the thermal noise driving force. (2 points)
n

1

Pan Pearl River Delta Physics Olympiad

2007 年泛珠三角及中华名校物理竞赛

Q3 The Lorentz-Lorenz Relation (22 points) 题 3

The dielectric constant ? ( ? ) of a dielectric medium is given by the so called Lorentz-Lorenz Relation

? (? ) ? 1 ? (? ) ? n

??0 ?

1 3

K (? ) ,

where n is a number and K is a material-related constant that depends explicitly on the frequency ? 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

? (? ) ? 1 ? (? ) ? n

??0 ?

1 3

K ( ? ) ，其中 K 与所加电场的频率? 以及介质的物质

R

E ( t ) ? A co s(? t ) . Find the induced dipole moment of the atom. (6

2

Pan Pearl River Delta Physics Olympiad

2007 年泛珠三角及中华名校物理竞赛

3

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