Problem $1.$ In this problem, we aim to become more familiar with the geometry of vector fields in space, focusing specifically on electric fields generated by electric charges. In lectures, we discussed that one of the most significant types of vector fields in physics and engineering is the electric field E$.$ When an electric charge q is placed at the point x$0=($x$0$; y$0$; z$0)$ in space, it generates a vector force field E$($x; y; z$)$ given by: E$($x; y; z$)=$ q $40$ x $$ x$0$ jx $$ x$0$j$3$ ; y $$ y$0$ jx $$ x$0$j$3$ ; z $$ z$0$ jx $$ x$0$j$3$ ; where q is the electric charge, $0$ is the permittivity of free space, and j x $$ x$0$j represents the distance from point x to the charge at x$0.$ This distance is also sometimes denoted as kx $$ x$0$ k $.$ a$)$ Prove that E is a conservative force field with the potential function u: u$($x; y; z$)=$q $40$ jx $$ x$0$j ; in the sense E $=$ ru$.$ In this sense, the function U$($x; y; z$)=$u$($x; y; z$)=$ q $4"0$ jx $$ x$0$j is known as the potential energy of a charge located at $($x; y; z$).$ b$)$ Suppose two electric charges q and $$q are located at the points $(1$;$0$;$0))$ and $(1$;$0$;$0),$ respectively. Determine the electric field E and the associated potential energy U at an arbitrary point $($x; y; z$)$ different from $(1$; $0$; $0)$ or $(1$; $0$; $0).$