Electric Flux

A field can pass straight through a surface, slide along it, or do anything in between, and nothing about the field alone separates those cases — the difference lives in the surface's orientation. So every patch of surface is handed a unit normal n̂, and the patch becomes a vector dA = n̂ dA: direction along the normal, magnitude the patch's area in m². A flat surface needs only one such vector, A = A n̂. A 0.40 m × 0.30 m rectangle lying in the xy plane has area 0.12 m², so A = (0.12 m²) ẑ. Two normals are available at every point, n̂ and −n̂, and the geometry prefers neither. On an open surface you choose one and you state the choice. On a closed surface — one that bounds a volume, with an inside and an outside — convention takes the outward normal everywhere, and that is what makes a closed-surface flux unambiguous.

For a uniform field on a flat surface, Φ_E = EA cos θ, where θ is the angle between E and n̂. Read it two ways. Project the field: E cos θ is the part running through the surface, while E sin θ runs along it and crosses nothing. Or project the area: A cos θ is the area of the surface's shadow cast along the field direction, so the flux is the field multiplied by the area the field actually sees. Take E = 250 N C⁻¹ and A = 0.12 m². Head-on, θ = 0, gives 30 N m² C⁻¹. At θ = 30° it is 26 N m² C⁻¹; at 60°, 15 N m² C⁻¹; at 90°, exactly zero — even though the field at the surface is as strong as it ever was. Nothing about E changed; only how much of it crosses.

Suppose the field points along +x with a magnitude that grows with height, Eₓ = cy with c = 400 N C⁻¹ m⁻¹, and the surface is the square 0 ≤ y ≤ 0.50 m, 0 ≤ z ≤ 0.50 m in the plane x = 0, normal +x̂. No single value of E describes it, so cut the square into strips chosen so that E⋅n̂ is constant along each one: horizontal strips of height dy and width 0.50 m, area dA = 0.50 dy. Then dΦ_E = (400y)(0.50 dy), and Φ_E = 200 ∫ y dy from 0 to 0.50 = 200 × 0.125 = 25 N m² C⁻¹. Check it: the field averages 400 × 0.25 = 100 N C⁻¹ over the square, and 100 × 0.25 m² = 25 N m² C⁻¹. Choosing strips on which E⋅n̂ does not change is the whole art, and it is the skill the Gaussian-surface topics later turn into a method.

Reverse the normal and the flux reverses, so an open surface's flux carries a sign only once you have said which face is the front. A closed surface removes the choice: outward is positive, so negative flux is field entering and positive flux is field leaving. Put a cube of side 0.20 m in a uniform 250 N C⁻¹ field along +x, two faces square to the field. Four faces have n̂ perpendicular to E, so E⋅n̂ = 0 and they carry nothing at all. The upstream face has n̂ = −x̂ and gives 250 × 0.040 × cos 180° = −10 N m² C⁻¹. The downstream face has n̂ = +x̂ and gives +10 N m² C⁻¹. The net flux out of the closed cube is zero — every line that entered has left.

Centre a sphere of radius r on a point charge q. On that sphere E is radial, so parallel to n̂ at every point, and its magnitude q/(4πε₀r²) is the same at every point, which lets the integral collapse to a product: Φ_E = E × 4πr² = q/ε₀. The radius has vanished. For q = 4.0 nC, Φ_E = (4.0 × 10⁻⁹ C)/(8.854 × 10⁻¹² C² N⁻¹ m⁻²) = 4.5 × 10² N m² C⁻¹ on any concentric sphere. At r = 0.30 m that is a field of 4.0 × 10² N C⁻¹ over 1.13 m²; at r = 0.60 m it is 99.9 N C⁻¹ over 4.52 m²; both give 4.5 × 10² N m² C⁻¹. The 1/r² in the field and the r² in the area cancel exactly. Generalising that one cancellation to any surface and any charge is the next topic's business.

Draw field lines with their areal density proportional to E and the flux through a surface becomes a count: the net number of lines piercing it, positive where they leave through n̂ and negative where they enter. That is why tilting a loop lets fewer lines through, and why the cube above nets zero. But one number for an entire surface is a lossy summary. Φ_E = 0 through a closed surface does not mean E = 0 anywhere on it: the cube sat in a 250 N C⁻¹ field throughout. A large flux does not say which part of the surface the field crossed, nor whether it crossed evenly or all in one place. Flux answers how much, net, and nothing else — which is exactly why it needs a law to make it useful.