Question

What is the electric potential vtot at the center of the square? make the usual assumption that the potential tends to zero far away from a charge. express your answer in terms of q, d, and appropriate constants?

Answer 1

Missing figure of the problem: http://tsephysics.weebly.com/uploads/5/1/9/3/51934203/477140_orig.jpg

Solution:
Assuming the potential is zero at infinite distance from the charge, then the potential at a certain distance r from a single point charge is 
$V(r)=k_e \frac{q}{r}$
where $k_e=8.99\cdot 10^9 Nm^2C^{-2}$ is the Coulomb's constant.

In our problem, we just have to superimpose the potential generated by every charge. The diagonal of the square is $\sqrt{2} d$, therefore the distance between each charge and the center of the square is $\frac{ \sqrt{2} }{2} d$.
So, the total potential is:
$V=V_1+V_2+V_3+V_4=$
$=k_e \frac{q}{ \frac{ \sqrt{2}d }{2} }+ k_e \frac{2q}{ \frac{ \sqrt{2}d }{2} }+k_e \frac{5q}{ \frac{ \sqrt{2}d }{2} }-k_e \frac{3q}{ \frac{ \sqrt{2}d }{2} }=$
$=5 \sqrt{2} k_e \frac{q}{d}$