What is the definition of electric flux through a surface?

• A. Φ_E = ∮ E · dA
• B. Φ_E = ∮ E dA
• C. Φ_E = ∮ dA · E
• D. Φ_E = ∮ E × dA

Answer: (A)

Electric flux through a surface measures how much electric field passes through that surface. To compute it, treat the area element as a vector dA pointing normal to the surface with magnitude equal to the patch area. The dot product E · dA picks out the component of the field that goes through the surface, and integrating this over the whole surface gives the total flux: Φ_E = ∬_S E · dA. For a closed surface, this is written with a closed-surface integral ∮ E · dA. The sign reflects whether E is exiting or entering the surface. Using a cross product would not produce a scalar flux, and omitting the dot product would ignore the field component normal to the surface. This definition underpins Gauss’s law, relating the flux to the enclosed charge.