Electric flux over a surface in an electric field

Options

A. Positive
B. Negative
C. Zero
D. Any of these

Answer

To solve the problem of determining the electric flux over a surface in an electric field, we can follow these steps:

### Step 1: Understand the Concept of Electric Flux
Electric flux (Φ) through a surface is defined as the product of the electric field (E) and the area (A) of the surface, taking into account the angle (θ) between the electric field and the normal (perpendicular) vector to the surface. Mathematically, it is expressed as:
\[ \Phi = E \cdot A \cdot \cos(\theta) \]

### Step 2: Define the Surface and Electric Field
Consider a Gaussian surface, which we can visualize as a cylinder with three distinct parts: the curved surface, the top face, and the bottom face. Assume that the electric field is directed upwards.

### Step 3: Analyze Each Part of the Surface
1. **Top Face**:
- The area vector (dS) is directed upwards (same direction as the electric field).
- The angle θ between E and dS is 0°.
- Therefore, the flux through the top face is:
\[ \Phi_1 = E \cdot A \cdot \cos(0°) = E \cdot A \cdot 1 = E \cdot A \quad (\text{positive}) \]

2. **Curved Surface**:
- The area vector is perpendicular to the electric field.
- The angle θ between E and dS is 90°.
- Thus, the flux through the curved surface is:
\[ \Phi_2 = E \cdot A \cdot \cos(90°) = E \cdot A \cdot 0 = 0 \quad (\text{zero}) \]

3. **Bottom Face**:
- The area vector (dS) is directed downwards (opposite direction to the electric field).
- The angle θ between E and dS is 180°.
- Therefore, the flux through the bottom face is:
\[ \Phi_3 = E \cdot A \cdot \cos(180°) = E \cdot A \cdot (-1) = -E \cdot A \quad (\text{negative}) \]

### Step 4: Conclusion
From the analysis of the three parts of the Gaussian surface, we find that:
- The flux through the top face is positive.
- The flux through the curved surface is zero.
- The flux through the bottom face is negative.

Thus, the electric flux over a surface in an electric field can be positive, negative, or zero. Therefore, the answer to the question is:
**Any of these values (positive, negative, or zero).**