Write only on A$4$ paper. Show your work in detail. Use SI units. $1$:$1$ plagiarism will be strictly penalized.

$1.$ On a remote planet in the outer rim of the galaxy, a series of complex electromagnetic phenomena are threatening the stability of the region and creating a disturbance in the force. As part of a secret research team led by two Jedi Knights, you have been summoned to analyze and solve the mysteries surrounding these events. $[$NOTE: An accurately drawn vector diagram, properly labeled, is required where necessary. Use vector resolution to add vectors. Also, the approximations for the infinite cases of continuous charges you learned in the class are implied by default.$]$

$($a$)$ Upon reaching the planet's orbit, a young Padawan reports an anomaly where Gauss's Law seems to break down. To understand this new realm, you must first write down the conditions required for Gauss's Law to remain valid to him. Explain the meaning behind the first Maxwell's equations in both its integral and differential forms.

$($b$)$ Deep within an abandoned cave on the planet, you discover an artifact in the shape of a cylinder, with an infinite line of positive charge running through its center. The charge per unit length on the line is $\backslash (\backslash$lambda$=5\backslash$mu $\backslash$mathrm$\{$Cm$\}^\{-1\}\backslash ).($i$)$ What is the electric flux through the cylinder due to this infinite line of charge? $($ii$)$ What is the flux through the cylinder if its radius is increased to $\backslash ($ r$=0\mathrm{.}575\backslash$mathrm$\{$~m$\}\backslash )?($iii$)$ What is the flux through the cylinder if its length is increased to $\backslash ($ l$=0\mathrm{.}905\backslash$mathrm$\{$~m$\}\backslash )?$ NOTE: You must show a diagram of how you get to the flux calculating formula.

$($c$)$ You find a laboratory inside the cave where three charged planes, having surface charge densities $\backslash (\backslash$sigma$\_\{1\}=-1\mathrm{.}5\backslash$sigma$\_\{2\},\backslash$sigma$\_\{2\}=+\backslash$sigma$,\backslash$sigma$\_\{3\}=-0\mathrm{.}5\backslash$sigma$\_\{2\}\backslash )$ are suspended mid$-$air in a parallel configuration, horizontally apart, each influencing the electric fields around them. The plates divide the horizontal axis into four regions labeled $\backslash ($ A$,$ B$,$ C $\backslash ),$ and $\backslash ($ D $\backslash ).$ Your task is calculating the electric fields $\backslash (\backslash$vec$\{$E$\}\_\{$B$\}\backslash )$ and $\backslash (\backslash$vec$\{$E$\}\_\{$D$\}\backslash )$ in regions $\backslash ($ B $\backslash )$ and $\backslash ($ D $\backslash ).$

$($d$)$ To demonstrate how charged particles accelerate to a Padawan, you release an electron into a region of space between two infinite, nonconducting charged sheets. The charge densities on the sheets are different in each of the following three scenarios. Your goal is to help the Padawan determine the electron's acceleration in each case and rank the situations according to the magnitude of the electron's acceleration, from greatest to least.

$($e$)$ Just on the exit of that same cave, you encounter a charged conducting drum. Just above the surface of this drum, the electric field is measured to have a magnitude of $\backslash ($ E$=2\mathrm{.}3\backslash$times $10^\{5\}\backslash$mathrm$\{$NC$\}^\{-1\}\backslash ).$ Your mission is to determine the surface charge density on the drum, a key piece in unlocking the mysteries of its electromagnetic properties and bringing balance to the Force.