**Correct Option: A. The surface is parallel to the electric field lines.** ### Detailed Explanation: To understand why the correct answer is A, we need to delve into the concept of electric flux and how it relates to the orientation of surfaces in an electric field. **1. Understanding Electric Flux:** Electric flux (\( \Phi_E \)) through a surface is defined as the product of the electric field (\( E \)) and the area (\( A \)) of the surface, as well as the cosine of the angle (\( \theta \)) between the electric field lines and the normal (perpendicular) to the surface. The formula for electric flux is given by: \[ \Phi_E = E \cdot A \cdot \cos(\theta) \] Where: - \( \Phi_E \) is the electric flux, - \( E \) is the magnitude of the electric field, - \( A \) is the area of the surface, - \( \theta \) is the angle between the electric field lines and the normal to the surface. **2. Analyzing the Condition of Zero Electric Flux:** When the electric flux through a surface is zero (\( \Phi_E = 0 \)), it implies that: \[ E \cdot A \cdot \cos(\theta) = 0 \] This equation can be satisfied in two scenarios: - The electric field strength \( E \) is zero (which is not the case here since we are considering a uniform electric field). - The cosine term is zero, which occurs when \( \theta = 90^\circ \) (the surface is perpendicular to the electric field lines) or when the area vector is oriented such that no field lines pass through it. However, if the surface is perpendicular to the electric field lines, the flux would not be zero; instead, it would be maximized. Therefore, the only viable conclusion is that the surface must be oriented in such a way that it does not "capture" any electric field lines, which occurs when the surface is parallel to the electric field lines. **3. Why Other Options Are Incorrect:** - **Option B: The surface is perpendicular to the electric field lines.** - This option is incorrect because if the surface were perpendicular to the electric field lines, the electric flux would be at its maximum, not zero. The electric field lines would pass through the surface, contributing to a positive flux. - **Option C: The surface is at an angle of 45 degrees to the electric field lines.** - This option is also incorrect. If the surface were at a 45-degree angle, the cosine of 45 degrees is \( \frac{1}{\sqrt{2}} \), which would yield a non-zero electric flux. Thus, the flux would not be zero. - **Option D: The surface has no electric field lines passing through it.** - While this option might seem plausible, it is not specific enough. A surface can have no electric field lines passing through it if it is parallel to the field lines. Therefore, this option does not accurately describe the relationship between the surface and the electric field lines as clearly as option A does. ### Summary of Key Points: - Electric flux is calculated using the formula \( \Phi_E = E \cdot A \cdot \cos(\theta) \). - A zero electric flux indicates that the surface is oriented such that it does not capture any electric field lines. - The only configuration that results in zero flux in a uniform electric field is when the surface is parallel to the electric field lines. - Other orientations (perpendicular or at an angle) would result in non-zero flux. ### Revision Summary: - Electric flux is zero when the surface is parallel to the electric field lines. - The formula for electric flux involves the angle between the electric field and the surface normal. - Perpendicular surfaces yield maximum flux, while angled surfaces yield partial flux. - Understanding the orientation of surfaces relative to electric fields is crucial for solving problems involving electric flux.