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
$$ F~=~k_e \frac{q_1q_2}{r^2}~? $$
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)?
Best Answer
The short answer is yes, and in fact you only need one single Maxwell equation, Gauss's law, together with the Lorentz force, to get Coulomb's law.
More specifically, you need Gauss's law in its integral form, which is equivalent to the differential form for well-behaved fields because of Gauss's theorem. Thus, you use the law $$ \nabla\cdot\mathbf{E}=\rho/\epsilon_0\quad\Leftrightarrow\quad \oint_S\mathbf{E}\cdot\mathrm{d}\mathbf{a}=Q/\epsilon_0, $$ where $Q$ is the total charge enclosed by the (arbitrary) surface $S$.
To derive Coulomb's law, consider the electric field of a single point particle, with nothing else in the universe. Because of isotropy (which must be added as an additional postulate), the electric field at a sphere of radius $r$ centred on the charge must be radial and with the same magnitude throughout. That means the integral is trivial and the electric field must be $$\mathbf{E}=\frac{Q}{4\pi\epsilon_0 r^2}\hat{\mathbf{r}}.$$
Coupled with Lorentz's force law at zero velocity for the test particle (since Coulomb's law only holds in electrostatics) this yields Coulomb's law.
It is not obvious that this highly symmetric situation can give the general electrostatic force for multiple particles. This follows from the superposition principle, which is very much at the heart of classical electrodynamics, and which can be obtained from the linearity of Maxwell's equations. This gives you the field for a single source; add the fields for all the individual sources and you'll get the field for the collection of sources.