When no charge is confined with in the Gauss's surface, it implies that -

Options

A. E = 0
B. `vec(E)` and `vec(ds)` are parallel
C. `vec(E)` and `vec(ds)` are mutually perpendicular
D. `vec(E)` and `vec(ds)` are inclined at some angle

Answer

To solve the question, "When no charge is confined within the Gauss's surface, it implies that -", we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Gauss's Law**:
Gauss's Law states that the electric flux (Φ_E) through a closed surface is proportional to the charge (q_enclosed) enclosed within that surface. Mathematically, it is expressed as:
\[ \Phi_E = \frac{q_{\text{enclosed}}}{\epsilon_0} \]
where ε₀ is the permittivity of free space.

2. **Applying the Condition of No Charge**:
If there is no charge enclosed within the Gauss's surface, then:
\[ q_{\text{enclosed}} = 0 \]
Substituting this into Gauss's Law gives:
\[ \Phi_E = \frac{0}{\epsilon_0} = 0 \]
This means that the total electric flux through the closed surface is zero.

3. **Interpreting Zero Electric Flux**:
The electric flux (Φ_E) is defined as the dot product of the electric field (E) and the area vector (A) of the surface:
\[ \Phi_E = \int \mathbf{E} \cdot d\mathbf{A} \]
Since Φ_E = 0, this implies that the integral of the electric field over the closed surface is zero.

4. **Understanding the Dot Product**:
The dot product being zero indicates that the electric field vector (E) and the area vector (A) are perpendicular to each other. This can be expressed as:
\[ \mathbf{E} \cdot d\mathbf{A} = E \cdot A \cdot \cos(\theta) = 0 \]
Here, θ is the angle between the electric field vector and the area vector.

5. **Conclusion**:
Therefore, when there is no charge within the Gauss's surface, it implies that the electric field is such that it is perpendicular to the area vector at all points on the surface. This means that the electric field lines do not penetrate the surface.

### Final Answer:
When no charge is confined within the Gauss's surface, it implies that the electric field is perpendicular to the area vector of the surface.