Chapter 13.1, Problem

Assume from electrostatics the equations $\nabla \cdot E=\rho /{ϵ}_0$ and $E=-\nabla \varphi (E=$ electric field, $\rho =$ charge density, ${ϵ}_0=$ constant, $\varphi =$ electrostatic potential). Show that the electrostatic potential satisfies Laplace’s equation (1.1) in a charge-free region and satisfies Poisson’s equation (1.2) in a region of charge density $\rho$.