Potential from a Point Charge

Suppose that you have a point charge $q$. We would like to calculate the electric potential at any point in space due to this point charge.

First, we will find the electrostatic potential energy. Let us say that we have a test charge $q_2$ at a distance $d$ from the first charge.

From the definition of potential energy, we know the following: $$U=\int_{\infty}^{d}\vec{F}\cdot\vec{dl}$$ Since the electrostatic force is conservative (I will not prove this here), we may choose any path that we wish to calculate this integral. The most natural choice is a radial line. Using this simplification, the force and displacement vectors will be parallel, so we may discard the vector notation: $$U=\int_{\infty}^{d}F\cdot(-dl)$$ Since we are going from infinity to the point, the $dl$ is negative. We may reverse the integration to fix this: $$U=\int_{d}^{\infty}F\cdot dl$$ Now, we may use Coulomb's Law, and plug in the expression for $F$: $$U=\int_{d}^{\infty}\frac{kqq_2}{l^2}\cdot dl$$ Moving the constants out of the integral, we are left with a trivial integral: $$U=kqq_2\int_{d}^{\infty}\frac{dl}{l^2}$$ $$U=\frac{kqq_2}{d}$$ Now, we may use the definition of electric potential to finish off the calculation: $$V=\frac{U}{q_2}$$ $$V=\frac{kq}{d}$$ Notice that the electric potential is independent of the test charge. This also holds in general, for any configuration of charges, not just for a point charge.