Are Gauss's Law and Coulomb's Law Equivalent?

[Enjolras1789]

Gauss's law may be used to derive Coulomb's law. I've never seen it done the other way around. Is this possible? Because I have not seen it, I infer that these forms are not logically equivalent.

 

[Galileo]

They are the same. Take a point charge. Since the Coulomb field is everywhere divergenceless except at the position of the charge, you can show it doesn't matter what shape your surface containing the charge is, the flux through any surface containing the charge is the same. You can easily evaluate the flux using a sphere with the charge at the origin. This gives Gauss' law for a point particle, but you can extend it to any charge distribution using superposition.

 

[neutrino]

Almost every intro textbook in EM (at least the more popular ones) introduce Coulomb Law and use it to derive the Gauss Law. It's usually done for the case for the point-particle, and then generalised to a continuous charge distribution.