The correct option for the definition of electric flux (\(\Phi_E\)) through a surface in a uniform electric field is: **D. \(\Phi_E = E \cdot A \cdot \cos(\theta)\)** ### Detailed Explanation 1. **Understanding Electric Flux**: - Electric flux (\(\Phi_E\)) is a measure of the electric field (\(E\)) passing through a given area (\(A\)). It quantifies how much electric field lines penetrate a surface. - The concept of electric flux is crucial in understanding Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed by that surface. 2. **Defining the Variables**: - \(E\): The magnitude of the electric field (measured in volts per meter, V/m). - \(A\): The area of the surface through which the electric field lines are passing (measured in square meters, m²). - \(\theta\): The angle between the electric field vector and the normal (perpendicular) vector to the surface. The normal vector is a vector that is perpendicular to the surface at the point of interest. 3. **The Formula**: - The formula for electric flux is given by: \[ \Phi_E = E \cdot A \cdot \cos(\theta) \] - Here, \(\cos(\theta)\) accounts for the orientation of the surface relative to the electric field. If the surface is perpendicular to the electric field (\(\theta = 0^\circ\)), \(\cos(0) = 1\), and the flux is maximized. Conversely, if the surface is parallel to the electric field (\(\theta = 90^\circ\)), \(\cos(90) = 0\), and the flux is zero. 4. **Why Option D is Correct**: - Option D correctly incorporates the angle \(\theta\) into the calculation of electric flux. It reflects the fact that only the component of the electric field that is perpendicular to the surface contributes to the flux through that surface. ### Why the Other Options are Incorrect or Weaker: - **Option B: \(\Phi_E = E \cdot A\)**: - This option is incorrect because it does not account for the angle \(\theta\). It assumes that the electric field is always perpendicular to the surface, which is not the case in general. Therefore, it fails to provide an accurate representation of the electric flux when the angle is not zero. ### Common Pitfalls: - **Ignoring the Angle**: A common mistake is to forget to consider the angle between the electric field and the surface normal. Always remember that the orientation of the surface affects the amount of electric field passing through it. - **Assuming Uniformity**: The formula applies to uniform electric fields. In non-uniform fields, the calculation of electric flux would require integration over the surface. ### Revision Summary: - Electric flux (\(\Phi_E\)) measures the electric field passing through a surface. - The correct formula is \(\Phi_E = E \cdot A \cdot \cos(\theta)\), where \(\theta\) is the angle between the electric field and the surface normal. - The \(\cos(\theta)\) term adjusts for the orientation of the surface relative to the electric field. - Always consider the angle when calculating electric flux to avoid errors.