If `oint_(s) E.ds = 0` Over a surface, then

Options

A. The electric field inside the surface and on it is zero.
B. the electric field inside the surface is necessarily uniform
C. the number of flux lines entering the surface must be equal to the number of flux lines leaving it.
D. all charges must necessarily be outside the surface.

Explanation

To solve the problem, we need to analyze the equation given and apply the principles of electric flux and Gauss's law.

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the integral given in the question:
\[ \oint_{S} \mathbf{E} \cdot d\mathbf{s} = 0 \]
This represents the electric flux (\(\Phi\)) through a closed surface \(S\).

2. **Electric Flux Definition**:
The electric flux through a surface is defined as:
\[ \Phi = \oint_{S} \mathbf{E} \cdot d\mathbf{s} \]
If this integral equals zero, it indicates that the net electric field lines entering the surface are equal to the net electric field lines exiting the surface.

3. **Applying Gauss's Law**:
According to Gauss's law, the electric flux through a closed surface is related to the charge enclosed (\(Q_{enc}\)) by the equation:
\[ \Phi = \frac{Q_{enc}}{\epsilon_0} \]
where \(\epsilon_0\) is the permittivity of free space.

4. **Setting the Flux to Zero**:
Since we have established that \(\Phi = 0\), we can set the equation from Gauss's law to zero:
\[ \frac{Q_{enc}}{\epsilon_0} = 0 \]
This implies that:
\[ Q_{enc} = 0 \]
Therefore, there is no net charge enclosed within the surface.

5. **Physical Interpretation**:
The condition that the electric flux is zero means that the number of electric field lines entering the surface is equal to the number of electric field lines leaving the surface. This can occur when the surface is placed in a uniform electric field or when there are charges outside the surface that create an equal number of field lines entering and exiting.

6. **Conclusion**:
From the above analysis, we conclude that if \(\oint_{S} \mathbf{E} \cdot d\mathbf{s} = 0\), then:
- The charge enclosed within the surface is zero.
- The electric field lines entering the surface are equal to those exiting the surface.

### Final Answer:
The correct options based on the analysis are:
- All charges must necessarily be outside the surface.
- The electric field inside the surface and on it is not necessarily zero, but the net flux is zero.