Problem $5$: Potential energy and electric potential in multinle$\_$chares cvstems

Three point charges are arranged on the corners of a rectangle as shown.

A$.$ Label the distances between each pair of charges $r_{12},r_{13},$ and $r_{23}.$ In terms of these distances and the charges $q_1,q_2,$ and $q_3,$ write an expression for the total electric potential energy of the system.

$($Hint: How many separate pairs of charges are there?$)$

B$.$ What is the distance $r_{13}?$

C$.$ Now use your answers above to compute the total electric potential energy of the system.

D$.$ Suppose that $q_1$ was moved straight downward to a new distance $r_{12}=2cm($the other charges remain in place$).$

Which terms in $U_E,$ total would change?

Would $U_{E,\text{t}\text{o}\text{t}\text{a}\text{l}}$ increase or decrease? $($Hint: $q_2$ is closer to $q_1$ than $q_3$ is$,$ and it also has a larger absolute value than $q_3$ does.$)$

Now consider the potential difference $V$ between the initial and final position of $q_1.$ Would $V$ be positive, negative, or zero?

$5$