The Gauss' theorem for gravitational field may be written as

Options

A. `oint g*dS = (m)/(G)`
B. `- oint g*dS = 4 pi mG`
C. `oint g*dS = (m)/(4 piG)`
D. `- oint g*dS = (m)/( G)`

Explanation

To rewrite Gauss' theorem for the gravitational field, we start by recalling the form of Gauss' theorem for electric fields and then adapt it to gravitational fields.

### Step-by-Step Solution:

1. **Understanding Gauss' Theorem for Electric Fields:**
Gauss' theorem for electric fields states that:
\[ \oint \mathbf{E} \cdot d\mathbf{s} = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( \mathbf{E} \) is the electric field, \( d\mathbf{s} \) is a differential area vector on a closed surface, \( Q_{\text{enc}} \) is the charge enclosed by the surface, and \( \epsilon_0 \) is the permittivity of free space.

2. **Comparing Electric and Gravitational Fields:**
In the case of gravitational fields, we denote the gravitational field intensity as \( \mathbf{g} \) and the mass enclosed as \( M \). The gravitational field intensity due to a mass \( M \) at a distance \( r \) is given by:
\[ \mathbf{g} = \frac{G M}{r^2} \]
where \( G \) is the gravitational constant.

3. **Rewriting Gauss' Theorem for Gravitational Fields:**
By analogy, we can write Gauss' theorem for gravitational fields as:
\[ \oint \mathbf{g} \cdot d\mathbf{s} = \frac{M_{\text{enc}}}{G} \]
where \( M_{\text{enc}} \) is the mass enclosed by the surface.

4. **Relating to the Electric Field Equation:**
From the electric field equation, we can relate the constants:
\[ \epsilon_0 = \frac{1}{4 \pi G} \]
Thus, substituting this into our gravitational Gauss' theorem gives:
\[ \oint \mathbf{g} \cdot d\mathbf{s} = 4 \pi G M_{\text{enc}} \]

5. **Including the Direction of Gravitational Field:**
Since the gravitational field is always attractive, we include a negative sign to indicate this nature:
\[ \oint \mathbf{g} \cdot d\mathbf{s} = -4 \pi G M_{\text{enc}} \]

### Final Expression:
Thus, the Gauss' theorem for the gravitational field can be written as:
\[ \oint \mathbf{g} \cdot d\mathbf{s} = -4 \pi G M_{\text{enc}} \]