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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?

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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.

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  • $\begingroup$ 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? $\endgroup$ – A4Treok Oct 16 '17 at 22:06
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    $\begingroup$ The Laplace equation is satisfied at a point where there is no net charge. $\endgroup$ – velut luna Oct 16 '17 at 22:07
  • $\begingroup$ Can two (opposite) charges occupy the same space? Wouldn't it be required that there is no charge at all? $\endgroup$ – A4Treok Oct 16 '17 at 22:10
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    $\begingroup$ Classical EM is an effective theory for macroscopic scales. $\endgroup$ – velut luna Oct 16 '17 at 22:12

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