Electric Flux
How much field passes through a surface.
Lesson at a glanceShow the key points
- Area vector: magnitude ΔS along the positive normal.
- ΔΦ = E⃗·ΔS⃗ = EΔS cos θ: maximum face-on, zero edge-on, negative beyond 90°.
- Φ = ∫E⃗·dS⃗ for any surface; for a closed surface ∮E⃗·dS⃗ with the outward normal positive.
- A uniform field gives zero total flux through any closed surface.
Gauss’s law ties the field on a closed surface to the charge inside it. To state it we first need a way to measure “how much field passes through a surface”. That measure is called flux.
Flux through a small flat surface
Take a small flat patch of area ΔS in a uniform field E⃗. Pick one side of the patch and draw the perpendicular on that side: this is the positive normal. Let the field make an angle θ with it.
- Field along the normal (θ = 0): ΔΦ = EΔS, the largest possible.
- Field opposite to the normal (θ = π): ΔΦ = −EΔS.
- Field along the surface (θ = π/2): ΔΦ = 0. Nothing passes through.
Think of the field lines: flux is proportional to the number of lines that pierce the surface. Tilt the plate and fewer lines get through, exactly as cos θ says.
A rectangle 10 cm × 20 cm sits in a uniform field of 300 N/C. Its positive normal makes 60° with the field. Find the flux.
A field of 250 N/C points along +x. A square of edge 20 cm lies in the y–z plane with its positive normal along +x. Find the flux.
Curved surfaces and non-uniform fields
Flux is a scalar, so the flux through a surface made of pieces is the sum of the fluxes through the pieces. Break any surface into tiny, nearly flat bits, small enough that E barely changes across each, and add:
A closed surface encloses a volume: a sphere, or a cylinder together with its two flat end caps. (A hemisphere, or a cylinder without its caps, is open.) For a closed surface we write a small circle on the integral, , and always take the outward normal as positive. Field leaving the volume counts positive; field entering counts negative.
A closed cylinder of radius 3.0 cm has its axis along a uniform field of 400 N/C. Find the flux through the curved side, each end, and the whole surface.
- 1Area vector: magnitude ΔS, along the positive normal.
- 2ΔΦ = E⃗·ΔS⃗ = EΔS cos θ; maximum face-on, zero edge-on, negative past 90°.
- 3Any surface: Φ = ∫E⃗·dS⃗. Closed surface: ∮E⃗·dS⃗ with the outward normal positive.
- 4Flux is a scalar; unit N m²/C = V m.
Check yourself
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