Are the Maxwell's equations enough to derive the law of Coulomb?

Question

Are the 8 Maxwell's equations enough to derive the formula for the electromagnetic field created by a stationary point charge, which is the same as the [law of Coulomb](https://en.wikipedia.org/wiki/Coulomb%27s_law) $$ F~=~k_e \\frac{q_1q_2}{r^2}~? $$ If I am not mistaken, due to the fact that Maxwell's equations are differential equations, their general solution must contain arbitrary constants. Aren't some boundary conditions and initial conditions needed to have a unique solution. How is it possible to say without these conditions, that a stationary point charge does not generate magnetic field, and the electric scalar potential is equal to $$\\Phi(\\mathbf{r})=\\frac{e}{r}.$$ If the conditions are needed, what kind of conditions are they for the situation described above (the field of stationary point charge)?

Answer 1

If I remember correctly, we have the boundary condition that the potential energy at an infinite distance is zero for a test charge moved from infinity to the distance r from the point charge. Then writing the charge distribution as a Dirac delta function in position vector r for the point charge we obtain the expression for the electric field vector for a point charge of magnitude + q upon integration

The integral version of Gauss's equation can thus be rewritten as

This yields Coulombs law for the point charge in free space.