AP Physics C: Electricity and Magnetism · Unit 8 · Topic 3 of 4 · 5 min lesson

Electric Flux

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.

Quiz this topic · 4 questions

The big idea

Electric flux measures how much field passes through a surface: Φ_E = ∫E · dA, which for a flat surface in a uniform field is EA cos θ, with θ measured from the surface’s normal. Through a closed surface, the net flux depends only on the charge inside.

1. Flux through a surface

Tilt the surface and less field passes through it. Turn it edge-on (θ = 90°) and the flux is zero.

Φ = ∫E · dAdefinition (N·m²/C)

Φ = EA cos θflat surface, uniform E

Closed surface: dA points outwardfield leaving is +, entering is −

∮E · dA = Q_enc/ε₀Gauss’s law

2. Closed surfaces

Think of flux as counting field lines. Every line that starts on a charge inside has to leave through the surface somewhere, so the total depends only on Q_enc, not the surface’s size or shape.

Charges outside contribute lines that enter and then leave, adding zero.

A charge at the center of a cube sends equal flux through each of the 6 faces: q/(6ε₀) each.

A uniform field through a box with no charge inside: what enters one side leaves the other, net flux 0.

3. Example, then try it

Example

A uniform 400 N/C field passes through a flat 0.30 m² surface whose normal is 30° from the field. Find the flux. What if the surface is turned so its normal is perpendicular to E?

Φ = (400)(0.30) cos 30° ≈ 104 N·m²/C.

Normal ⊥ E: cos 90° = 0, so Φ = 0.

Try it · work it on paper first

A 3.0 nC point charge sits at the center of a cube 0.20 m on a side. Find the net flux through the cube and through one face. What changes if the charge moves off center (still inside)? What if a second, −3.0 nC charge is added inside?

Net flux: Φ = q/ε₀ = (3.0 × 10⁻⁹) ÷ (8.85 × 10⁻¹²) ≈ 340 N·m²/C. The cube’s size doesn’t matter.

One face: by symmetry, 340 ÷ 6 ≈ 57 N·m²/C.

Off center: the total is still 340 N·m²/C, but the faces no longer share it equally.

Add −3.0 nC: Q_enc = 0, so the net flux is 0, though the field itself isn’t zero.

Answer: 340 N·m²/C total, 57 per face; off center the total is unchanged; with −3.0 nC added, zero.

4. Common mistakes

Measuring θ from the surface.θ is from the normal.

“Zero net flux means zero field.”It means zero net charge inside.

Thinking a bigger Gaussian surface changes the flux.Only the enclosed charge matters.

Check yourself

A Gaussian sphere surrounds a point charge. Its radius is doubled. What happens to the net flux through it?

Φ = Q_enc/ε₀. The same charge is enclosed, so the flux is the same. (E drops as 1/r², but the area grows as r².)

Ready to practice?

4 questions on Electric Flux, each with a hint and an explanation for every wrong answer.

Quiz this topic All of Unit 8