### Correct Option: A. The electric flux is zero when perpendicular and maximum when parallel. #### Detailed Explanation: **Understanding Electric Flux:** Electric flux (\( \Phi_E \)) is a measure of the electric field (\( \vec{E} \)) passing through a given area (\( A \)). It is mathematically defined as: \[ \Phi_E = \vec{E} \cdot \vec{A} = E A \cos(\theta) \] Where: - \( \Phi_E \) is the electric flux, - \( \vec{E} \) is the electric field vector, - \( \vec{A} \) is the area vector (which has a magnitude equal to the area and a direction normal to the surface), - \( \theta \) is the angle between the electric field lines and the area vector. **Case 1: Area Vector Perpendicular to Electric Field Lines (\( \theta = 90^\circ \))** - When the area vector is perpendicular to the electric field lines, \( \theta = 90^\circ \). - The cosine of 90 degrees is zero: \( \cos(90^\circ) = 0 \). - Therefore, the electric flux through the surface is: \[ \Phi_E = E A \cos(90^\circ) = E A \cdot 0 = 0 \] This means that no electric field lines pass through the surface, resulting in zero electric flux. **Case 2: Area Vector Parallel to Electric Field Lines (\( \theta = 0^\circ \))** - When the area vector is parallel to the electric field lines, \( \theta = 0^\circ \). - The cosine of 0 degrees is one: \( \cos(0^\circ) = 1 \). - Therefore, the electric flux through the surface is: \[ \Phi_E = E A \cos(0^\circ) = E A \cdot 1 = E A \] This indicates that the maximum number of electric field lines pass through the surface, resulting in maximum electric flux. ### Why Other Options Are Incorrect: **B. The electric flux is maximum when perpendicular and zero when parallel.** - This statement is incorrect because it reverses the conditions. The maximum flux occurs when the area vector is parallel to the electric field, not perpendicular. **C. The electric flux remains constant regardless of the angle between the area vector and the field lines.** - This option is incorrect because electric flux is dependent on the angle \( \theta \). As shown in the calculations, the flux changes based on the orientation of the area vector relative to the electric field. **D. The electric flux is minimum when perpendicular and maximum when parallel.** - This option is misleading because it states that the flux is minimum when perpendicular, which is true (it is zero), but it does not clearly state that the flux is maximum when parallel. The wording could lead to confusion about the nature of maximum and minimum values. ### Summary of Key Points: - Electric flux is calculated using the formula \( \Phi_E = E A \cos(\theta) \). - When the area vector is perpendicular to the electric field, the flux is zero. - When the area vector is parallel to the electric field, the flux is at its maximum value \( E A \). - The angle between the area vector and the electric field is crucial in determining the electric flux. This understanding is essential for solving problems related to electric fields and flux in physics.