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）︰2014/04/15 考試時限（分鐘）：120 試題 : 1. A small circular hole of radius R has been cut in the middle of an infinite, flat, non-conducting surface that has uniform charge density σ. A Z axis, with its origin at the hole's center, is perpendicular to the surface. In unitvector notation, what is the electric field at point P at height z? https://i.imgur.com/2UyDscK.png \begin{tikzpicture} \filldraw[gray] (4.5, 1.5) -- (-1.5, 1.5) -- (-4.5, -1.5) -- (1.5, -1.5) -- cycle; \filldraw[gray, fill=white] (0, 0) ellipse(0.6 and 0.3); \draw[->] (0, 0) -- node[midway, above]{$R$} (0.6, 0); \draw (0, -0.3) -- (0, 2) node[anchor=south]{$z$}; \filldraw (0, 1.7) circle(1pt) node[anchor=east]{$P$}; \end{tikzpicture} 2. What are the potential differences (a) $V_A-V_B$ (b) $V_B-V_C$ (c) $V_C-V_D$ and (d) $V_A-V_C$? https://i.imgur.com/Up0paRd.png \begin{circuitikz} \draw (0, 0) to (3, 0) node[anchor=north west]{$D$} to [R, l_=$R_2$](5, 2) node[anchor=west]{$B$} to [R, l_=$R_1$](3, 4) node[anchor=south west]{$A$} to (0, 4) to (0, 2) to [battery1, l=$\uparrow\mathscr E$](0, 0); \draw (3, 0) to [R, l=$R_1$](1, 2) node[anchor=east]{$C$} to [R, l=$R_2$](3, 4); \draw (1, 2) to [R, l=$R_3$](5, 2); \end{circuitikz} 3. A cross section of a long cylindrical conductor of radius a containing a long cylindrical hole of radius b. The central axes of the cylinder and hole are parallel and are distance d apart; current i is uniformly distributed over the tinted area. (a) What is the magnitude of the magnetic field at the cen- ter of the hole? (b) If a long wire is allocated at b which is parallel to the cylinder, what is the force between the cylinder and the wire? Suppose the length of the wire is $\ell$ and the current is i. https://i.imgur.com/aJfv8yR.png \begin{tikzpicture} \filldraw[gray] (0, 0) circle(2); \filldraw (0, 0) circle(1pt); \draw[->] (0, 0) -- node[midway, above right]{$a$} ( {-sqrt(2)}, {sqrt(2)}); \filldraw[white] (1.4, 0) circle(0.4); \filldraw (1.4, 0) circle(1pt) node[anchor=south east]{$b$}; \draw[->] (1.4, 0) -- ({1.4+sqrt(2)/5}, {sqrt(2)/5}); \draw (0, 0) -- (0, -2.6); \draw (1.4, 0) -- (1.4, -2.6); \draw[<->] (0, -2.3) -- node[midway, below]{$d$} (1.4, -2.3); \end{tikzpicture} 4. A rectangular coil of N turns and of length a and width b is rotated at fre- quency f in a uniform magnetic field. The coil is connected to co-rotating cylinders, against which metal brushes slides to make contact. Show that the emf induced in the coil is given (as a function of time t) by ε = 2πfNabB sin(2πft) https://i.imgur.com/izxWvxX.png \begin{circuitikz} \foreach \x in {0, 0.5, ..., 5}{ \foreach \y in {-1, -0.5, 0, 0.5, 1}{ \draw (\x, \y) node{$\color{gray}{\times}$}; } } \draw (0, 1) node[anchor=south east]{$\vec B$}; \draw[thick] (5.5, 0.05) -- (4.5, 0.05) -- (4.5, 0.55) -- (0.55, 0.55) -- (0.55, -0.55) -- (4.5, -0.55) -- (4.5, -0.05) -- (6, -0.05); \draw (0.55, -0.6) -- (0.55, -1.5); \draw (4.5, -0.6) -- (4.5, -1.5); \draw[<->] (0.55, -1.4) -- node[midway, below]{$a$} (4.5, -1.4); \draw (-0.5, 0.55) -- (0.5, 0.55); \draw (-0.5, -0.55) -- (0.5, -0.55); \draw[<->] (-0.4, 0.55) -- node[midway, left]{$b$} (-0.4, -0.55); \draw (6, 0) to (6, 0.9) to (6.5, 0.9) to (6.5, -1) to (8, -1) to [R, l=$R$] (8, 1) to (5.5, 1) to (5.5, 0); \draw[fill=white] (5.4, -0.3) rectangle(5.6, 0.3); \draw[very thin] (5.5, 0.3) -- (4.5, 1.3) -- (4.3, 1.3) node[anchor=east]{ Sliding contacts}; \draw[fill=gray] (5.42, 0.3) rectangle(5.58, 0.4); \filldraw (5.4, 0.05) circle(1pt); \draw[fill=white] (5.9, -0.3) rectangle(6.1, 0.3); \draw[fill=gray] (5.92, 0.3) rectangle(6.08, 0.4); \filldraw (5.9, -0.05) circle(1pt); \draw[very thin, dashed] (-1.5, 0) -- (6.5, 0); \draw[-stealth] (-1, 3pt) arc[ start angle=30, end angle=330, x radius=3pt, y radius=6pt]; \end{circuitikz} -- 雖然我把render的png檔也放出來 由於圖片可能會失效 這裡放上純文字也能瀏覽的tikz code 讀者可以自行用tex compiler編譯出類似的結果 對了tikz有夠難用[rdrrC] -- 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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