(6) [10 points] Consider a closed box of ideal gas divided into two parts by a gate. The left part has a volume of V and contains N particles in thermal equilibrium, while the right part has a volume of  2V and contains zero particles (i.e., a vacuum). The number of the accessible microstates of the gas is $\Omega{1}$. If the gate is suddenly removed, the gas fills the entire box and reaches a new equilibrium with a different number of the accessible microstates $\Omega{2}$. What is the ratio $\Omega{2}/\Omega{1}$?

(A) 1

(B) $2\ln N$

(C) $N^{2}$

(D) $3^{N}$

(E) N

(7) [10 points] The Bohr radius is the characteristic size of a hydrogen atom, which can be expressed by a combination of fundamental constants. Choose the correct expression for the Bohr radius. Here, $\hbar$ is the reduced Planck constant, k is the Coulomb constant, $m_{e}$ is the electron mass, $e$ is the electron charge, and $G$ is the gravitational constant.(A) $\frac{\hbar^{2}}{km_{e}e^{2}}$(B) $\frac{G\hbar^{2}m_{e}}{k^{2}e^{2}}$(C) $\frac{\hbar^{2}}{Gm_{e}^{3}e^{2}}$(D) $\frac{\hbar^{2}}{m_{e}^{2}e^{2}\sqrt{Gk}}$(E) $\sqrt{\frac{\hbar^{2}}{Gm_{e}e^{2}}}$

(8) [10 points] Consider a positively charged particle with charge q and mass m initially at rest in a uniform electric field of $E{0}$ along the +y-axis and a uniform magnetic field $B{0}$ along the +z-axis (i.e., the electric and magnetic fields are perpendicular to each other). Once released, the particle undergoes a cycloid motion under the Lorentz force. What is the maximum displacement of the particle along the y-axis?(A) $\frac{mE{0}}{qB{0}^{2}}$(B) $\frac{2mE{0}}{qB{0}^{2}}$(C) $\frac{mE{0}}{2qB{0}^{2}}$(D) $\frac{mE{0}}{qB{0}^{3}}$(E) $\frac{mE{0}}{\sqrt{2}qB{0}^{2}}$

(9) [10 points] A charged particle with charge q and mass m is initially at the origin with a initial velocity $v=v_{0}\hat{z}$ along the z-axis. The particle is in a nonuniform magnetic field: $B(x,y)=-y\hat{x}+x\hat{y}$. What is the work done by the magnetic field along the trajectory when the particle moves a distance L along the z direction?(A) $mv_{0}^{2}/2$(B) $mv_{0}^{2}/L$(C) 0(D) $mv_{0}^{2}/q$(E) $v_{0}^{2}L/q$

(10) [10 points] An electron is confined in an infinite 1D potential well. Suppose the infinite potential well is suddenly lowered to a finite height $U_{0}$ that is still significantly larger than the initial ground-state energy. How would the de Broglie wavelength λ and kinetic energy of the electron E change accordingly?(A) Both increase(B) Both decrease(C) λ increases and E decreases(D) λ decreases and E increases(E) λ increases and E remains unchanged

1. 游標卡尺無法直接量測的是 (A)深度 (B)錐度 (C)外徑 (D)內徑。

2. 1/20 公制游標卡尺其精度為 (A)0.02mm (B)0.03mm (C)0.05mm (D)0.01mm。

3. 1/20 公厘精度的游標卡尺，其原理為 (A)本尺 20 公厘，游尺(副尺)30 等分 (B)本尺 19 公厘，游尺 20 等分 (C)本尺 49 公厘，游尺 50 等分 (D) 本尺 50 公厘，游尺 49 等分。

4. 游標卡尺的內測顎是用於 (A)量內徑 (B)量外徑 (C)當圓規用 (D) 當劃線針用。

5. 游標卡尺量測外徑時，工件應靠在 (A)離本尺愈遠愈好 (B)離本尺愈 近愈好 (C)任何位置都一樣 (D)依工件形狀而定。

6. 游標卡尺，本尺刻度是 1 公厘，游尺(副尺)為 19/20 公厘，其精度為若 干公厘？ (A)0.05mm (B)0.04mm (C)0.03mm (D)0.02mm。

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