Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up.
Sign up to join this community
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 $\rho$.
One cannot say because the total net charge inside a sphere is zero, so we can apply the equation inside the sphere.
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.
