Chapter $11$ The Uniform Plane Wave

$|$ Problem Set $|$

P$1.$ The electric field of an electromagnetic wave propagating through air without a source is as follows.

$E=a_x{E}_{0}cos(t-kz)$

Here, $$ and $k$ are constants. Please answer the following questions regarding this electromagnetic wave.

$($a$)$ Verify that $E(z,t)$ satisfies the wave equation denoted as $gra{d}^{2}E-\frac{de{l}^{2}E}{del{t}^2}=0$ under specific conditions.

$($b$)$ Express the electric field in the form $f(z-vt)$ and determine the propagation speed and direction of the electromagnetic wave.

$($c$)$ Does the direction of wave propagation align with the direction of electromagnetic power flow as indicated by the Poynting vector? Is the velocity consistent with $v=\frac{1}{\sqrt[\phantom{2}]{17t}}?$

P$2.$ Given that the permittivity of the glass is $=2\mathrm{.}56{}_0$ and the permeability is $={}_0,$ determine the propagation speed of light traveling through glass. Additionally, compare this calculated result with the speed of light traveling through air.

P$3.$ Express the following quantities, which vary in a sinusoidal shape over time, as complex phasors. It is assumed that the constants $a,E_0$ are all real numbers.

$E(z,t)=a_x{E}_{0}cos(t-2z)+a_y{E}_{0}sin(t-2z)$

P$4.$ Represent the time$-$harmonic field depicted as a complex phasor in a time domain format. Assume angular frequency is $[ra\frac{d}{s}].$

$H=a_x(1-j)+e^{-j2z}{a}_y(1-j\sqrt[\phantom{2}]{3})e^{+j2z}$

P$5.$ A time$-$harmonic electromagnetic wave with a peak electric field amplitude of $E_0$ is propagating through a lossless medium characterized by permeability $$ and permittivity $.$ Prove that the time$-$averaged power density carried by this electromagnetic wave is as follows. Here, $=\sqrt[\phantom{2}]{\frac{}{}}$ denotes the intrinsic impedance of the medium.

$S_{av}=\frac{|E_0{|}^2}{2}[\frac{W}{m^2}]$

$[2024-2|$ Field & Wave Electromagnetics $|10424]$