Say we have:
$ F(x,y) = \dfrac{ϵq_1q_2}{|r|^2} \hat{u}$
I am attempting to find the formula in terms of the position of the charge $q_2$ in $(x,y)$, while leaving everything else constant. (thanks Peter!)
I have come up with:
$ F(x,y) = \dfrac{ϵq_1q_2}{x^2+y^2}\dfrac{x}{\sqrt{x^2+y^2}}\hat{i} + \dfrac{ϵq_1q_2}{x^2+y^2}\dfrac{y}{\sqrt{x^2+y^2}}\hat{j}$
I also am considering:
$ F(x,y) = \dfrac{ϵq_1x}{x^2+y^2}\hat{i} + \dfrac{ϵq_1y}{x^2+y^2}\hat{j}$
Ultimately I will be finding the divergence and testing if the field is conservative.