(Solution) - Show that the average field inside a sphere of radius -(2025 Original AI-Free Solution)

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Show that the average field inside a sphere of radius R, due to all the charge within the sphere, is where p is the total dipole moment. There are several ways to prove this delightfully simple result. Here's one method:
(d) While you're at it, show that the average field over the sphere due to all the charges outside is the same as the field they produce at thecenter.