Only the curved wall contributes for cylindrical symmetry
You will be able to: Derive interior and exterior fields for an infinite uniformly charged solid cylinder.
How does a Gaussian cylinder reveal the field of a charged wire?
Wrap an imaginary cylinder around a long straight charged object. The radial field crosses its curved wall but runs parallel to its end caps. That tells you which areas belong in the flux sum.
A useful starting point: Match a spherical surface to spherical charge symmetry →
Words and symbols before equations
- Coaxial
- Sharing the same central axis.
- Gaussian length ℓ
- Length of the imaginary cylinder; it cancels from the final field formula.
- Volume density ρ
- Charge per volume in C/m³.
- Curved area 2πrℓ
- The side area crossed normally by a radial field.
What this picture assumes
An infinite uniform solid cylinder of physical radius R. Gaussian cylinder is coaxial, radius r and length ℓ. At r = 0 use the continuous field limit E = 0. End-cap flux is zero. The cross-section diagram does not imply a finite source length.
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 = 0.7854 nC, E_r = 28.27 N/C. Flux 88.83 N·m²/C; curved area 3.142 m². Cap flux is zero.
- 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 an infinite uniform cylinder of physical radius R and density ρ, symmetry gives a purely radial field depending only on r. The end-cap normals point along the axis, so E·dA = 0 on both caps.
For r < R, Q_enc = ρπr²ℓ. Thus E_r(2πrℓ) = ρπr²ℓ/ε₀, giving E_r = ρr/(2ε₀). For r ≥ R, Q_enc = ρπR²ℓ and E_r = ρR²/(2ε₀r).
Define λ = ρπR². The exterior becomes λ/(2πε₀r), agreeing with the infinite-line integral. The real-wire approximation requires sufficient length and observation away from ends; changing ℓ changes both flux and enclosed charge, not E.
A worked example, step by step
An ideal solid cylinder has ρ = +1 nC/m³ and R = 1 m. Find E at r = 0.5 m and r = 2 m using the unit’s ε₀.
- For r = 0.5 m, use E_r = ρr/(2ε₀).
- E_r = (10⁻⁹)(0.5)/(2ε₀) = 28.27 N/C outward.
- For r = 2 m, E_r = ρR²/(2ε₀r) = 28.27 N/C outward.
- The equal values occur at these particular radii; the interior grows linearly and the exterior decays as 1/r.
Do not include cap area in EA when the radial field is parallel to the caps.
Why does the imaginary cylinder length cancel?
Compare with an explanation
Both enclosed charge and curved surface area are proportional to that length.
Predict. Change one thing. Explain.
Change the Gaussian length at fixed r. Predict how enclosed charge and flux change while the field stays fixed. Then move r inside and outside the physical radius.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At r = 0.5 m: Q_enc = 0.7854 nC, E_r = 28.27 N/C. Flux 88.83 N·m²/C; curved area 3.142 m². Cap flux is zero.
An infinite uniform solid cylinder of physical radius R. Gaussian cylinder is coaxial, radius r and length ℓ. At r = 0 use the continuous field limit E = 0. End-cap flux is zero. The cross-section diagram does not imply a finite source length.
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 questionFor an infinite cylinder with ρ = +2 nC/m³ and R = 0.5 m, derive E at r = 1 m and explain whether doubling Gaussian length changes it.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Q_enc = ρπR²ℓ and Φ = E 2πrℓ.
- 1 point: E = ρR²/(2ε₀r).
- 1 point: E = (2 × 10⁻⁹)(0.25)/(2ε₀) = 28.27 N/C outward.
- 1 point: Doubling ℓ doubles both sides of the flux balance; E is unchanged.
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 1Which part of the Gaussian surface carries flux?
The curved wall.
RECALL 2What does λ equal for the uniform solid cylinder?
ρπR².
RECALL 3Does the infinite-cylinder result describe wire ends?
No. End effects break the assumed symmetry.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Only the curved wall contributes for cylindrical symmetry
- Φ = E_r 2πrℓ; cap flux = 0.
- Inside: E_r = ρr/(2ε₀).
- Outside: E_r = ρR²/(2ε₀r) = λ/(2πε₀r).
Remember: Do not include cap area in EA when the radial field is parallel to the caps.
Conditions: An infinite uniform solid cylinder of physical radius R. Gaussian cylinder is coaxial, radius r and length ℓ. At r = 0 use the continuous field limit E = 0. End-cap flux is zero. The cross-section diagram does not imply a finite source length.
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.
Want to work through this with a tutor?
Bring your question about Only the curved wall contributes for cylindrical symmetry. Your explanation and answers remain free to access.