$(25$ points$)$ Two parallel circular plates with the same radius $=$a are made of PEC and are separated by a distance of $h$ and a material with conductivity $$ and permittivity $lon.$ Assuming a positive charge $+Q$ uniformly distributed on the upper plate where $z=h$ and a negative charge $-Q$ uniformly distributed on the lower plate where $z=0,$ solve the following problems.

$($a$)(5$ points$)$ Find the electric field between the two plates;

Electric Field: vec$(E)(z)=$

$($b$)(5$ points$)$ Find the voltage difference between the two plates;

$($c$)(5$ points$)$ Find the total current flowing from the upper to the lower plate;

$($d$)(5$ points$)$ Find an expression for the resistance between the two spheres.

Resistance: $R=$

$($e$)(5$ points$)$ Validate the point form of Gauss's law for the electric flux density. $($Hint: Use the electric field solved earlier, show that the left$-$hand side of Gauss's law matches the right$-$hand side.$)$