2015 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 2015
(c) Consider the case that the lens is exactly aligned with the source (? = 0). The image of S appears to be a ring known as an Einstein ring. Derive the expression for the angular radius ?E of the Einstein ring. (2 points)
考虑透镜与光源对準的情况（? = 0）。 S的影象呈环形，称为爱因斯坦环。试推导爱因斯坦环的角半径?E的表达式。 （2分）
(d) Calculate the Einstein radius for the following typical values: 试以下列的典型值，计算爱因斯坦半径： M = 0.3 solar mass, Ds = 10 kpc. Dl = 3 kpc.
Give your answer in milli-arc-seconds. You may use the following constants: 请以milli-arc-seconds表达你的答案。您可以使用以下参量：
G = 6.67 ? 10?11 Nm2kg?2, 1 solar mass = 1.99 ? 1030 kg, c = 3 ? 108 ms?1, 1 kpc = 3.09 ? 1019 m, 1 radian = 206265 arc seconds. (1 point) （1分）
(e) When the lens and the source are not exactly aligned, there will be two images of S. It is convenient to express the angles ? and ? in multiples of the Einstein radius ?E. Hence we define u???E and y???E. Derive the expressions for the angular positions y of the two
images in terms of u. (2 points)
当透镜和光源不完全对齐时，S将有两个影像。为方便起见，我们以爱因斯坦半径?E的倍数表达角? 和?。因此我们定义u???E和y???E。试推导两个影像的角位置y，以
u表示。（2分）
(f) To study the effect of the finite size of star S, we introduce Cartesian coordinates on the plane normal to ES and through S, with the y axis lying in the plane containing E, L and S. Consider the corners (0, u + ?) and (?, u) of a square on the surface of star S (? << u). Calculate the coordinates of the two corners of the two images when viewed from Earth. (2 points)
为研究星S有限大小的影响，我们在垂直于ES和通过S的平面上，引入一平面直角坐标，其中y轴位于包含E，L和S的平面中。考虑星S表面上一个正方形的角(0, u + ?) and (?, u) (? << u)。试计算从地球观察时，这两个影像的两个角的坐标。（2分） (g) Calculate the areal magnifications of the two images of star S in terms of u. Following the practice in astronomical observations, give your answer in absolute values. (2 points)
试计算星S的两个影像的面积放大率，请以u表达。按照天文观测的习惯，请以绝对值为答案。（2分）
(h) In practice, since the images cannot be resolved, astronomers measure the sum of the magnifications of the two images. Derive the expression for the total magnification. Describe its behavior when star S is remote (u approaches infinity) and when S approaches perfect alignment with L and E (u approaches 0). (3 points)
实际上，由于影象不易分辨，天文学家只测量两个影像的放大率的总和。试推导总放大率的表达式。试描述星S在远处时（u趋近无穷大），及星S趋近对準L与E时(u 趋近0)，总放大率的行为。（3分）
(i) A planet P of star L has mass m and is located in the plane of E, L and S at the same distance Dl from Earth. EP and EL makes an angle ?p. Derive an equation for the angle ? taking into account the gravitational lensing effects of both star L and planet P. Expressions in the
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2015 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 2015
equation should be written in terms of the parameters Ds, Dl, G, M, c, ?, m and ?p, assuming that all angles are small. Simplify the equation by introducing the mass ratio qrescaled positions up??mM and the
?p?E, u???E, y???E. (3 points)
星L旁有一行星P位于E、L和S的平面上，其质量为m，与地球距离跟星L同为Dl，EP与EL间角度为?p 。考虑到星L和行星P两者的引力透镜作用，试推导角?的方程式，式中的表达式应以Ds、Dl、G、 M、 c、?、 m和?p表达。可假设所有角度都很小。引入质量比q?mM和重整位置up??p?E, u???E, y???E，以简化方程式。（3分）
(j) In typical exoplanet detections, there is a motion of star S relative to star L. As star S approaches the closest distance to star L and moves away, u decreases with time to a minimum value u0 and increases again. By plotting the magnification of the image of star S versus time, one observes a smooth and relatively broad peak in the magnification curve due to gravitational lensing by star L. In addition, one can observe a side peak due to the presence of the planet. For q << 1, estimate the width of this side peak, that is, the range of u in which the side peak is significant. (1 point)
在典型的系外行星检测中，星S对于星L有相对运动。星S趋近星L至最短距离，然后离开，过程中u随时间降到最小值u0然后再增加。把星S影像的放大率与时间的关系绘成图表，放大率曲线上可以看到一个平滑和较宽的主峰，是由星L的引力透镜作用形成的。另外，我们可以观察到一个侧峰，是由行星形成的。对于q << 1，试估计这个侧峰的宽度，也就是可以显著看到侧峰的u数值范围。 （1分）
(k) For q << 1, consider the situation that light rays pass very near to planet P, so that the gravitational lensing by star L becomes relatively insignificant. Calculate the position of star S where the total magnification of its image diverges, and the behavior of the total magnification in the neighborhood of this location. (3 points)
当q << 1时，考虑光线非常靠近行星P的情况，在这情况下星L的引力透镜作用相对很弱。试计算当星S图像的总放大率发散时星S的位置，和这位置附近总放大率的行为。（3分）
2. Cosmic Gravitational Waves (28 points) 宇宙引力波（28分）
In March 2014, scientists operating gravitational wave detectors in the South Pole claimed that they found evidences of gravitational waves originated from the early universe in the cosmic microwave background radiation. While the evidence is still being debated, it is interesting to understand how gravitational waves interact with electromagnetic (EM) waves. To approach this issue, we start by considering how molecules scatter EM waves.
2014年3月，操作南极引力波探测器的科学家，声称在宇宙微波背景辐射中，发现来自早期宇宙的引力波的证据。虽然证据还存在争议，但了解引力波如何作用于电磁（EM）波是一个有趣的课题。为了处理这个问题，我们首先考虑分子是如何散射电磁波。
(a) An oscillating electric dipole consists of charges oscillating at an angular frequency ?. Specifically, the charges are Q(t) = ?Q0cos?t, located at (x, y, z) = (0, 0, ?s) respectively. What is the current between them? (1 point)
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2015 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 2015
一个振动的电偶极子，包含以角频率?振动的电荷。具体来说，电荷分别为Q(t) = ?Q0cos?t, 位于(x, y, z) = (0, 0, ?s)。它们之间的电流是什么？ （1分）
(b) In spherical coordinates, we denote the components of the magnetic field as Br, B? and B?, as shown in figure (A). Calculate B?(r, ?, t) according to Biot-Savart’s law at time t and distance r from the origin making an angle ? with the z axis. Note that due to the finite speed of light c, the magnetic field at a distant location is due to the time-changing current at an earlier instant. Hence the retarded magnetic field takes the form B(r,t)?B0(r)cos(?t?kr??), where
k??c is the wavenumber, and ? is the phase shift. Express your answer in terms of the
magnitude of the dipole moment p ? 2Qs in the limit s approaches 0. Below, your answer to this part will be denoted as BBS(r, ?, t). (3 points)
在球坐标中，我们以Br, B? 和B?,表示磁场的分量，如图（A）所示。根据毕奥 - 萨伐尔定律，试计算磁场B?(r, ?, t)，其中r为位置与原点的距离，?为位置与z轴形成的角，t为时间。注意，由于光以有限速率c传播，在远处的磁场是源于某一较早时刻的电流（电流随时间变化）。因此，延迟磁场的形式为B(r,t)?B0(r)cos(?t?kr??)，其中
k??c是波数，而?是相移。答案请以偶极矩p ? 2Qs表达（取s趋于0的极限）。下
面，你在这部的答案将被表示为BBS(r, ?, t)。（3分）
(A)
z Br (B)
dr
B? ? r ? B? y
r ? d? x
(c) However, Biot-Savart’s law is only applicable to steady state currents. It is incomplete even after including the retarded nature of the oscillating current. By considering the wave nature of the magnetic field, the complete expression of the magnetic field is given by
但是，毕奥 - 萨伐尔定律只适用于稳态电流。甚至考虑了振动电流的滞后性质后，它还是不完整的。通过考虑磁场的波动性，完整的磁场表达式是
B(r,?,t)?BBS(r,?,t)?Bwave(r,?,t),
where其中
B
wave(r,?,t)??r??dl?r?It?. ?????4?c??dt?c??r?0??dDerive an expression for the B? component of Bwave at (r, ?, t). (3 points)
试推导Bwave 在 (r, ?, t)的B? 分量的表达式。（3分）
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2015 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 2015
(d) Compare the amplitudes of BBS and Bwave at large distance r. Derive the condition of r such that BBS becomes negligible when compared with Bwave. (2 points)
比较BBS和Bwave在距离r很大时的幅度。试推导BBS相比Bwave变得微不足道时，关于r的条件。（2分）
(e) At large distance r, the electric field at (r, ?, t) is mainly due to the electromagnetic induction by the magnetic field Bwave. By considering the electromotive force along the circuit shown in figure (B), derive the relation between
?B??t?E??r and
?B??t. Here,
?E??r is known as the partial
derivative of E? with respect to r, meaning that other variables such as ? and t are considered fixed. Similarly,
is the partial derivative of B? with respect to t, with other variables
such as r and ? being fixed. You may assume that only the E? component of the electric field is significant at large distance r. (3 points)
在距离r很大时，在(r, ?, t)的电场主要是源于Bwave的电磁感应。通过考虑沿著图（B）中闭路的电动势，试推导
?E??r与
?B??t之间的关系。这里，
?E??r被称为E?相对于r的偏是B?相对于t的偏导数，
导数，意味着其他变量如? 和t被假定为固定的。同样地，
?B??t当中假定其他变量如r和? 为固定的。你可以假设在距离r很大时，电场仅有E?分量是显著的。（3分）
(f) At large distance r, the electric field is given by E?(r,?,t)points)
在距离r很大时，电场为E?(r,?,t)?A(?)rcos(?t?kr)?A(?)rcos(?t?kr). Find A(?). (2
。试找出A(?)。（2分）
(g) The magnitude and direction of the power per unit area of the EM wave are given by the Poynting vector. Calculate the time-averaged power per unit area at large distance r. This will be denoted as the radiation intensity I(r). (3 points)
电磁波每单位面积传播功率的大小和方向，是由Poynting矢量给定的。试计算在距离r很大时，每单位面积按时间平均的传播功率。这将被表示为辐射强度I(r)。（3分） (h) When an EM wave is incident on a molecule, its electric field E will drive the molecule into an oscillating dipole moment given by p = ?E, where ? is the polarizability of the molecule. In turn, the oscillating dipole will radiate power. This is called a scattering process. Consider an EM wave incident from the x direction, given by Ei = Ex0cos(?t ? kx). If Ex0 is polarized at an angle ?x with the z axis, calculate:
当电磁波射向一分子时，其电场E会使该分子产生振动偶极矩p = ?E，其中?是该分子的极化度。随之振动偶极子会辐射功率。这就是所谓的散射过程。考虑电磁波从x方向入射，由Ei = Ex0cos(?t ? kx)给出。若Ex0的偏振方向与z轴成角度?x，试计算： (h1) the intensity Ix(r) of the radiation scattered to the z direction, 散射至z方向的辐射强度Ix(r)，
(h2) the electric field polarization of the scattered wave along that direction, 沿该方向的散射波的电场偏振方向，
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2015 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 2015
(h3) the intensity ?Ix(r)? of the radiation scattered to the z direction for an unpolarized incident beam (that is, the polarization angle ?x has a uniform distribution). (3 points) 非偏振入射光束（即偏振角?x均匀分布）散射至z方向的辐射强度?Ix(r)?。（3分） (i) Next, consider an EM wave incident from the y direction, given by Ei = Ey0cos(?t ? ky). If Ey0 is polarized at an angle ?y with the z axis, calculate:
接下来，考虑电磁波从y方向入射，由Ei = Ey0cos(?t ? ky)给出。若Ey0的偏振方向与z轴成角度?y，试计算：
(i1) the electric field polarization of the scattered wave along the z direction, 沿z方向的散射波的电场偏振方向，
(i2) the intensity ?Iy(r)? of the radiation scattered to the z direction for an unpolarized incident beam (that is, the polarization angle ?y has a uniform distribution). (2 points)
非偏振入射光束（即偏振角?y均匀分布）散射至z方向的辐射强度?Iy(r)?。（2分） (j) During the rapid expansion of the early universe, gravitational waves are formed. They consist of quadrupolar temperature oscillations，meaning that the directions of the maxima and minima of the oscillations are separated by an angle of ?/2. Hence to analyze their effects on EM waves, we consider two incoherent incident beams of EM waves of the same frequency ?/2?, one from the x direction and the other from the y direction. The amplitudes of their electric fields are Ex0 and Ey0 respectively. Suppose the EM radiations in the x and y directions correspond to temperatures T + ?T and T respectively (?T << T and is positive). What is the ratio
Ix(r)Iy(r)? (1 point)
早期宇宙的迅速膨胀，形成引力波。它引起温度的振动，呈四偶极分布。这意味着振动的最大值和最小值的方向以?/2角度分开。因此，要分析它们对电磁波的影响，我们考虑两束频率同为?/2?的非相干入射光，一束来自x方向，另一束则来自y方向，其电场的幅度分别是Ex0和Ey0。假设在x和y方向的电磁辐射分别对应于温度T + ?T 和T（?T << T，且是正的）。比例
Ix(r)Iy(r)是什么？ （1分）
(k) The degree of polarization of the scattered radiation is given by下式是散射辐射的偏振度
??
Ix(r)?Iy(rIx(r)?Iy(r.
Calculate ?. What is the direction of the electric field polarization in the scattered wave? (2 points)
试计算?。散射辐射中电场的偏振方向是什么？（2分）
《THE END完》
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2015年泛珠三角及中华名校物理奥林匹克邀请赛试题及答案
