108 年 國立中央大學光電科學與工程學系半導體光電科技碩士班《電磁學》

What is the $y$ component of the force acting on $Q_2$?

(A) $5\times10^{-2}$ newtons
(B) $5$ newtons
(C) $0.5$ newtons
(D) $5\times10^{-3}$ newtons
(E) $5\times10^{-4}$ newtons

For the above question, what is the magnitude of the force experienced by $Q_1$?

(A) $0.9$ newtons
(B) $9\times10^{-2}$ newtons
(C) $9$ newtons
(D) $9\times10^{-3}$ newtons
(E) $9\times10^{-4}$ newtons

A straight wire carrying a current $I$ in the $z$ direction is located along the $z$ axis in free space. If the wire is infinitely long and infinitely thin, what is the magnetic field intensity at the point $(0.5L,0,0)$ in rectangular coordinate system?

(A) $\frac{I}{\pi L}$
(B) $\frac{2I}{\pi L}$
(C) $\frac{I}{2\pi L}$
(D) $\frac{I}{4\pi L}$
(E) $\frac{I}{3\pi L}$

In a perfect dielectric medium with the permeability of $\mu$, there exists an electric field given as

$\mathbf{E}=E_0\cos(\omega t-kz)\mathbf{a}_x,$

where $E_0$ is the peak value, $\omega$ is the frequency, $k$ is a constant, and $\mathbf{a}_x$ is the unit vector in $x$ direction.

What is the $x$ component of the magnetic field in the region?

(A) $\frac{E_0k}{\omega\mu}\cos(\omega t-kz)$
(B) $0$
(C) $\frac{E_0}{\omega\mu}\cos(\omega t-kz)$
(D) $\frac{E_0k}{\omega\mu}\cos(\omega t-kz)$
(E) None of the above.

For an infinite plane carrying uniform surface charge density $\rho_s$ in rectangular coordinate system, what is the magnitude of the electric field intensity at any point on the $z$ axis?

$\varepsilon_0$: the permittivity in free space.

(A) $\frac{2\rho_s}{\varepsilon_0}$
(B) $\frac{4\rho_s}{\varepsilon_0}$
(C) $0$
(D) $\frac{\rho_s}{\varepsilon_0}$
(E) $\frac{\rho_s}{2\varepsilon_0}$

On a metallic sphere of radius $R=30\ \mathrm{cm}$ in free space (permittivity: $8.85\times10^{-12}\ \mathrm{F/m}$), charge $Q=1\ \mu\mathrm{C}$ is uniformly distributed over the surface.

What is the electric field just above the surface of the sphere?

(A) $1\ \mathrm{V/cm}$
(B) $10\ \mathrm{V/cm}$
(C) $100\ \mathrm{V/cm}$
(D) $1000\ \mathrm{V/cm}$
(E) $10000\ \mathrm{V/cm}$

Two metallic plates are separated by a distance $d$, and the medium between the plates is homogeneous and has finite conductivity $\sigma$.

If the two plates are of the cross-sectional area $A$ and infinite conductivity, what is the resistance of the region between the plates when a potential difference $V$ is applied to the two plates?

(A) $\frac{2d}{\sigma A}$
(B) $\frac{V}{d}$
(C) $\frac{V}{\sigma d}$
(D) $\frac{\sigma V}{Ad}$
(E) None of the above.

What is $\vec{J}_1$?

(A) $1385.6\ \mathrm{A/m^2}$
(B) $1.732\ \mathrm{A/m^2}$
(C) $1\ \mathrm{A/m^2}$
(D) $1853.7\ \mathrm{A/m^2}$
(E) $1.237\ \mathrm{A/m^2}$

For the above question, what is $\theta_1$?

(A) $39.96^\circ$
(B) $89.96^\circ$
(C) $49.96^\circ$
(D) $59.96^\circ$
(E) $69.96^\circ$

Repeating question (8), what is the surface charge density at the interface?

(A) $-0.68\ \mathrm{mC/m^2}$
(B) $-0.78\ \mathrm{mC/m^2}$
(C) $-0.88\ \mathrm{mC/m^2}$
(D) $-0.98\ \mathrm{mC/m^2}$
(E) $-0.58\ \mathrm{mC/m^2}$

What are Coulomb and Lorenz gauges? Please check the following two cases, which of them are in Coulomb and/or Lorenz gauges.

(a) $V=3t$, $\vec{A}=A_0\sin(kx-\omega t)\hat{y}$; where $k$ and $A_0$ are constants. ($5\%$)

(b) $V(\vec{r},t)=0$, $\vec{A}(\vec{r},t)=-\frac{qt}{4\pi\varepsilon_0r^2}\hat{r}$, where $q$ and $\varepsilon_0$ are constants. ($5\%$)

You get an extremely thin quarter waveplate, made by a crystal (extraordinary refractive index $n_e=1.671$, ordinary refractive index $n_o=1.521$, optical axis along $x$ axis), for light with wavelength of $600\ \mathrm{nm}$, as shown in Figure 1.

Figure 1:
🖼️【此處有附圖，請對照原卷】

(a) Plot the relative direction of the optical axis and the polarization of the light to obtain a circularly polarized light. ($4\%$)

(b) Calculate the smallest thickness of the crystal. ($6\%$)

Please explain what the Poynting vector is.

Calculate the power transported down the cable shown in Figure 2, using Poynting vector. These two conductors are held at potential difference $V$ and carry current $I$ (down one and back up the other).

Figure 2:
🖼️【此處有附圖，請對照原卷】

Suppose we have a wave guide of rectangular shape with height $a$ and width $b$ ($a\geq b$), and we are interested in the propagation of TM waves. [Figure 3]

Figure 3:
🖼️【此處有附圖，請對照原卷】

(a) What is a TM wave? Please give the boundary condition, and explain it briefly. ($4\%$)

(b) Find the fields, $B_x(x,y)$, $B_y(x,y)$ and $B_z(x,y)$. Please give your derivation process. ($8\%$)

(c) Explain the limitation of $m$ and $n$ in $\mathrm{TM}_{mn}$, where $m$ and $n$ mean the factors of integer for the wave number along $x$ and $y$ directions ($k_x$ and $k_y$). ($4\%$)

(d) Derive the cut-off frequency, and explain its physical meaning in detail. ($4\%$)

The components of EM wave in waveguides for $E_x$, $E_y$, $B_x$, and $B_y$, as well as the uncoupled equations.

$E_x=\frac{i}{(\omega/c)^2-k^2}\left(k\frac{\partial E_z}{\partial x}+\omega\frac{\partial B_z}{\partial y}\right),$

$E_y=\frac{i}{(\omega/c)^2-k^2}\left(k\frac{\partial E_z}{\partial y}-\omega\frac{\partial B_z}{\partial x}\right),$

$B_x=\frac{i}{(\omega/c)^2-k^2}\left(k\frac{\partial B_z}{\partial x}-\frac{\omega}{c^2}\frac{\partial E_z}{\partial y}\right),$

$B_y=\frac{i}{(\omega/c)^2-k^2}\left(k\frac{\partial B_z}{\partial y}+\frac{\omega}{c^2}\frac{\partial E_z}{\partial x}\right),$

$\left[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\left(\frac{\omega}{c}\right)^2-k^2\right]\begin{pmatrix}E_z\\B_z\end{pmatrix}=0,$

uncoupled equations, where $\omega$, $c$, and $k$ are angular frequency, light velocity, and wave number, respectively.