Can Coulomb's law be used to derive Gauss's law? How?
I shall outline a better method. Not just by assuming the surface is spherical with area
So, to start with I shall prove it for the field due to a single point charge. (At O)
Since superposition principle applies to electric fields, this can be generalised to any number of charges quite easily.
There it is, given below.
Consider a charged particle
Take an arbitrary surface element
Let the normal on
Now, by Coulomb's law,
Thus, the flux through
Integrating over the entire surface,
But, the net solid angle subtended by a closed surface at an internal point is always
Which is Gauss law in integral form.
One may derive the differential form from Coulomb's law as well by taking the Divergence of the electric field and then proceeding with vector calculus methods.
The differential form looks something like this,