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Consider the Earth to be a flat infinite plane having linear mass density equal to the mass density of the actual earth.

  1. Can there be an analogous Gauss' law that can give the gravitational field for any point on a particular Gaussian surface.

  2. If yes, what would be the substitute for the permittivity constant that appears in the Gauss' law for an electric field?

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Any inverse square law can be substituted by a Gauss law. In Gravitation, gravitational field

$$E_{g}(R)=\frac{GM}{R^{2}}$$

Think of a sphere of Radius $R$ around the object of mass $M$ (This can be generalized to any shape). This gravitational flux coming out of it is $$\phi_{g}= E_{R}\times4\pi R^{2}=4\pi GM$$

So the gauss law will read as

$$\phi_{g}=4\pi G\times M_{enclosed}$$

Similarly for electric charge

$$\phi_{e}=\frac{q_{enclosed}}{\epsilon_{0}}$$

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