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:
$$\oint \vec E \cdot d \vec a = \frac{q_{encl}}{\epsilon_0} \leftrightarrow \oint \vec g \cdot d \vec a = -4\pi Gm_{encl}.$$
In differential form you get
$$\nabla \cdot \vec E = \frac{\rho_e}{\epsilon_0} \leftrightarrow \nabla \cdot \vec g = -4\pi G \rho_m$$
where $$\rho_e$$ is the charge density, and $$\rho_m$$ is the mass density.

 

[charng]

Yes, Gauss' Law in its equivalent form can be used to determine the gravitational field. This is known as Gauss' Law for Gravity and it states that the flux of the gravitational field through a closed surface is equal to the enclosed mass divided by the square of the distance from the center of mass.

In order to use Gauss' Law for Gravity to determine the gravitational field, we first need to choose a closed surface around the object in question. This surface could be any shape, but it is often chosen to be a sphere with the object at its center.

Next, we calculate the flux of the gravitational field through this closed surface by integrating the gravitational field over the surface. This gives us the total gravitational flux passing through the surface.

Finally, using the equation for Gauss' Law for Gravity, we can equate this flux to the enclosed mass divided by the square of the distance from the center of mass. This allows us to solve for the gravitational field at any point outside the object.

In summary, Gauss' Law for Gravity provides a useful tool for determining the gravitational field at any point outside an object by relating it to the enclosed mass. This can be applied to a variety of situations, such as finding the gravitational field of a planet or a star.