作者xavier13540 (柊 四千)
看板NTU-Exam
标题[试题] 102-2 黄晖理 普通物理学甲下 第四次小考
时间Tue Apr 15 14:49:21 2025
课程名称︰普通物理学甲下
课程性质︰数学系大一必带
课程教师︰黄晖理
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2014/05/27
考试时限（分钟）：60
试题 :
1. Figure shows an optical fiber in which a central plastic core of index of re-
fraction $n_1$ is surrounded by a plastic sheath of index of refraction $n_2$
(Note: $n_1 > n_2$). Light can travel along different paths within the cen-
tral core, leading to different travel times through the fiber. This causes
an initially short pulse of light to spread as it travels along the fiber,
resulting in information loss. Consider light that travels directly along the
central axis of the fiber and light that is repeatedly reflected at the cri-
tical angle along the core-sheath interface, reflecting from side to side as
it travels down the central core. If the fiber length is L, what is the dif-
ference in the travel times along these two routes? (25 pt.)
https://i.imgur.com/oZNe4Uy.png
\begin{tikzpicture}
\filldraw[lightgray] (0, 0) ellipse(.3 and .7);
\filldraw[lightgray] (0, -.7) rectangle (3, .7);
\filldraw[lightgray] (3, 0) ellipse(.3 and .7);
\filldraw[gray] (0, -.6) rectangle(3.1, .6);
\filldraw[darkgray] (0, 0) ellipse(.2 and .6);
\draw[fill=gray] (3.1, 0) ellipse(.2 and .6);
\draw (0, .6) arc[
start angle=90, end angle=270, x radius=.2, y radius=.6];
\draw[white] (0, -.6) arc[
start angle=-90, end angle=90, x radius=.2, y radius=.6];
\draw (0, .6) -- (3.1, .6);
\draw (0, -.6) -- (3.1, -.6);
\draw[very thin, dashed] (-1, 0) -- (3.5, 0);
\draw[-stealth] (-1, -.7) -- (0, 0) -- (1.5, .6) -- (3.25, -.1);
\draw (1, 0) node[anchor=north east]{$n_1$};
\draw[very thin] (1.35, .65) -- (1, 1) node[anchor=south east]{$n_2$};
\end{tikzpicture}
2. First-order reflection from the reflection planes shown occurs when an x-ray
beam of wavelength λ makes an angle θ with the top face of the crystal.
What is the unit cell size $a_0$? (25 pt.)
https://i.imgur.com/0pIyzX9.png
\begin{tikzpicture}
\foreach \x in {0, 1, 2, 3}{
\foreach \y in {0, 1, 2}{
\filldraw (\x, \y) circle(1pt);
}
}
\draw[<->, >=stealth] (0, 1) -- node[midway, fill=white]{$a_0$} (0, 2);
\draw[<->, >=stealth] (0, 1) -- node[midway, fill=white]{$a_0$} (1, 1);
\draw (0, 1) -- (1, 0);
\draw (0, 2) -- (2, 0);
\draw (1, 2) -- (3, 0);
\draw (2, 2) -- (3, 1);
\draw (-.5, 2) -- (3.5, 2);
\draw[-stealth] (1, 3) -- node[midway, above right]{X rays} (1.5, 2);
\draw (1.2, 2) arc[start angle=180, end angle={180-atan(2)}, radius=.3];
\draw (1.1, {2+.2*(sqrt(5)-1)}) node{$\theta$};
\end{tikzpicture}
3. (a) Please derive the Lorentz transformation between S and S' system (15
pt.), (b) and the velocity addition under the Lorentz transformation between
u and u'. (10 pt.) (Note: The four-coordinate (t', x', y', z') detected from
S', and the four-coordinate (t, x, y, z) at S. Frame S' has velocity (v, 0,
0) relative to frame S and the speed of light is c. Assume the origin of S
and S' coincide at t = t' = 0.)
https://i.imgur.com/jVdUedn.png
\begin{tikzpicture}
\draw (0, 0) -- (.5, 0) node[anchor=west]{$x$};
\draw (0, 0) -- (0, 3) node[anchor=south]{$y$}
node[anchor=north west]{$S$};
\draw (1, 0) -- (3, 0) node[anchor=west]{$x'$};
\draw (1, 0) -- (1, 3) node[anchor=south]{$y'$}
node[anchor=north west]{$S'$};
\draw[thick, -stealth] (1, 2) -- node[midway, below]{$\vec v$} (1.5, 2);
\filldraw (2, 1) circle(2pt);
\draw (2, 1) -- (2.3, 1.6) -- (2.5, 1.6) node[anchor=west]{Particle};
\draw[thick, -stealth] (2, 1) -- (3, 1)
node[anchor=south west]{$\vec u'$ as measured from $S'$}
node[anchor=north west]{$\vec u$ as measured from $S$};
\end{tikzpicture}
4. Figure shows the concept of Compton scattering. (a) What prompts Compton to
do that experiment? (10 pt.) (b) Show the Compton shift is $\Delta\lambda
= \frac h{mc}(1-\cos\phi)$. (15 pt.)
https://i.imgur.com/kdnaeXB.png
https://i.imgur.com/KzIaSaO.png
\begin{tikzpicture}
\draw[very thin] (-3, 0) -- (3, 0) node[anchor=west]{$x$};
\draw[very thin] (0, -2) -- (0, 2) node[anchor=south]{$y$};
\filldraw (0, 0) circle(1pt) node[anchor=south west]{Electron}
node[anchor=north west]{$\vec v=0$};
\draw[domain=0:2, samples=101, variable=\x] plot(\x-3, {sin(360*\x)/5});
\draw[-stealth] (-3, 0) -- node[midway, above]{X ray} (-.8, 0)
node[midway, below]{$\lambda$};
\draw (0, -2) node[anchor=north]{
An x ray heads toward a target electron.};
\end{tikzpicture}
\begin{tikzpicture}
\draw[very thin] (-3, 0) -- (3, 0) node[anchor=west]{$x$};
\draw[very thin] (0, -2) -- (0, 2) node[anchor=south]{$y$};
\draw[dashed] (0, 0) -- (.6, -.8);
\filldraw (.6, -.8) circle(1pt) node[anchor=south west]{Electron};
\draw[-stealth] (.6, -.8) -- (.9, -1.2) node[anchor=north east]{$\vec v$};
\draw (.3, 0) arc[start angle=0, end angle={asin(-.8)}, radius=.3];
\draw (.4, -.2) node{$\theta$};
\draw[dashed, -stealth] (0, 0) -- node[midway, above]{X ray} (3, .875)
node[anchor=south]{$\lambda'$};
\draw[domain=1:3, samples=101, variable=\t] plot(
{(120*\t-7*sin(360*\t))/125}, {(35*\t+24*sin(360*\t))/125});
\draw (.9, 0) arc[start angle=0, end angle={asin(.28)}, radius=.9];
\draw (1.12, .16) node{$\phi$};
\draw (0, -2) node[anchor=north, align=left]{
Intermediate energy is transferred.\\
Or it can scatter at some intermediate angle $\phi$.};
\end{tikzpicture}
--
第01话 似乎在课堂上听过的样子 第02话 那真是太令人绝望了
第03话 已经没什麽好期望了 第04话 被当、21都是存在的
第05话 怎麽可能会all pass 第06话 这考卷绝对有问题啊
第07话 你能面对真正的分数吗 第08话 我，真是个笨蛋
第09话 这样成绩，教授绝不会让我过的 第10话 再也不依靠考古题
第11话 最後留下的补考 第12话 我最爱的学分
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※ 编辑: xavier13540 (36.230.44.40 台湾), 04/15/2025 14:55:39