Electric flux and Gauss's law

Electric flux measures how much electric field passes through a surface, like counting the field lines that cross it.

For a flat area S in a uniform field: Φ = E · S = ES cos θ, where θ is the angle between E and the normal (the perpendicular) to the surface. Unit: N m²/C.

• Field straight through (θ = 0): Φ = ES, the largest.
• Field along the surface (θ = 90°): Φ = 0; no lines pass through.

Example: E = 200 N/C, perpendicular to an area of 0.5 m²: Φ = 100 N m²/C.

For a closed surface, the normal points outwards. Lines coming out count as positive flux, lines going in as negative.

Gauss's law: the total flux out of any closed surface equals the charge inside divided by ε₀:

Φ = q_inside / ε₀

Example: a closed surface holds +5 µC and –2 µC. Φ = 3 × 10⁻⁶ / 8.854 × 10⁻¹² ≈ 3.4 × 10⁵ N m²/C.

Consequences to remember:

• The shape and size of the surface do not matter, only the charge inside.
• Charges outside the surface add nothing to the net flux: every line that enters also leaves.
• Zero net flux means zero net charge inside, not necessarily no charges at all (+q and –q also give zero).
• A charge at the centre of a cube sends one sixth of its flux through each face, by symmetry.