Is Gauss' Law Applicable to Gravitational Fields?

[iitjee10]

Can gauss law in its equivalent form be used to determine the gravitational field??

If so how??

 

[adriank]

Yes. Notice how gravity corresponds with electrostatics; for point masses/charges you have
$$\vec E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat r \leftrightarrow \vec g = -G \frac{m}{r^2} \hat r.$$
Thus you have the correspondences $$q \leftrightarrow m, 1/4\pi\epsilon_0 \leftrightarrow -G$$.

From Gauss's law (in integral form) for electrostatics, you can get the corresponding equation for gravity:
$$\oint \vec E \cdot d \vec a = \frac{q_{encl}}{\epsilon_0}<br /> \leftrightarrow<br /> \oint \vec g \cdot d \vec a = -4\pi Gm_{encl}.$$
In differential form you get
$$\nabla \cdot \vec E = \frac{\rho_e}{\epsilon_0}<br /> \leftrightarrow<br /> \nabla \cdot \vec g = -4\pi G \rho_m$$
where $$\rho_e$$ is the charge density, and $$\rho_m$$ is the mass density.