### Correct Option: A. Zero ### Detailed Explanation: **Understanding Electric Flux:** Electric flux (\( \Phi_E \)) through a surface is a measure of the electric field (\( \vec{E} \)) passing through that surface. It is mathematically defined as: \[ \Phi_E = \vec{E} \cdot \vec{A} = E \cdot A \cdot \cos(\theta) \] Where: - \( \Phi_E \) is the electric flux, - \( \vec{E} \) is the electric field vector, - \( \vec{A} \) is the area vector of the surface, - \( A \) is the magnitude of the area, - \( \theta \) is the angle between the electric field lines and the area vector. **Area Vector and Electric Field:** The area vector (\( \vec{A} \)) is defined to be perpendicular to the surface and has a magnitude equal to the area of the surface. When the area vector is perpendicular to the electric field lines, the angle \( \theta \) between the electric field vector and the area vector is \( 90^\circ \). **Calculating Electric Flux:** Using the formula for electric flux, we can substitute \( \theta = 90^\circ \): \[ \Phi_E = E \cdot A \cdot \cos(90^\circ) \] Since \( \cos(90^\circ) = 0 \), we find: \[ \Phi_E = E \cdot A \cdot 0 = 0 \] Thus, the electric flux through the surface is zero when the area vector is perpendicular to the electric field lines. ### Why Other Options Are Incorrect: **Option B: Equal to the product of the electric field strength and the area** - This option suggests that the electric flux is simply \( E \cdot A \). However, this is only true when the area vector is parallel to the electric field lines (i.e., \( \theta = 0^\circ \)). Since in this case \( \theta = 90^\circ \), this option is incorrect. **Option C: Equal to the area of the surface divided by the permittivity of free space** - This option is misleading. The electric flux is not defined in terms of the area divided by the permittivity of free space. The permittivity of free space (\( \varepsilon_0 \)) is relevant in the context of electric fields and charge distributions, but it does not apply to the calculation of electric flux in this scenario. **Option D: Negative of the product of the electric field strength and the area** - This option implies that the electric flux is negative, which would only be the case if the area vector were oriented opposite to the electric field (i.e., \( \theta = 180^\circ \)). Since the area vector is perpendicular to the electric field in this scenario, this option is also incorrect. ### Summary of Key Points: - Electric flux is calculated using the formula \( \Phi_E = E \cdot A \cdot \cos(\theta) \). - When the area vector is perpendicular to the electric field lines, \( \theta = 90^\circ \), leading to zero electric flux. - The other options misinterpret the relationship between the electric field, area, and angle, leading to incorrect conclusions. ### Revision Summary: - Electric flux (\( \Phi_E \)) is zero when the area vector is perpendicular to the electric field. - The formula for electric flux includes the cosine of the angle between the electric field and area vector. - Understanding the orientation of vectors is crucial in determining electric flux. - Always check the angle \( \theta \) when applying the electric flux formula.