Understanding Charge Density when using Laplace's Equation

I'm following Griffith's Intro to Electrodynamics, and when discussing Laplace's Equation, it states

Very often we are interested in finding the potential in a region where $\rho = 0$.

So, if you had a sphere with equal but opposite charge densities distributed over the northern and southern hemispheres, would Laplace's equation hold only over the entire sphere? But you couldn't analyze a hemisphere using the same method?

The Laplace equation is a differential equation which tells us how the potential varies locally at a point, so is the volume charge density $\rho$.
It tells you that if you have $\rho=0$ at a point, then the potential satisfies Laplace equation at this point.
For your case, you cannot apply the Laplace equation at any point inside the sphere, because $\rho \ne 0$ at every point. Instead, the potential satisfies the Poisson's equation.
• The only place you would have $\rho = 0$ at a single point would be places where there is absolutely no charge- is that right? Does that mean Laplace's equation describes the potential in open space? Oct 16, 2017 at 22:06