Pan Pearl River Delta Physics Olympiad 2009 2009年泛珠三角及中华名校物理奥林匹克邀请赛 Part-1 (Total 5 Problems) 卷-1（共5题）

(9:00 am – 12:00 pm, 02-04-2009)

Math hints数学提示:

n

∫xdx=

dx1n+1

x+C; ∫=lnx+C. n+1x

Q.1 (8 points) 题1（8分）

Two large parallel square conductor plates C1 and C2 of side length a are separated by two types of media. The cross section is shown in the figure. The distance between the plates is d. Medium-2 is a rectangular insulator slab with dielectric constant ε2, thickness d/2, and side lengths a and a/2, respectively. The rest of the space between the

C1

plates is filled with medium-1 with conductivity σ1 and dielectric constant ε1. Find the resistance and capacitance between the two plates. 两块边长为 a的正方形平行导电板 C1 和 C2 之间夹着两种介C2质，其截面如图所示。导电板间的距离为 d。介质-2 为一厚度

为d/2，边长分别为a 和 a/2的绝缘长方形板，其介电常数为ε2。导电板之间其余的空间充满了介质-1，其介电常数为 ε1，电导率为σ1 。求两导电板之间的电阻和电容。

Q.2 (10 points) 题2（10分）

A galaxy is made of many stars revolving around its center on circular orbits. One way

scientists study galaxies is to measure the relation between the revolving speed v of a star in the galaxy and its distance r from the center. Expressing the relation as v∝rn, scientists pay special attention to the exponential power n.

星系由很多绕中心作圆形轨道运行的恒星组成。科学家研究星系的一个方法是测量恒星在星系中的运行速度v和离星系中心的距离r。用v∝rn这样的关系来表达，科学家们特别关心指数n。

(a) If the gravity force on the star under investigation is mainly from a massive black hole

at the galaxy center, derive the value of n. (4 points)

若作用于恒星的引力主要来自星系中心的巨型黑洞，求n的值。（4分） (b) Suppose there is no black hole at the center and the stars in the galaxy are evenly

distributed such that the mass of the galaxy is evenly distributed in a flat disk. Consider a star in the disk at distance r from the center. The total gravity force on the star from the stars enclosed by its circular orbit can be treated as if they are concentrated at the center. The total gravity force on the star from the stars outside the orbit is zero. (This is like the case of a uniform density sphere.) Derive the value of n. (4 points)

若星系中心无黑洞，星系的恒星分布均匀，整个星系可当作是一个质量均匀分布的扁圆盘。对于一个离中心为r的恒星来讲，来自轨道内的恒星的引力与把它们集中在星系中心的一样，来自轨道外的恒星的引力合力为零。（类似于均匀质量球的情形）求n的值。（4分）

(c) What scientists found from real galaxies is that for most of them, n is larger than 1.

Give your brief explanation to this scientific puzzle. (2 points) 科学家们发现，很多实际星系的n >1。请简单解释。（2分）