# Gauss law for gravitational field

Gauss's law is fundamental law of electrostatics. But Can we apply Gauss's law for Gravitational field also?

• Gauß' law is simply that $\int\boldsymbol{\nabla}\cdot\boldsymbol{A}\,\mathrm{d}V = \oint\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{S}$ for any vector field $\boldsymbol{A}$. Jun 10 '15 at 14:18
– pwf
Jun 10 '15 at 18:33

Indeed, we have that:

$$\nabla \cdot \vec{g}=-4\pi G\rho$$

Where $\rho$ is the mass density. Integrating both sides, we have:

$$\iiint_{\Sigma}(\nabla \cdot \vec{g})\:\mathrm{d}V=-4\pi G\iiint_{\Sigma}\rho(\vec{r})\:\mathrm{d}^{3}\vec{r}$$

But by Gauss' Law (the divergence theorem), we have:

$$\oint_{\partial \Sigma}\vec{g}\cdot\mathrm{d}\vec{A}=-4\pi G\iiint_{\Sigma}\rho(\vec{r})\:\mathrm{d}^{3}r$$

Which is the same as the Gauss' Law for electromagnetism, so we can apply it in the same way.

I thought it might be useful to provide a (admittedly contrived) example of where this is useful.

For instance, if we imagine a narrow tunnel drilled through the earth, and dropping something through it, we note that we can find the gravity at a radius $r$ from the center by applying Gauss' Law:

$$4\pi r^{2} \vec{g} = -4\pi G \rho_{\oplus} \cdot \frac{4 \pi r^{3}}{3}\hat{r}$$

Where $\rho_{\oplus} = \frac{M_{\oplus}}{V_{\oplus}} = \frac{3M_{\oplus}}{4\pi R_{\oplus}^{3}}$ is the density of the earth. Thus:

$$\vec{g} = -\frac{GM_{\oplus}r}{R_{\oplus}^{3}}\hat{r}$$

We note that the force on an object of mass $m$ is given by:

$$\vec{F} = m\vec{g} \implies \frac{\mathrm{d}^{2}\vec{r}}{\mathrm{d}t^{2}}=\vec{g}$$

Therefore, we have:

$$r(t)=R_{\oplus}\cos\left(\sqrt{\frac{GM_{\oplus}}{R_{\oplus}^{3}}}t\right)$$

I.e. the object will exhibit Simple Harmonic Motion about the center of the earth.

According to my knowledge, Gauss Law can be applied to any function where the quantity(the force) is inversely proportional to distance squared.Now, is the force required to be conservative?

• It is not a particularly good way to answer a question with a question. Jun 10 '15 at 16:16