109 年 國立臺灣大學電信工程學研究所甲組《電磁學與電磁波》

An infinite planar current sheet, $J_s$, is placed at the interface ($z=0$) of two perfect dielectric materials, where $J_s$ is a time-varying surface current density given by
$J_s = 0.2 \cos(8\pi \times 10^9 t) \hat{x} \quad (\text{A/m})$
The values of permittivity in these two regions are given by $\epsilon_1 = 4\epsilon_0$ and $\epsilon_2 = 9\epsilon_0$ for $z<0$ and $z>0$, respectively. (Both cases assume $\mu = \mu_0$).

(1) (4 points) Find the intrinsic impedances, $\eta_1$ and $\eta_2$, in these two regions of $z<0$ and $z>0$, respectively. Also find the corresponding phase constants, $\beta_1$ and $\beta_2$. (Note that the speed of light is $c = 3 \times 10^8$ m/s).

(2) (8 points) Find the electric field $E(z,t)$ and the magnetic field $H(z,t)$ in both regions $z > 0$ and $z < 0$.

(3) (4 points) Specify the boundary conditions for $E(z,t)$ and $H(z,t)$, respectively at $z=0$ when the fields in the two regions go close to the interface at $z=0$.

(4) (4 points) Please show if the time-average power densities in these two regions are equal.

Consider two parallel infinite plane sheets held a distance $d$ apart (the material inside is air). The surface charge density is $\rho_{s0}$ at $z=0$, and is $-\rho_{s0}$ at $z=d$.

(1) (6 points) Find the electric field everywhere. Calculate the voltage between the two plane sheets. Find the capacitance per area.

(2) (6 points) Now the air is replaced by a dielectric material, $\epsilon = 2.25\epsilon_0$. Repeat the computations in (1).

(3) (3 points) Find the polarization current density, $J_p$.

Consider an electric field propagating radially outward in free space, which is expressed in spherical coordinates by
$\vec{E} = \frac{E_0}{r} \sin[\omega(t - r/c)] \hat{r} \quad (\text{V/m})$
where $(r, \theta, \phi)$ is the variables of spherical coordinate system with $(\hat{r}, \hat{\theta}, \hat{\phi})$ being the unit vectors of their axes. We assume that $\vec{E}$, $\vec{H}$ and their propagation direction follow the rule of plane wave.

(a) (5 points) Please find the propagation direction, and afterward determine the magnetic field, $\vec{H}$.

(b) (5 points) Find the instantaneous power density ($\vec{P} = \vec{E} \times \vec{H}$). Specify the direction of power propagation.

(c) (5 points) Integrate the power density in (b) through a spherical surface with radius $r$ to find the power ($P = \oint_S \vec{P} \cdot d\vec{s}$). Also calculate the time average power. (Note: $d\vec{s} = r^2 \sin\theta d\theta d\phi \hat{r}$)

Consider a transmission line with characteristic impedance $Z_0$, propagation constant $\beta$, and length $l$. It is connected to a short-circuited load at one end.

(a) (5 points) Show that the input impedance looking into the transmission line (from the other end) can be expressed as $Z_{in} = jZ_0 \tan(\beta l)$.

(b) (5 points) Assume that the short-circuited load is replaced by an arbitrary load $Z_L$, as shown in Figure 1, please show that the input impedance looking into the transmission line can be expressed as
$Z_{in} = Z_0 \frac{Z_L + jZ_0 \tan(\beta l)}{Z_0 + jZ_L \tan(\beta l)}$
(Hint: use $Z(d)=Z_0 \frac{Z_L + jZ_0 \tan(\beta d)}{Z_0 + jZ_L \tan(\beta d)}$, where $\Gamma(d)$ is the reflection coefficient at location $d$)

Figure 1.
🖼️【此處有附圖，請對照原卷】

(c) (5 points) Let $Z(d)$ be the line impedance at location $d$ looking towards the load. Show that
$Z(d) = Z_0 \frac{1+\Gamma(d)}{1-\Gamma(d)}$
where $\Gamma(d)$ is the reflection coefficient at location $d$.

Figure 2 shows a plane wave of TE polarization incident upon a planar interface at $x=0$, the electric field is $\vec{E}_i(\vec{r}) = \hat{y} E_0 e^{-j k_x x - j k_z z}$, the reflected electric field is $\vec{E}_r(\vec{r}) = \hat{y} R E_0 e^{-j k'_x x - j k'_z z}$, and the transmitted electric field is $\vec{E}_t(\vec{r}) = \hat{y} T E_0 e^{-j k''_x x - j k''_z z}$, where $R$ is the reflection coefficient and $T$ is the transmission coefficient.

(a) (5 points) Derive the incident, reflected and transmitted magnetic fields by using $\vec{H}(\vec{r}) = \frac{1}{j\omega\mu} \nabla \times \vec{E}(\vec{r})$.

(b) (5 points) Impose the boundary condition that the tangential electric and magnetic fields are continuous at $x=0$ to solve $R$ and $T$.

(c) (5 points) If $\epsilon_1 > \epsilon_2$, derive the formula of the critical angle $\theta_c$ at which total reflection occurs.

Figure 2.
🖼️【此處有附圖，請對照原卷】

The vector potential $\vec{A}(\vec{r})$ induced by a Hertzian dipole $I l \delta(\vec{r})$ at the origin can be represented in the spherical coordinate as
$\vec{A}(\vec{r}) = -\frac{\mu_0 I l}{4\pi r} e^{-j\beta r} e^{j\omega t} \hat{r}$
(a) (5 points) Derive the magnetic field in the spherical coordinate by using $\vec{H}(\vec{r}) = \frac{1}{\mu_0} \nabla \times \vec{A}(\vec{r})$.

(b) (5 points) Derive the electric field in the spherical coordinate by using $\vec{E}(\vec{r}) = \frac{1}{j\omega\epsilon_0} \nabla \times \vec{H}(\vec{r})$.

(c) (5 points) In the far field ($kr \gg 1$), write down the dominant term in the expressions of $\vec{H}(\vec{r})$ in (a) and $\vec{E}(\vec{r})$ in (b).

(d) (5 points) Derive the time-average power density by using the far-field expressions in (c).