Integrate the enclosed charge before using Gauss’s law
You will be able to: Integrate a spherical density and use the result to determine the field.
What if a sphere’s charge density varies with radius?
Imagine a charged insulating ball with less charge per volume near its center and more near its edge. Enclosing half its radius does not then enclose one eighth of its total charge. You must account for both shell volume and density.
A useful starting point: A pillbox has two faces: keep the factor of two →
Words and symbols before equations
- Volume density ρ(r)
- Charge per unit volume, allowed to depend on radius.
- Thin shell volume dV
- For radius s and thickness ds, dV = 4πs² ds.
- Dummy variable s
- The integration coordinate inside the source; r is the fixed Gaussian radius.
- Power n
- In the family ρ(s) = ρ₀(s/R)ⁿ, n controls how strongly charge is concentrated outward.
What this picture assumes
Spherical insulating distribution ρ(s) = ρ₀(s/R)ⁿ for 0 ≤ s ≤ R, zero beyond R. Exponents 0–2 are illustrative. Changing n at fixed edge density changes total charge. Vacuum outside; r = 0 uses the continuous zero-field limit.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- At r = 0.5 m: Q_enc = 0.07854 nC, E_r = 2.827 N/C. Total charge 2.513 nC; density is radial, so spherical symmetry remains.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
A radial density preserves spherical symmetry even when it is not uniform. For a physical radius R, Q_enc(r) = ∫₀ʳρ(s)4πs² ds inside. Use a different letter for the source radius to keep the upper limit distinct.
For ρ(s) = ρ₀(s/R)ⁿ with n ≥ 0, integrate to obtain Q_enc = 4πρ₀ rⁿ⁺³/[(n + 3)Rⁿ]. Then E_r = Q_enc/(4πε₀r²) = ρ₀ rⁿ⁺¹/[(n + 3)ε₀Rⁿ]. At n = 0 this reduces to the uniform solid sphere.
Outside, enclose the entire charge Q_total = 4πρ₀R³/(n + 3), giving E_r = kQ_total/r². Compare sources carefully: holding edge density ρ₀ fixed while changing n does not hold total charge fixed.
A worked example, step by step
Let R = 1 m and ρ(s) = (1 nC/m³)(s/R)². Find Q_total and E at r = 0.5 m.
- Use n = 2 and dV = 4πs² ds.
- Q_total = 4πρ₀R³/5 = 2.513 nC.
- E_r = ρ₀r³/(5ε₀R²) at interior points.
- At 0.5 m, E_r = (10⁻⁹)(0.125)/(5ε₀) = 2.827 N/C outward.
Do not multiply the local density by the whole volume when density varies. Integrate ρ(s)dV.
Does nonuniform density prevent Gauss’s law from giving E?
Compare with an explanation
Not if the density remains spherically symmetric. Angular dependence would generally spoil this shortcut.
Predict. Change one thing. Explain.
Compare n = 0, 1 and 2 at fixed edge density. Track Q_total as well as the interior field; explain why the total charge decreases when density is lower near the center.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At r = 0.5 m: Q_enc = 0.07854 nC, E_r = 2.827 N/C. Total charge 2.513 nC; density is radial, so spherical symmetry remains.
Spherical insulating distribution ρ(s) = ρ₀(s/R)ⁿ for 0 ≤ s ≤ R, zero beyond R. Exponents 0–2 are illustrative. Changing n at fixed edge density changes total charge. Vacuum outside; r = 0 uses the continuous zero-field limit.
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 with R = 1 m has ρ(s) = (2 nC/m³)(s/R). Find total charge and E at its surface. State the far-field distance dependence.
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Compare with the answer and four-point rubric
- 1 point: Use n = 1: Q_total = 4πρ₀R³/4.
- 1 point: Q_total = 2π nC = 6.283 nC.
- 1 point: E(R) = ρ₀R/(4ε₀) = 56.55 N/C outward.
- 1 point: Outside the sphere E = kQ_total/r², falling as 1/r².
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 1Why use a dummy variable in the integral?
To distinguish the source shell location from the fixed observation radius.
RECALL 2What symmetry survives radial density variation?
Spherical symmetry.
RECALL 3At fixed edge density, does increasing n keep Q fixed?
No. Less charge occupies the inner volume, so total charge decreases.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Integrate the enclosed charge before using Gauss’s law
- Q_enc = ∫₀ʳρ(s)4πs² ds inside a spherical source.
- For ρ = ρ₀(s/R)ⁿ: E_r = ρ₀rⁿ⁺¹/[(n + 3)ε₀Rⁿ] inside.
- Outside: E_r = kQ_total/r².
Remember: Do not multiply the local density by the whole volume when density varies. Integrate ρ(s)dV.
Conditions: Spherical insulating distribution ρ(s) = ρ₀(s/R)ⁿ for 0 ≤ s ≤ R, zero beyond R. Exponents 0–2 are illustrative. Changing n at fixed edge density changes total charge. Vacuum outside; r = 0 uses the continuous zero-field limit.
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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