A closed surface measures net enclosed charge
You will be able to: Apply Gauss’s law while distinguishing enclosed charge from all field sources.
Does zero net flux mean zero field everywhere?
A uniform electric field enters one side of an empty box and leaves the other. Its net flux is zero, although the field on both faces is nonzero. A charge inside the box would change the balance.
A useful starting point: Add local flux when the field varies across a surface →
Words and symbols before equations
- Gaussian surface
- An imaginary closed three-dimensional surface used for a flux calculation.
- Enclosed charge Q_enc
- The algebraic sum of all charge inside that surface.
- Closed-surface integral ∮
- A sum over every part of a closed surface using outward normals.
- Vacuum permittivity ε₀
- Use ε₀ = 1/(4πk) ≈ 8.84 × 10⁻¹² C²/(N·m²) with this unit’s rounded k.
What this picture assumes
A central point charge and an external point charge at x = 4 m. The spherical Gaussian radius remains below 4 m. Net flux uses only enclosed charge; the rightmost surface-point field includes both sources. The external charge breaks spherical field symmetry.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Net flux 226.2 N·m²/C. At the rightmost surface point x = R, Eₓ = 15 N/C. This local value cannot be multiplied by the full area when the external source breaks symmetry.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Gauss’s law states ∮E·dA = Q_enc/ε₀. E includes the contribution of every source, inside and outside the surface. Only enclosed net charge appears on the right because an outside source gives zero net closed-surface flux.
The law always relates flux and enclosed charge. It does not generally let you replace the left side with EA: field strength and direction can vary over a surface. That shortcut requires justified symmetry.
A zero enclosed total may mean no charge, equal positive and negative enclosed charges, or another canceling sum. It does not imply zero local field. Gauss’s law is one of Maxwell’s equations; no physical shell needs to be built.
| Question | Net closed-surface flux | Electric field at a point |
|---|---|---|
| Depends on | Net enclosed charge | All source charges |
| Zero means | Net enclosed charge is zero | Vector contributions cancel there |
| Units | N·m²/C | N/C |
A worked example, step by step
A closed surface encloses +3 nC and −1 nC. An additional +5 nC charge lies outside. Find the total outward flux.
- Add only enclosed charges for Q_enc = +2 nC.
- Apply Φ = Q_enc/ε₀ using ε₀ = 1/(4π × 9 × 10⁹).
- Φ = 2 × 10⁻⁹ × 4π × 9 × 10⁹ = 72π ≈ 226 N·m²/C.
- The outside charge changes local E but contributes zero net flux through this closed surface.
E in Gauss’s law includes all sources; zero net flux does not mean zero field.
Does enlarging a surface change flux if it encloses the same charges?
Compare with an explanation
No. The net flux stays Q_enc/ε₀, even though local field and surface area may change.
Predict. Change one thing. Explain.
Change the enclosed and external charges independently. Compare total flux with the illustrative field at the right side of the Gaussian sphere. The outside source is always beyond the sphere.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Net flux 226.2 N·m²/C. At the rightmost surface point x = R, Eₓ = 15 N/C. This local value cannot be multiplied by the full area when the external source breaks symmetry.
A central point charge and an external point charge at x = 4 m. The spherical Gaussian radius remains below 4 m. Net flux uses only enclosed charge; the rightmost surface-point field includes both sources. The external charge breaks spherical field symmetry.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant charge, vector superposition, electric field, flux or symmetry relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA sphere encloses +4 nC and −4 nC while a positive charge remains outside. Find net flux and explain why E need not be zero on the sphere.
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Compare with the answer and four-point rubric
- 1 point: Q_enc = +4 − 4 = 0 nC.
- 1 point: Gauss’s law gives net Φ = 0.
- 1 point: Both enclosed charges and the external source contribute local fields.
- 1 point: Those local fields can be nonzero while the signed surface contributions cancel.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What is Q_enc?
The signed sum of charges inside the closed surface.
RECALL 2What does an outside charge contribute to net flux?
Zero, though it changes local E.
RECALL 3Is every Gaussian surface an equipotential surface?
No. It is simply a chosen closed mathematical surface.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A closed surface measures net enclosed charge
- ∮E·dA = Q_enc/ε₀.
- Outward normals define positive flux.
- EA shortcuts require symmetry, not just a closed shape.
Remember: E in Gauss’s law includes all sources; zero net flux does not mean zero field.
Conditions: A central point charge and an external point charge at x = 4 m. The spherical Gaussian radius remains below 4 m. Net flux uses only enclosed charge; the rightmost surface-point field includes both sources. The external charge breaks spherical field symmetry.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 1 (official Unit 8) · Objectives 8.6.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.6, objectives 8.6.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 1: Electric Charges, Fields, and Gauss’s Law, numbered Unit 8 in the official combined Physics C sequence. Topics 8.1–8.6 retain their official identifiers. Quantitative force examples use at most four point charges. Field integrals use the specified rods, ring, arc and infinite wire; Gauss-law field calculations use spherical, cylindrical or planar symmetry. Optional projected 3D views clarify area normals and geometry; camera rotation never changes the physics. Checked with the Fall 2026 clarifications. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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