114 年 國立臺灣聯合大學系統（清華、政治、陽明交通、中央四校聯招）研究所電機類《電磁學》

1. Which of the following statements is true?
   (A) Dynamic electromagnetic forces satisfy Newton's third law
   (B) Static electromagnetic forces satisfy momentum conservation
   (C) Given the electric field $E = -\nabla V - \frac{\partial A}{\partial t}$ and charge q, the physical meaning of qA is energy.
   (D) All of the above are true

2. Which of the following cannot support TEM modes (ignoring the fringing effect)?
   (A) Two-wire line
   (B) Rectangular waveguide
   (C) Coaxial cable/line
   (D) Microstrip line

3. Which of the following statements about the vector potential $A$ is always true?
   (A) $\nabla \cdot A = B$
   (B) $\nabla \cdot A = 0$
   (C) $\nabla^2 A = -\mu_0 J$
   (D) None of the above

4. There is a dielectric-filled parallel-planar metallic waveguide oriented in the way that guided waves propagate along z direction, and the separation d of two parallel is along y axis. Ignoring the fringing effect, the following field component of fundamental mode supported by the waveguide is non-zero:
   (A) $E_x$
   (B) $E_y$
   (C) $H_x$
   (D) $H_y$

5. In the complex notation (also called the phasor notation), we can express a monochromatic plane wave as $E = E_0 e^{j(kx-\omega t)}$. How do we calculate the time-averaged Poynting vector in this notation?
   (A) $E \times H$
   (B) $Re[E \times H^*]$
   (C) $\frac{1}{2} Re[E \times H^*]$
   (D) None of the above

6. Consider a plane wave traveling in a lossy material. The electric field is given by $E = E_0 e^{1 + 0.1i(\mu m^{-1})}$. The intensity of the plane wave can be expressed as $I = I_0 e^{-\alpha z}$, where $\alpha$ is the absorption coefficient. Please calculate $\alpha$.
   (A) 2 ($\mu m^{-1}$)
   (B) 0.2 ($\mu m^{-1}$)
   (C) 0.1 ($\mu m^{-1}$)
   (D) None of the above

7. If $A = r \cos\phi \hat{r} + \sin\phi \hat{\phi}$, evaluate $\oint A \cdot dl$ around the path shown in the following figure.
   🖼️【此處有附圖，請對照原卷】
   (A) 0
   (B) $\frac{27}{4}(\sqrt{2}+1)$
   (C) $\frac{27}{4}(\sqrt{2}-1)$
   (D) None of the above

8. The finite sheet $0 \le x \le 1$ and $0 \le y \le 1$ on the $z=0$ plane has a surface charge density $\rho_s = xy(x^2+y^2)^{3/2}$ nC/m$^2$. Evaluate the electric field at $(0, 0, 5)$.
   🖼️【此處有附圖，請對照原卷】
   (A) $(-3 \hat{x} - 3 \hat{y} + 22.5 \hat{z})$ V/m
   (B) $(-3 \hat{x} - 3 \hat{y} + 11.25 \hat{z})$ V/m
   (C) $(-1.5 \hat{x} - 1.5 \hat{y} + 22.5 \hat{z})$ V/m
   (D) $(-1.5 \hat{x} - 1.5 \hat{y} + 11.25 \hat{z})$ V/m

9. In the following figure, two conducting cones of infinite extent with $\theta_1$ and $\theta_2$ are separated by an infinitesimal gap at $R = 0$. If $0 < \theta_1 < \theta_2 < \frac{\pi}{2}$, $V(\theta_2) = 0$, and $V(\theta_1) = 10$ V, evaluate the potential distribution $V(R, \theta)$ between the cones.
   🖼️【此處有附圖，請對照原卷】
   (A) $10 \frac{[\ln(\tan(\theta/2)\cot(\theta_1/2))]}{[\ln(\tan(\theta_2/2)\cot(\theta_1/2))]} V$
   (B) $10 \frac{[\ln(\tan(\theta/2)\cot(\theta_1/2))]}{[\ln(\tan(\theta_2/2)\cot(\theta_1/2))]} V$
   (C) $10 \frac{[\ln(\tan(\theta/2)\cot(\theta_1/2))]}{[\ln(\tan(\theta_2/2)\cot(\theta_1/2))]} V$
   (D) $10 \frac{[\ln(\tan(\theta/2)\cot(\theta_1/2))]}{[\ln(\tan(\theta_2/2)\cot(\theta_1/2))]} V$

Consider a plane wave that is incident obliquely on an interface between two dielectric media of refractive indices $n_i$ and $n_t$, as shown in the figure. We express the incident wave and transmitted wave as $\mathbf{E}_i=\mathbf{E}_{0i}e^{i(\mathbf{k}_i\cdot\mathbf{r}-\omega t)}$ and $\mathbf{E}_t=\mathbf{E}_{0t}e^{i(\mathbf{k}_t\cdot\mathbf{r}-\omega t)}$, respectively. Here, $\mathbf{k}_i$, $\mathbf{k}_r$, and $\mathbf{k}_t$ are the wave vectors of the incident wave, reflected wave, and transmitted wave, respectively. These wave vectors are all on the same plane. Which one of the following relations is correct?

🖼️【此處有附圖，請對照原卷】

(A) $k_i\cos\theta_i=k_t\cos\theta_t$

(B) $k_i\sin\theta_i=k_t\sin\theta_t$

(C) $k_i n_i\sin\theta_i=k_t n_t\sin\theta_t$

(D) None of the above

Consider an ideal lossless transmission line with the characteristic impedance $R_0=50\,\Omega$. The transmission line is connected to a load composed of a resistor $R=25\,\Omega$ and a capacitor $C=0.2\,\mathrm{pF}$ in series. The signal frequency is $10\,\mathrm{GHz}$. If we want to use the Smith chart for determining the signal reflectivity at the load interface, please refer to the following figure and determine inside which quadrant of the Smith chart the data point will be located:

🖼️【此處有附圖，請對照原卷】

(A) I

(B) II

(C) III

(D) IV

A TE wave propagating in a dielectric-filled rectangular metallic waveguide of unknown permittivity has dimensions $a=5\,\mathrm{cm}$ and $b=2\,\mathrm{cm}$. Its $x$ component of electric field is given by

$E_x=-10\cos(40\pi x)\sin(100\pi y)\sin(2.4\pi\times10^{10}t-52.9\pi z)\ \mathrm{V/m}$

Which of the following statements is correct?

(A) The mode number is $\mathrm{TE}_{23}$.

(B) The relative permittivity $\varepsilon_r$ of the filled dielectric material is $2$.

(C) The cutoff frequency of the TE mode is below $11\,\mathrm{GHz}$.

(D) None of the above

The quality factor $Q$ for a hollow resonant cavity operating in the $\mathrm{TE}_{101}$ mode is about $6000$, and the frequency of the $\mathrm{TE}_{101}$ mode is $15\,\mathrm{GHz}$. What is the bandwidth of the cavity at the $\mathrm{TE}_{101}$ mode?

(A) $2.50\,\mathrm{MHz}$

(B) $1.67\,\mathrm{MHz}$

(C) $1.33\,\mathrm{MHz}$

(D) None of the above

A disk of surface charge density $\sigma_0$ and radius $a$ is on the $x$-$y$ plane and spinning about its center with angular velocity $\boldsymbol{\omega}=\omega_0\hat{\mathbf{z}}$. What is the $\mathbf{B}$ at its center?

(A) $0$

(B) $\dfrac{\mu_0\sigma_0\omega_0a}{4}\,\hat{\mathbf{z}}$

(C) $\dfrac{\mu_0\sigma_0\omega_0a}{3}\,\hat{\mathbf{z}}$

(D) None of the above

A wire is defined by the parametric equations $x=a\cos u$, $y=a\sin u$, and $z=a\sin u$ for $u\in[0,2\pi]$. A current $I$ flows counterclockwise on the wire (top view, from $+z$ to $-z$). What is the magnetic dipole moment of the wire?

(A) $I\pi a^2\hat{\mathbf{x}}+I\pi a^2\hat{\mathbf{z}}$

(B) $-I\pi a^2\hat{\mathbf{y}}+I\pi a^2\hat{\mathbf{z}}$

(C) $I\pi a^2\hat{\mathbf{y}}+I\pi a^2\hat{\mathbf{z}}$

(D) None of the above

A DC voltage source is connected across a long, straight conducting wire with radius $a$ and conductivity $\sigma$. The wire carries a direct current $I$ along the $-z$ direction. Determine the Poynting vector on the surface of the wire associated with the electromagnetic fields.

(A) $-\dfrac{I^2}{\sigma\pi^2a^3}\,\hat{\mathbf{r}}$

(B) $-\dfrac{I^2}{2\sigma\pi^2a^3}\,\hat{\mathbf{r}}$

(C) $\dfrac{I^2}{\sigma\pi^2a^3}\,\hat{\mathbf{r}}$

(D) $\dfrac{I^2}{2\sigma\pi^2a^3}\,\hat{\mathbf{r}}$

An AC voltage source $V_0\sin(\omega t)$ (V) is connected across a parallel-plate capacitor $C$. The capacitor has circular plates with a radius $a$. Determine the $\mathbf{H}$ at a distance $r$ from the axis of the capacitor, where $r<a$.

(A) $\dfrac{CV_0\omega}{\pi a}\cos(\omega t)\,\hat{\boldsymbol{\phi}}\ \mathrm{(A/m)}$

(B) $\dfrac{CV_0\omega}{2\pi a}\cos(\omega t)\,\hat{\boldsymbol{\phi}}\ \mathrm{(A/m)}$

(C) $\dfrac{CV_0\omega}{\pi r}\cos(\omega t)\,\hat{\boldsymbol{\phi}}\ \mathrm{(A/m)}$

(D) $\dfrac{CV_0\omega}{2\pi r}\cos(\omega t)\,\hat{\boldsymbol{\phi}}\ \mathrm{(A/m)}$

Consider an ideal lossless coaxial cable transmission line. The dielectric material between the two conductors of the coaxial cable has a dielectric constant $\varepsilon=2.25\varepsilon_0$. The cable is open-circuited at one end and is excited with a sinusoidal signal at $1\,\mathrm{GHz}$ from the other end. We want to determine at which position inside the cable the root-mean-square (rms) of the net voltage will have the minimum value. If the distance between the nearest minimum rms voltage point and the open-circuited end is $L$, determine which is the closest value for $L$ among the following choices:

(A) $4\,\mathrm{cm}$

(B) $5\,\mathrm{cm}$

(C) $6\,\mathrm{cm}$

(D) $7\,\mathrm{cm}$

Consider an ideal lossless transmission line with the characteristic impedance $R_0=100\,\Omega$. A $\lambda/8$ section ($\beta l=\pi/4$) of this transmission line is connected to a load with impedance $Z_L$, and the input impedance seen from the input end of the $\lambda/8$ transmission line with the load is exactly $Z_i=50\,\Omega$. Please determine which is the nearest value for the real part of the load impedance, $\operatorname{Re}[Z_L]$, among the following choices:

(A) $50\,\Omega$

(B) $60\,\Omega$

(C) $70\,\Omega$

(D) $80\,\Omega$

Consider a lossy distortionless transmission line. The characteristic impedance is $50\,\Omega$, the phase velocity is $2\times10^8\,\mathrm{m/s}$, and the operation frequency is $1\,\mathrm{GHz}$. For the four transmission line parameters $R$, $L$, $G$, and $C$ of the distortionless transmission line, we already know that the value of $R$ is $0.1\,\Omega/\mathrm{m}$ and the signal voltage propagates in the $z$ direction as $V(z)=V(0)\exp(-\alpha z-j\beta z)$ without the reflected signal. Please determine which is the nearest value for the ratio $\beta/\alpha$ among the following choices:

(A) $10^3$

(B) $10^4$

(C) $10^5$

(D) $10^6$