$1)$ Point charges $5$ nC and $-2$ nC are located at $(2,0,4)$ and $(-3,0,5),$ respectively.

$($a$)$ Determine the force on a $1$ nC point charge located at $(1,-3,7).$

$($b$)$ Find the electric field E at $(1,-3,7).$

$2)$ Planes x $=2$ and y $=-3,$ respectively, carry charges $10$ nC$/$m$2$ and $15$ nC$/$m$2.$ If the line x $=0,$ z $=2$ carries

charge $10$ nC$/$m$,$ calculate E at $(1,1,-1)$ due to the three charge distributions.

$3)$ A uniform line charge of $16$ nC$/$m is located along the line defined by y $=2,$ z $=5.$

Find E at PA$(4,2,-3),$ PB $(-1,-5,2),$ and PC$(-2,4,5).$

$4)$ A line charge of uniform charge density $0$ C$/$m and of length l$,$ is oriented along the z axis at $$ l$/2<$ z $<$ l$/2.$

$($a$)$ Find the electric field strength E in magnitude and direction at any position along the x axis.

Hint: If the line does not extend from infinity, and instead is finite, then we use the integral formula for E and find

a new distance vector integrated over the new finite length.