How Do Laplace and Poisson's Equations Derive from Gauss's Law?

[black_kitty]

Hi!

I must derive Laplace & Poisson's equation from the Gauss law and condition for potentiality of an electrostatic field and I have no idea how to do it!

 

[olgranpappy]

In the case of Laplace's equation, I suggest that you start off by writing down Gauss' Law in differential form in the absence of any charge. Then write down the equation for the electric field in terms of the electric potential. Then plug the latter into the former.

In the case of Poisson's equation, do the exact same thing but without setting the charge density to zero.