作者xavier13540 (柊 四千)
看板NTU-Exam
標題[試題] 102-2 黃暉理 普通物理學甲下 第四次小考
時間Tue Apr 15 14:49:21 2025
課程名稱︰普通物理學甲下
課程性質︰數學系大一必帶
課程教師︰黃暉理
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/05/27
考試時限（分鐘）：60
試題 :
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話 似乎在課堂上聽過的樣子 第02話 那真是太令人絕望了
第03話 已經沒什麼好期望了 第04話 被當、21都是存在的
第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊
第07話 你能面對真正的分數嗎 第08話 我，真是個笨蛋
第09話 這樣成績，教授絕不會讓我過的 第10話 再也不依靠考古題
第11話 最後留下的補考 第12話 我最愛的學分
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※ 編輯: xavier13540 (36.230.44.40 臺灣), 04/15/2025 14:55:39