PlanetPhysics/Derivation of Coulomb's Law From Gauss Law

As an example of the statement that Maxwell's equations completely define electromagnetic phenomena, it will be shown that Coulomb's law may be derived from Gauss' Law for electrostatics. Consider a point charge of charge $$q$$; we can obtain an expression for the Electric Field at a point in space due to this charge by surrounding it with a "virtual" sphere of radius $$R$$, and then using the Gauss' law in integral form: $$ \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac {q}{\epsilon_0}\mbox{ .} $$ The surface integral on the the right-hand-size of the equation can be written in spherical polar coordinates over the "virtual" sphere, considering the point charge at its centre. Under the assumption that the electric field is spherically symmetric, its value over the sphere surface is constant. Then, we can write $$ \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \int^{2\pi}_0 \int^\pi_0 R^2 E \sin \theta \,d\theta \,d\phi; $$ hence, $$ 4\pi R^2 E = \frac{q}{\epsilon_0}, $$ or $$ E = \frac{q}{4\pi\epsilon_0 R^2}. $$ The usual form can then be recovered from the Lorentz force law $$\mathbf{F} = \mathbf{E}q + \mathbf{v} \times \mathbf{B}$$, noting the absence of magnetic field.