Add local flux when the field varies across a surface
You will be able to: Evaluate a surface integral by slicing a rectangular patch into strips.
What replaces EA when the field is not constant?
One side of a rectangular patch can have a stronger normal field than the other. Multiplying its area by the field at an arbitrary point misses that variation. Add narrow strips instead.
A useful starting point: Flux counts the field through a surface, not along it →
Words and symbols before equations
- dA
- A small area element with a chosen normal.
- Surface integral
- A sum of local dot products E·dA over a surface.
- Normal component E_n
- The signed component perpendicular to a surface at a given point.
- Average normal field
- The flux divided by total area; it need not equal the field at an edge.
What this picture assumes
A rectangle in the xy plane with normal +z and E_z(x) = E₀ + bx² on the patch. Only on-surface normal values are specified; no claim is made about the complete off-surface vector field. Signed strips may cancel. Midpoint sums approximate the exact integral.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Area 2 m²; exact flux 28 N·m²/C; 10-strip sum 27.98 N·m²/C. Average normal field 14 N/C.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For a rectangle in the xy plane, choose normal +z, width w along y and x from 0 to L. A narrow strip has dA = w dx. If the field on the patch is E_z(x) = E₀ + bx², then dΦ = (E₀ + bx²)w dx.
Integrate: Φ = w∫₀ᴸ(E₀ + bx²) dx = w(E₀L + bL³/3). The average normal field is E₀ + bL²/3, different from the far-edge field E₀ + bL².
This calculation needs only the specified field on the surface. A full physical source arrangement also determines the off-surface components. The model is a surface-integration exercise, not a claim that the field has only a z component everywhere in space.
A worked example, step by step
On a rectangle with L = 2 m and w = 1 m, E_z(x) = 10 + 3x² N/C, with x in meters. Calculate flux through normal +z.
- A strip from x to x + dx has area w dx = (1 m) dx.
- Φ = w∫₀²(10 + 3x²) dx.
- Φ = 1[10x + x³]₀² = 20 + 8 = 28 N·m²/C.
- Average normal field is 28/(2 × 1) = 14 N/C; the edge value 22 N/C is not the average.
Do not pull a position-dependent field outside an integral. The dot product is local.
When may E come outside the area integral?
Compare with an explanation
When the relevant normal component is constant over the surface, with the geometry handled correctly.
Predict. Change one thing. Explain.
Increase the strip count and compare the midpoint sum with the exact integral. Then change b, including negative values, and explain how signed contributions can cancel.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Area 2 m²; exact flux 28 N·m²/C; 10-strip sum 27.98 N·m²/C. Average normal field 14 N/C.
A rectangle in the xy plane with normal +z and E_z(x) = E₀ + bx² on the patch. Only on-surface normal values are specified; no claim is made about the complete off-surface vector field. Signed strips may cancel. Midpoint sums approximate the exact integral.
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 patch has w = 2 m, L = 1 m and E_z = 6 − 3x² N/C. Use normal +z. Find flux and average normal field.
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Compare with the answer and four-point rubric
- 1 point: dΦ = 2(6 − 3x²) dx.
- 1 point: Φ = 2[6x − x³]₀¹.
- 1 point: Φ = 10 N·m²/C.
- 1 point: Area is 2 m², so average normal field is 5 N/C.
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 being summed in a flux integral?
The signed normal field times each small area.
RECALL 2Can opposite contributions cancel?
Yes. Flux is signed.
RECALL 3Does more numerical resolution change the exact physical flux?
No. It improves the approximation to the same integral.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Add local flux when the field varies across a surface
- Φ_E = ∫ E·dA.
- For this patch: Φ = w(E₀L + bL³/3).
- Average normal field = Φ/(wL).
Remember: Do not pull a position-dependent field outside an integral. The dot product is local.
Conditions: A rectangle in the xy plane with normal +z and E_z(x) = E₀ + bx² on the patch. Only on-surface normal values are specified; no claim is made about the complete off-surface vector field. Signed strips may cancel. Midpoint sums approximate the exact integral.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 1 (official Unit 8) · Objectives 8.5.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.5, objectives 8.5.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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