Electric flux

Electric flux measures how much electric field passes through a surface.

Uniform Field and Flat Surface

For a uniform field and a flat surface, the flux equals the dot product of the electric field vector and the area vector, where θ is the angle between the field E⃗ and the area vector A⃗ (the surface normal):

$\Phi_{\mathrm{E}}$ = EAcosθ

• The area vector is perpendicular to the surface, with magnitude equal to the area
• For a closed surface, the area vector always points outward

Sign of the Flux

The sign of the flux is determined by the dot product:

• Positive when field lines exit the surface
• Negative when field lines enter the surface
• Zero when the field runs parallel to the surface (i.e. lies in the plane of the surface)

Surface Integral

When the field varies across a surface, or the surface is curved, the total flux requires a surface integral:

$\Phi_{\mathrm{E}}$ = ∫ E⃗ · dA⃗

This integral sums the flux contribution from every infinitesimal patch of area.

Closed Surface in a Uniform Field

For a closed surface in a uniform field with no enclosed charge, every field line that enters also exits, so the net flux is zero. A nonzero net flux through a closed surface means charge is enclosed. This concept is the foundation for Gauss's law, which directly relates the total flux through a closed surface to the enclosed charge:

$\Phi_{\mathrm{E}}$ = ∮ E⃗ · dA⃗ = $q_{\mathrm{enc}}$ / ε₀