Electric Flux and Gauss’s Theorem

Lesson

Part 1 — how much field passes through a surface, and a theorem that links it to the charge inside.

In this lesson you will

Calculate electric flux through a plane surface in a uniform field
State Gauss’s theorem and use it to find the flux through closed surfaces
Choose a suitable Gaussian surface for a symmetric charge distribution

Concept

Flux of a vector field

Think of water flowing through a wire loop: the amount passing through depends on the flow speed, the loop’s area and how it is tilted. The area vector $\Delta\vec S$ has magnitude equal to the area and direction normal to it (outward for a closed surface).

Flux is largest when the loop faces the flow and zero when it lies edge-on.
Electric flux applies the same idea to the electric field.

Diagram

Electric flux

For a plane area $\Delta S$ in a uniform field: $\Delta\Phi = \vec E\cdot\Delta\vec S = E\,\Delta S\cos\theta$ For a non-uniform field or a curved surface, add up small pieces: $\Phi = \oint \vec E\cdot d\vec S$ . Unit: N m² C⁻¹ (= V m). Flux is a scalar.

θ = 0°: $\Phi = E\Delta S$ (maximum).
θ = 90°: $\Phi = 0$ (field parallel to the surface).
θ = 180°: $\Phi = -E\Delta S$ (field entering against the outward normal).

Cookies