108 年 國立成功大學光電科學與工程研究所甲組《電磁學》

1. (a) A point charge $q$ is at a distance $d$ above an infinite grounded conducting plate. Find the induced surface charge density and total charge $q_{\text{total}}$ on the conducting plane.

1. (b) Similar to (a), a point charge $q$ is at distance $d$ above a semi-infinite dielectric region ($z<0$) with dielectric constant $\epsilon_r = 1+\chi_e$. Find the induced surface charge density and total charge $q_{b, \text{total}}$ at $z=0$ interface. (use image charge method, find $V$ $\sigma = -\epsilon_0 \frac{\partial V}{\partial z}|_{z=0}$ $E_z = E_{q,z} + E_{b,z}$ $E_{q,z} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ $E_{b,z} = -\frac{1}{4\pi\epsilon_0} \frac{q_b}{r^2}$ $E_z|_{z=0} = E_{q,z}|_{z=0} + E_{b,z}|_{z=0}$ $\sigma_b = P_b \cdot \hat{n} = \epsilon_0 \chi_e E_{b,z}|_{z=0}$ )

2. (a) An uncharged metal sphere of radius R is placed in a uniform electric field $\vec{E} = E_0 \hat{x}$. Find the potential $V$ outside the sphere and the induced surface charge density $\sigma = -\epsilon_0 \frac{\partial V}{\partial r}|_{r=R}$ (8%)
   (use separation of variable in spherical Coordinate. $V(r, \theta) = \sum_{l=0}^{\infty} (A_l r^l + B_l r^{-(l+1)}) P_l(\cos \theta)$ with Legendre Polynomial $P_0(\cos \theta) = 1, P_1(\cos \theta) = \cos \theta, P_2(\cos \theta) = \frac{1}{2}(3\cos^2 \theta - 1), \int_{-1}^{1} P_l(x) P_{l'}(x) dx = \frac{2}{2l+1} \delta_{ll'}$ )

2. (b) A dielectric sphere of homogenous linear material with relative dielectric constant $\epsilon_r = 1+\chi_e$ is placed in an uniform electric field $\vec{E} = E_0 \hat{x}$. Find the potential $V$ and electric field $\vec{E}$ inside the sphere. (7%)
   (use separation of variable in spherical Coordinate. $V (r, \theta) = \sum_{l=0}^{\infty} (A_l r^l + B_l r^{-(l+1)}) P_l(\cos \theta)$ with Legendre Polynomial $P_0(\cos \theta) = 1, P_1(\cos \theta) = \cos \theta, P_2(\cos \theta) = \frac{1}{2}(3\cos^2 \theta - 1), \int_{-1}^{1} P_l(x) P_{l'}(x) dx = \frac{2}{2l+1} \delta_{ll'}$ )

3. A infinite-thin metal with sheet conductivity $\sigma(x) = \sigma_0 \delta(x)$ is located at $x=0$. For a normal incident plane wave propagating in x direction with E field polarized in y direction, find the transmission coefficient $t$ and reflection coefficient $r=?$ (use methods similar to the derivation of the transmission/reflection coefficient of normal incident to an dielectric interface, except now there is a sheet current density $J_s(x) = \sigma_s(x) E_x(x) = \sigma_0 \delta(x) E_x(x)$ in $j\omega\epsilon_0 E_z + \sigma_0 \delta(x) E_x = \nabla \times H = j\omega\mu_0 H_y$ )

4. Similar to previous problem, now place a perfect electric conductor (PEC) wall a distance d after the infinite-thin metal. A plane wave in normal incident toward the metal and the PEC wall. Under what condition (d=? and r=?), there will be no energy reflected (thus the incident wave is total absorbed by the metal). (Use Fabry-Perot ray-optics method to derive the total reflection coefficient Ar = Ar1+Ar2+Ar3+ .... of the combined metal sheet (r1,t1) and PEC wall (r2,t2) (10%)

You are fishing near a large lake in the afternoon. The reflection from the sun makes it hard for you to look into the water. You have a linear polarizer. What can you do? Explain the physics behind your answer. Please describe it in terms of the orientation of the transmission axis.

Consider $\omega=\frac{2\pi c}{\lambda_0}$ and $k=\frac{2\pi n}{\lambda_0}$. In a dispersive medium, find that the expression of the group velocity is as follows:

A hollow rectangular waveguide is used to transmit signals at a carrier frequency of $6\ \mathrm{GHz}$. Design the waveguide so that the cut-off frequency of the dominant TE mode is lower than the carrier by $25\%$ and the next mode is at least $25\%$ higher than the carrier frequency.

Prove that the equivalent permittivity of the plasma with an electron density $N$ is

where

is the plasma frequency, $\omega$ is the angular frequency of electromagnetic waves, $m$ is the electron mass, and $\varepsilon_0$ is the permittivity of free space.

Hint: You can regard the plasma as a free electron gas and assume a time-harmonic electric field $E$ in the $x$-direction at angular frequency $\omega$, and then write down the force equation for an electron with charge $-e$ and mass $m$.

Discuss the wave propagation for the cases of $\omega>\omega_p$ and $\omega<\omega_p$, respectively.