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Find the average potential over a spherical surface of radius $R$ due to a point charge $q$ located inside. (In this case Laplace's equation does not hold within the sphere)

This is a question from Introduction to Electrodynamics by Griffth.

I know that Laplace's equation takes the form $$ \nabla^2 V = 0 $$ and I'm wondering why this is not true within the sphere. If $V$ stands for potential then from Gauss's law we have $$ \nabla^2 V = -{\rho \over \epsilon_0} $$ and hence if the position is not exactly the same as the point charge, $\rho$ should be $0$ even within the sphere.

Is my reasoning wrong somewhere, or otherwise why does the question say Laplace's equation is not valid in this case?

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Griffith meaned the Laplace equation does not hold for every point inside the sphere. As you said yourself, the Laplacian of the potential is zero everywhere inside the sphere EXCEPT at the position of the point charge. In fact, the $\rho$ here is a delta function.

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  • $\begingroup$ Yes, It tipped me off too the first time, may be the textbook should be reworded. $\endgroup$
    – Courage
    Jan 6, 2016 at 11:05

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