12. Consider a plane electromagnetic wave expressed by its phasor form of $\mathbf{E} = \mathbf{E}_0 e^{i(k_x x + k_y y + k_z z)}$. The expression should satisfy the Helmholtz equation: $\nabla^2 \mathbf{E} + \omega^2 \mu \epsilon \mathbf{E} = 0$. As a result, which one of the following relations must be true?
(A) $k_x + k_y + k_z = \omega \sqrt{\mu \epsilon}$
(B) $k_x^2 + k_y^2 + k_z^2 = \omega^2 \sqrt{\mu \epsilon}$
(C) $k_x^2 + k_y^2 + k_z^2 = \omega^2 \mu \epsilon$
(D) None of the above.

13. What is the short-circuit impedance measured at the input terminals of an infinitely long transmission line with the parameters of $(R = 45 \times 10^{-3}\ \Omega/\mathrm{m}, L = 27 \times 10^{-8}\ \mathrm{H/m}, G = 5 \times 10^{-5}\ \mathrm{S/m}, C = 3 \times 10^{-10}\ \mathrm{F/m})$ at $5\ \mathrm{GHz}$? (A) $30\ \Omega$ (B) $50\ \Omega$ (C) $75\ \Omega$ (D) None of the above. 

14. An 1-GHz voltage source $(V_{s})$ is applied to a short-circuit quarter-wavelength lossless transmission line as the figure shown below. The characteristic impedance of the transmission line is $20\Omega$. Please evaluate the current of $I_{x}$. (A) $0.25\mathrm{A}$ (B) $0.25\angle 90^{\circ}\mathrm{A}$ (C) $0.25\angle 270^{\circ}\mathrm{A}$ (D) None of the above.

15. Please select the closest dB value of the reflection coefficient of a $75 - \Omega$ resistive load in a $50 - \Omega$ system $Z_{0}$. (A) $-6.9\mathrm{dB}$ (B) $-13.9\mathrm{dB}$ (C) $-1.39\mathrm{dB}$ (D) None of the above.

16. A 5-V dc voltage with $25 - \Omega$ source resistance is applied at $t = 0$ to the input terminal of a $25 - \Omega$ terminated lossless coaxial of $30\mathrm{m}$ length as the figure below. The initial condition of voltage and current of the cable are $0\mathrm{V}$ and $0\mathrm{A}$. The dielectric constant $(\varepsilon_{\mathrm{r}})$ of the insulating material in the cable is 4, and the characteristic impedance of the cable $(Z_0)$ is $50\Omega$. Please select the closest voltage value at $V_{L}$ at 1s. (A) $5\mathrm{V}$ (B) $2.5\mathrm{V}$ (C) $7.5\mathrm{V}$ (D) None of the above.

17. A guided TE mode of frequency $15\mathrm{GHz}$ is excited and propagates in metallic waveguide filled with polyethylene, of which $\varepsilon_{r} = 2.25$. The waveguide is designed with dimensions $a = 2\mathrm{cm}$ (x axis) and $b = 1\mathrm{cm}$ (y axis). If the x component of its electric field is given by $$ E _ {x} = - 5 0 \cos (5 0 \pi x) \sin (1 0 0 \pi y) e ^ {- j \beta z} \mathrm {V / m} $$ Which of the following statements is NOT correct? (A) The mode number is $\mathrm{TE}_{11}$. (B) The guide wavelength is $1.33 \times 10^{-2} \mathrm{~m}$. (C) The cutoff frequency is smaller than $12\mathrm{GHz}$. (D) $E_{z}$ is zero. 

18. Which of the following modes is the dominant mode in an air-filled metallic rectangular resonator for $a > b > d$? where $a, b,$ and $d$ are the dimensions in the x-, y-, and z-directions in respective, and the mode number are $m, n,$ and $p$ accordingly. Hint: Here we choose z-axis as the reference "the longitudinal direction" to define TE modes $(E_z = 0)$ and TM modes $(H_z = 0)$.  (A) $\mathrm{TM}_{110}$ (B) $\mathrm{TM}_{101}$ (C) $\mathrm{TE}_{101}$ (D) $\mathrm{TE}_{011}$

19. A parallel-plate waveguide made of perfectly conducting plates spaced $3\mathrm{cm}$ apart by polyethylene, $\varepsilon_{r} = 2.25$, is operated at frequency of $12\mathrm{GHz}$. Without considering the fringing effect on edges, which of the following supported modes has faster group velocity than others: (A) $\mathrm{TE}_3$ (B) $\mathrm{TM}_2$ (C) $\mathrm{TE}_1$ (D) TEM

20. An engineer intends to design a cubic cavity with side lengths $a = b = d = 25 \mathrm{~mm}$. To achieve the highest possible Q factor, he is evaluating three candidate materials—copper, aluminum, and silver—with electrical conductivities of $\sigma = 5.8 \times 10^{7} \mathrm{~S} / \mathrm{m}, \sigma = 3.5 \times 10^{7} \mathrm{~S} / \mathrm{m}, \sigma = 6.2 \times 10^{7} \mathrm{~S} / \mathrm{m}$, respectively. Which material should be chosen when only conductivity is considered: (A) Copper (B) Aluminum (C) Silver (D) All have the same Q factor. 

1. Consider a particle in a 2D rectangular box. Let the box widths be L and 2L, respectively, in the x- and y-dimensions. Which of the following expressions is the correct first excited state energy? $(h = \text{Planck constant}, m = \text{particle mass})$ (A) $\frac{h^2}{8mL^2}$ (B) $\frac{h^2}{16mL^2}$ (C) $\frac{h^2}{4mL^2}$ (D) $\frac{h^2}{2mL^2}$ (E) $\frac{h^2}{32mL^2}$

2. Consider the EM wave mode with wave vector k and frequency f, in a cavity of temperature T. The average number of photons in the mode, in the limit where $k_{B}T \gg hf$, is given by $(k_{B} = \text{Boltzmann constant}, h = \text{Planck constant})$ (A) $\left[\frac{k_B T}{h f}\right]^2$ (B) $\frac{1}{\left[\exp\left(\frac{k_B T}{h f}\right) - 1\right]}$ (C) $\frac{k_B T}{h f}$ (D) $\exp\left(\frac{-hf}{k_B T}\right)$ (E) $\frac{1}{\left[\exp\left(\frac{hf}{k_B T}\right) + 1\right]}$

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