Academy

Electric Flux

Electric flux through a surface. For a uniform field and flat surface: Φ = EA cosθ. SI unit: N m² C⁻¹ (or V·m).
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Derivation

Physical idea

Electric flux is a measure of how many field lines pass through a surface. A field parallel to the surface contributes nothing; a field perpendicular to it contributes maximally.

Flat surface, uniform field

For a flat surface of area AA with outward normal n^\hat{n}, in a uniform field E⃗\vec{E}:

Φ=E⃗⋅n^ A=EAcos⁡θ\Phi = \vec{E} \cdot \hat{n}\, A = EA\cos\theta

where θ\theta is the angle between E⃗\vec{E} and n^\hat{n}.

General definition

For a curved surface or non-uniform field, divide the surface into infinitesimal area elements dA⃗=n^ dAd\vec{A} = \hat{n}\, dA:

Φ=∫SE⃗⋅dA⃗=∫SEcos⁡θ dA\Phi = \int_S \vec{E} \cdot d\vec{A} = \int_S E\cos\theta\, dA ΦE=∫SE⃗⋅dA⃗\boxed{\Phi_E = \int_S \vec{E} \cdot d\vec{A}}

Closed surface convention

For a closed surface, n^\hat{n} always points outward. Flux is positive when field lines exit and negative when they enter.

Units

[Φ]=N C−1⋅m2=N m2C−1=V m[\Phi] = \text{N C}^{-1} \cdot \text{m}^2 = \text{N m}^2 \text{C}^{-1} = \text{V m}
Note
Flux is a scalar. It can be zero even when $\vec{E} \neq 0$ everywhere on the surface — if equal flux enters and exits.