Electric Flux

Electric flux measures the number of electric field lines passing through a given surface. It provides a quantitative way to describe how much of the electric field penetrates a surface.

Definition

Electric flux ($\Phi_E$) through a surface is defined as:

$$\Phi_E = \vec{E} \cdot \vec{A} = EA\cos\theta$$

Where:

Symbol Meaning Units
$\Phi_E$ Electric flux N·m²/C or V·m
$E$ Electric field magnitude N/C
$A$ Area of the surface m²
$\theta$ Angle between $\vec{E}$ and normal to surface degrees/radians

Interpretation

Flux as Field Lines

  • High flux: Many field lines pass through surface
  • Zero flux: No field lines pass through (surface parallel to field)
  • Negative flux: Field lines enter surface (angle > 90°)

Special Cases

Configuration Flux Explanation
$\vec{E}$ perpendicular to surface ($\theta = 0°$) $\Phi_E = EA$ Maximum flux
$\vec{E}$ parallel to surface ($\theta = 90°$) $\Phi_E = 0$ No field lines through
$\vec{E}$ at angle $\theta$ $\Phi_E = EA\cos\theta$ General case

Flux Through Closed Surfaces

For a closed surface (encloses a volume):

  • Net flux = (flux out) − (flux in)
  • Field lines leaving: positive contribution
  • Field lines entering: negative contribution

Net Flux and Enclosed Charge

The net flux through any closed surface depends only on the charge enclosed: $$\Phi_{net} = \frac{Q_{enclosed}}{\varepsilon_0}$$

This is Gauss's Law.

Examples

Example 1: Flux Through Rectangle

A uniform electric field $E = 200$ N/C passes through a square of side 0.10 m. The field makes 30° with the normal.

$$\Phi_E = EA\cos\theta = (200)(0.01)\cos(30°) = 1.73 \text{ N·m}^2\text{/C}$$

Example 2: Zero Flux

A surface lies parallel to uniform electric field lines. $$\theta = 90° \Rightarrow \Phi_E = EA\cos(90°) = 0$$

Example 3: Flux Through Cube

Point charge $q$ at center of cube:

  • Total flux through cube: $\Phi_{total} = \frac{q}{\varepsilon_0}$
  • Flux through one face: $\Phi_{face} = \frac{q}{6\varepsilon_0}$

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