Let `S` be an imaginary closed surface enclosing mass `m`. Let `dvecS` be an element of area on `S`, the direction of `dvecS` being outward from `S`. Let `vecE` be the gravitational intensity at `dvecS`. We define `phi=oint_(s)E*dS`, the integration being carried out over the entire surface `S`.

Options

A. `phi=-Gm`
B. `phi=-4piGm`
C. `phi=-(Gm)/(4pi)`
D. No relation of the type `(a)` , `(b)` or `(c )` can exist

Answer

To solve the problem, we need to calculate the gravitational flux \(\phi\) through the closed surface \(S\) that encloses a mass \(m\). The gravitational intensity \(\vec{E}\) at a distance \(r\) from the mass \(m\) can be expressed using Newton's law of gravitation.

### Step-by-Step Solution:

1. **Understanding Gravitational Intensity**:
The gravitational intensity \(\vec{E}\) at a distance \(r\) from a mass \(m\) is given by:
\[ \vec{E} = -\frac{Gm}{r^2} \hat{r} \]
where \(G\) is the gravitational constant and \(\hat{r}\) is the unit vector pointing radially outward from the mass.

2. **Defining the Area Element**:
The area element \(d\vec{S}\) on the closed surface \(S\) is directed outward, and its magnitude is \(dS\). Thus, we can write:
\[ d\vec{S} = dS \hat{n} \]
where \(\hat{n}\) is the outward unit normal vector.

3. **Calculating the Gravitational Flux**:
The gravitational flux \(\phi\) through the surface \(S\) is defined as:
\[ \phi = \oint_{S} \vec{E} \cdot d\vec{S} \]
Substituting the expressions for \(\vec{E}\) and \(d\vec{S}\):
\[ \phi = \oint_{S} \left(-\frac{Gm}{r^2} \hat{r}\right) \cdot (dS \hat{n}) \]

4. **Considering Symmetry**:
For a symmetric surface (like a sphere), at any point on the surface, the direction of \(\hat{n}\) is the same as \(\hat{r}\). Thus, we can simplify the dot product:
\[ \phi = -\frac{Gm}{r^2} \oint_{S} dS \]

5. **Calculating the Total Area**:
The total surface area \(A\) of a sphere of radius \(r\) is:
\[ A = 4\pi r^2 \]
Therefore, substituting this into the flux equation:
\[ \phi = -\frac{Gm}{r^2} \cdot 4\pi r^2 \]

6. **Final Expression for Flux**:
Simplifying the expression gives:
\[ \phi = -4\pi Gm \]

### Conclusion:
The gravitational flux \(\phi\) through the closed surface \(S\) is:
\[ \phi = -4\pi Gm \]