If the flux of the electric field through a closed surface is zero,

Options

A. the electric field must be zero everywhere on the surface
B. the electric field may be zero everywhere on the surface
C. the charge inside the surface must be zero
D. the charge in the vicinity of the surface must be zero.

Answer

To solve the question, we need to analyze the implications of the electric flux being zero through a closed surface according to Gauss's Law.

### Step-by-Step Solution:

1. **Understanding Electric Flux**:
Electric flux (Φ) through a closed surface is defined by Gauss's Law as:
\[ \Phi = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( \mathbf{E} \) is the electric field, \( d\mathbf{A} \) is the differential area vector, \( Q_{\text{enc}} \) is the charge enclosed by the surface, and \( \epsilon_0 \) is the permittivity of free space.

2. **Given Condition**:
The problem states that the electric flux through the closed surface is zero:
\[ \Phi = 0 \]

3. **Applying Gauss's Law**:
From Gauss's Law, if the electric flux is zero, we can write:
\[ \frac{Q_{\text{enc}}}{\epsilon_0} = 0 \]
This implies that:
\[ Q_{\text{enc}} = 0 \]
Therefore, the total charge enclosed by the surface must be zero.

4. **Implications for Electric Field**:
The electric field \( \mathbf{E} \) can be zero everywhere on the surface, but it is not necessarily true that it must be zero everywhere. For example, if there are equal amounts of positive and negative charges outside the surface, they can create an electric field that cancels out at the surface, resulting in zero net flux.

5. **Conclusion**:
- The electric field may be zero everywhere on the surface (Option B).
- The charge inside the surface must be zero (Option C).
- However, the electric field does not have to be zero everywhere; it can exist due to external charges.

### Final Answer:
The correct options are:
- B: The electric field may be zero everywhere on the surface.
- C: The charge inside the surface must be zero.