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A law in classical electromagnetism and Newtonian gravity which relates (charge) density to the divergence of a field, or alternatively the charge in a volume to the flux through the bounding surface.

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When I take a Gaussian surface inside an insulating solid sphere, why does the outer volume ...

Who says the outside field doesn't effect.The Gauss's law gives the net field due to entire charges inside or outside the Gaussian surface only the charge taken is what inside.
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