gauss law in gravitation

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
[tex]\vec E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat r \leftrightarrow \vec g = -G \frac{m}{r^2} \hat r.[/tex]
Thus you have the correspondences [tex]q \leftrightarrow m, 1/4\pi\epsilon_0 \leftrightarrow -G[/tex].

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