A cylindrical gap produces a logarithmic voltage
You will be able to: Derive the capacitance of long concentric cylindrical conductors.
Why is a coaxial capacitor different from a spherical one?
A coaxial cable has a conducting core and a surrounding conducting shield. Along a long straight section, equal opposite charges create a radial field in the gap whose strength falls as 1/r.
A useful starting point: Integrate the field between concentric spheres →
Words and symbols before equations
- Coaxial
- Sharing a common central axis.
- Length L
- Active length of the long cylindrical capacitor.
- Linear density λ
- Inner-conductor charge per length, Q/L, in C/m.
- Natural logarithm ln
- The function obtained by integrating 1/r; its argument b/a is dimensionless.
What this picture assumes
Long concentric vacuum cylindrical capacitor, equal opposite charges, end effects neglected. L is at least 30 times outer radius throughout the controls. Diagram is a transverse cross-section; the length is stated separately.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- C = 80.22 pF; inner-to-outer voltage = 12.47 V. Field just outside inner conductor = 8992 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
A cylindrical Gaussian surface of radius r and length L has side area 2πrL. Ignoring end effects, Eᵣ = Q/(2πε₀Lr). Symmetry makes the field radial.
The voltage between radii a and b is ∫ₐᵇEᵣdr = Q ln(b/a)/(2πε₀L). The logarithm comes from integrating 1/r; do not substitute the spherical 1/r² field.
Thus C = 2πε₀L/ln(b/a). Longer overlapping conductors have more capacitance. The model requires L much larger than b and uniform concentric geometry; the diagram is a transverse cross-section, with length stated separately.
A worked example, step by step
Take a = 2 mm, b = 4 mm and L = 1.0 m in vacuum. Find capacitance.
- The radius ratio is b/a = 2; units cancel.
- The natural logarithm is ln 2 ≈ 0.6931.
- C = 2π(8.85×10⁻¹²)(1.0)/0.6931 ≈ 80.22 pF.
- At 5 V, the inner charge magnitude is CΔV ≈ 401.1 pC.
Use ln(b/a), not ln(b−a), and keep the long-cylinder approximation explicit.
If both radii double at fixed ratio and length, what happens to C?
Compare with an explanation
The ideal long-cylinder capacitance is unchanged because ln(b/a) is unchanged.
Predict. Change one thing. Explain.
Double L at fixed radii and Q. Compare C and ΔV. Then change b/a while holding length fixed. Read both the cross-section and radial-field graph.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
C = 80.22 pF; inner-to-outer voltage = 12.47 V. Field just outside inner conductor = 8992 N/C.
Long concentric vacuum cylindrical capacitor, equal opposite charges, end effects neglected. L is at least 30 times outer radius throughout the controls. Diagram is a transverse cross-section; the length is stated separately.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant conductor equilibrium, charge conservation, capacitance or energy 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 questionDerive the coaxial capacitance and explain why scaling both radii by the same factor leaves C unchanged when L stays fixed.
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Compare with the answer and four-point rubric
- 1 point: Use Eᵣ2πrL = Q/ε₀.
- 1 point: Integrate the 1/r field from a to b to get the logarithmic voltage.
- 1 point: Divide Q by voltage to obtain C = 2πε₀L/ln(b/a).
- 1 point: Common radial scaling leaves b/a unchanged, so C is unchanged within the long-cylinder approximation.
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 geometry gives the 1/r field?
A long cylindrical Gaussian surface.
RECALL 2Why must b/a be dimensionless?
A logarithm takes a dimensionless ratio.
RECALL 3What effect is neglected?
The nonuniform fields near the cylinder ends.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A cylindrical gap produces a logarithmic voltage
- Eᵣ = Q/(2πε₀Lr).
- ΔV = Q ln(b/a)/(2πε₀L).
- C = 2πε₀L/ln(b/a).
Remember: Use ln(b/a), not ln(b−a), and keep the long-cylinder approximation explicit.
Conditions: Long concentric vacuum cylindrical capacitor, equal opposite charges, end effects neglected. L is at least 30 times outer radius throughout the controls. Diagram is a transverse cross-section; the length is stated separately.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 3 (official Unit 10) · Objectives 10.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.3, objectives 10.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 3: Conductors and Capacitors, numbered Unit 10 in the official combined Physics C sequence. Topics 10.1–10.4 retain their official identifiers. Models state the electrostatic conditions, geometry approximations and whether charge or voltage stays fixed. Capacitor geometries include parallel plates, concentric spheres and long coaxial cylinders. Dielectric comparisons assume a fully filling ideal linear material. The optional 3D plate view uses explicitly different gap and lateral scales to show the small separation. 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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