Question

Using general logic for electric field, the flux of $\overrightarrow{g}$ through any closed surface is given by :

• A. $\oint \overrightarrow{g}\cdot d\overrightarrow{s}=4\pi G{m}_{enclosed}$
• B. $\oint \overrightarrow{g}\cdot d\overrightarrow{s}=-4\pi G{m}_{enclosed}$
• C. $\oint \overrightarrow{g}\cdot d\overrightarrow{s}=G{m}_{enclosed}$
• D. $\oint \overrightarrow{g}\cdot d\overrightarrow{s}=-G{m}_{enclosed}$

Answer 1

Answer: (B)

$\oint \overrightarrow{g}\cdot d\overrightarrow{s}=-4\pi G{m}_{enclosed}$

From the inverse square dependence of forces, we can have an analogy of electric and gravitational forces.

The charge is analogous to mass.

Gravitation states $F\propto -m_1{m}_2$ and in Electrostatics, $F\propto q_1{q}_2$

The proportionality constant is $G$ in gravitation and $\frac{1}{4\pi {ϵ}_0}$ in electrostatics.

Thus, $G\sim \frac{1}{4\pi {ϵ}_0}$ and $m\sim q$

Gauss' law for electrostatics is $\oint \overrightarrow{E}.\overrightarrow{ds}=\frac{q_{\text{encl}}}{{ϵ}_0}=4\pi \frac{q_{\text{encl}}}{4\pi {ϵ}_0}$

The analogous quantity for electric field is $\overrightarrow{g}$

Thus, the Gauss' law for gravitation is $\oint \overrightarrow{g}.\overrightarrow{ds}=-4\pi G{m}_{\text{encl}}$