Definition

General definition

Φ_{E} = \int_{surface} E \cdot d A

Statement

For a curved surface or a varying field, cut it into patches and add the contributions.

Symbols and units

$Φ_{E}$	electric flux	N·m²/C
$E$	electric field	N/C
$d A$	area element, pointing outward on a closed surface	m²

When it applies

Any field, any surface. On a closed surface dA points outward.

Derivation

Any small enough patch is flat and sees a nearly uniform field:

d Φ_{E} = E \cdot d A

Add over the whole surface:

Φ_{E} = \int E \cdot d A

For a closed surface write the integral with a circle, and let dA point outward everywhere:

Φ_{E} = \oint E \cdot d A

Watch out for

On a closed surface, lines going out count positive and lines coming in count negative. That sign convention is what makes Gauss's law come out clean.

Connects to

Uniform field, flat surface Gauss's law