8. Which of the following equations describes charge conservation?
(A) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
(B) $\nabla \cdot \mathbf{D} = \rho$
(C) $\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}$
(D) $\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$

9. For electromagnetic waves propagating in conductors, we can derive the modified wave equation: $$ \nabla^{2} \mathbf{E} - \mu \epsilon \frac{\partial^{2} \mathbf{E}}{\partial t^{2}} - \mu \sigma \frac{\partial \mathbf{E}}{\partial t} = 0 $$ Consider a monochromatic plane wave as $\mathbf{E} = \mathbf{E}_0 e^{i(kz - \omega t)}$, where the wave number $k$ is a complex number. Please derive the expression for the wave number $k$. (A) $\omega \sqrt{\mu \epsilon} \left(1 + \frac{i\sigma}{\omega \epsilon}\right)^{\frac{1}{2}}$ (B) $i\omega \sqrt{\mu \epsilon} \left(1 + \frac{\sigma}{\omega \epsilon}\right)$ (C) $\omega \sqrt{\mu \epsilon} \left(1 + \frac{\omega \epsilon}{i\sigma}\right)^{1/2}$ (D) None of the above.

10. For plane electromagnetic waves propagating in vacuum, which one of the following descriptions is incorrect? Here $\mathbf{k}$ is the wave vector, and $\mathbf{S}$ is the Poynting vector. (A) $\mathbf{k} \perp \mathbf{E}$ (B) $\mathbf{S} \perp \mathbf{k}$ (C) $\mathbf{E} \perp \mathbf{H}$ (D) None of the above.

11. Consider a plane electromagnetic wave propagating in vacuum. The electric field is given by $\mathbf{E} = \mathbf{E}_0 e^{i(kx - \omega t)}$. The Poynting vector is labeled by $\mathbf{S}$. The intrinsic impedance $\eta$ (also called wave impedance) is $377\ \Omega$. The time-averaged Poynting vector $\langle S \rangle$ is $1\ \mathrm{W/m^2}$. Please calculate $E_0$, i.e., the amplitude of the electric field. Please choose the closest answer. (A) $54.2\ \mathrm{V/m}$ (B) $377\ \mathrm{V/m}$ (C) $19.4\ \mathrm{V/m}$ (D) $27.5\ \mathrm{V/m}$

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) $\mathrm{TE}_{011}$

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