B$.$ Problem Solving $(16$ points$)$

For a uniform plane wave propagating in a vacuum with a frequency of $10$ GHz$,$ the electric field phasor is given by:

E$($x$,$t$)=5$z$^$cos$(2$$109$t$2$$106$x$)$V$/$m$.\backslash$vec$\{$E$\}($x$,$t$)=5\backslash$hat$\{$z$\}\backslash$cos$(2\backslash$pi $\backslash$times $10^9$ t $-2\backslash$pi $\backslash$times $10^6$ x$)\backslash ,\backslash$text$\{$V$/$m$\}.$E$($x$,$t$)=5$z$^$cos$(2$$109$t$2$$106$x$)$V$/$m$.$

$($a$)$ Find the wavelength $\backslash$lambda of the wave. $(4$ points$)$

$($b$)$ Determine the propagation constant $\backslash$beta for the wave. $(4$ points$)$

$($c$)$ Calculate the frequency of the wave. $(4$ points$)$

$($d$)$ Determine the average power density SavgS$\_\{\backslash$text$\{$avg$\}\}$Savg$$ of the wave. $(4$ points$)$