If electric field around a surface is given by `|vec(E)|=(Q_(in))/(epsilon_0|A|)` where 'A' is the normal area of surface and `Q_(in)` is the charge enclosed by the surface. This relation of gauss's law is valid when

Options

A. Surface is equipotential
B. Magnitude of electric field is constant
C. Magnitude of electric field is constant & the surface is equipotential
D. For all Gaussian surfaces.

Explanation

To solve the question regarding the validity of Gauss's law expressed as \(|\vec{E}| = \frac{Q_{in}}{\epsilon_0 |A|}\), we need to analyze the conditions under which this relationship holds true.

### Step-by-Step Solution:

1. **Understanding Gauss's Law**: Gauss's law states that the electric flux \(\Phi_E\) through a closed surface is proportional to the charge \(Q_{in}\) enclosed by that surface. Mathematically, it is expressed as:
\[ \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{in}}{\epsilon_0} \]
where \(\vec{E}\) is the electric field and \(d\vec{A}\) is the differential area vector on the surface.

2. **Electric Flux Calculation**: The electric flux through a surface can be calculated as:
\[ \Phi_E = \int \vec{E} \cdot d\vec{A} \]
If the electric field is uniform and constant over the surface, we can simplify this to:
\[ \Phi_E = \vec{E} \cdot \vec{A} = E |A| \cos \theta \]
where \(\theta\) is the angle between the electric field vector and the area vector.

3. **Conditions for the Formula**: For the relation \(|\vec{E}| = \frac{Q_{in}}{\epsilon_0 |A|}\) to hold, we need:
- The electric field \(\vec{E}\) must be uniform across the surface.
- The angle \(\theta\) between \(\vec{E}\) and the area vector \(\vec{A}\) must be zero (i.e., \(\cos \theta = 1\)), which implies that the electric field is perpendicular to the surface.

4. **Conclusion**: Therefore, the relation is valid under the following conditions:
- The electric field is constant in magnitude and direction across the surface.
- The surface is an equipotential surface, meaning that the electric field lines are perpendicular to the surface.

### Final Answer:
The relation of Gauss's law \(|\vec{E}| = \frac{Q_{in}}{\epsilon_0 |A|}\) is valid when:
- The electric field is uniform (constant in magnitude and direction).
- The surface is equipotential (electric field lines are perpendicular to the surface).