An infinite line needs a finite potential reference
You will be able to: Integrate an infinite-line field using a finite reference and extend the result inside a uniform cylinder.
Why can’t an infinite charged wire use V = 0 at infinity?
A long-wire field decreases as 1/r. Its contribution to a potential difference involves a logarithm, which keeps growing without bound as the reference radius goes to infinity. Choose a finite reference instead.
A useful starting point: Equal distances simplify the center potential of an arc →
Words and symbols before equations
- Reference radius r₀
- A finite radius where potential is assigned zero.
- Exterior line density λ
- Charge per unit length, in C/m.
- Logarithmic potential difference
- A difference proportional to ln(r/r₀), a dimensionless ratio.
- Cylinder radius R
- A finite physical radius needed to describe a nonsingular interior charge distribution.
What this picture assumes
Infinite uniform source in vacuum. Thin line uses V(r₀)=0; uniform solid cylinder uses V(R)=0 and ignores the line-reference control. Cylinder total charge per length is λ. A thin line is singular at r=0; positive-radius controls exclude that location.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Thin line with V(r₀)=0: V=-12.48 V, E_r=9 N/C at r=2 m. No zero at infinity is assigned.
- 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 line, E_r = 2kλ/r. Integrate radially: V(r) − V(r₀) = −∫ᵣ₀ʳ2kλ ds/s = −2kλ ln(r/r₀). Setting V(r₀) = 0 fixes a useful finite reference.
For an infinite uniformly volume-charged solid cylinder, let λ be its total charge per length and choose V(R) = 0. Outside, V = −2kλ ln(r/R). Inside, E_r = 2kλr/R², so integration gives V = kλ(1 − r²/R²).
The cylinder’s potential and radial derivative match at R. The ideal thin line remains singular at its axis, while a uniform solid cylinder has finite center potential. Neither infinite source permits the finite-source convention V(∞) = 0.
A worked example, step by step
An infinite line has λ = +1 nC/m. Set V = 0 at r₀ = 1 m. Find V at r = 2 m and at 0.50 m.
- Use V = −2kλ ln(r/r₀).
- At 2 m, V = −18 ln2 = −12.48 V.
- At 0.50 m, V = −18 ln0.5 = +12.48 V.
- The positive-source field points outward toward decreasing potential; changing r₀ shifts the zero but not E.
Do not assign zero potential at infinity for an ideal infinite line or cylinder. Use a finite reference.
Does changing only r₀ change the line’s electric field?
Compare with an explanation
No. It adds a constant to V; the derivative is unchanged.
Predict. Change one thing. Explain.
Compare the ideal line with a uniform solid cylinder. Set the reference to the cylinder radius for a fair exterior comparison. Move the probe inward and explain why their interiors differ.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Thin line with V(r₀)=0: V=-12.48 V, E_r=9 N/C at r=2 m. No zero at infinity is assigned.
Infinite uniform source in vacuum. Thin line uses V(r₀)=0; uniform solid cylinder uses V(R)=0 and ignores the line-reference control. Cylinder total charge per length is λ. A thin line is singular at r=0; positive-radius controls exclude that location.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant potential, energy, work, field-gradient or line-integral 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 questionAn infinite uniform solid cylinder has λ = +2 nC/m and R = 1 m. With V(R)=0, find V at r = 0.5 m and r = 2 m.
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Compare with the answer and four-point rubric
- 1 point: Inside: V = kλ(1 − r²/R²).
- 1 point: V(0.5) = 18(1 − 0.25) = 13.5 V.
- 1 point: Outside: V = −2kλ ln(r/R).
- 1 point: V(2) = −36 ln2 = −24.95 V.
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 not use infinity as zero here?
The logarithmic potential difference diverges for the ideal infinite source.
RECALL 2Which model has finite center potential?
The finite-radius uniformly charged cylinder.
RECALL 3What is continuous at the uniform cylinder boundary?
Both V and its radial derivative, hence the field.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
An infinite line needs a finite potential reference
- Line: V(r) − V(r₀) = −2kλ ln(r/r₀).
- Uniform cylinder, V(R)=0: V_inside = kλ(1−r²/R²).
- Cylinder exterior: V = −2kλ ln(r/R).
Remember: Do not assign zero potential at infinity for an ideal infinite line or cylinder. Use a finite reference.
Conditions: Infinite uniform source in vacuum. Thin line uses V(r₀)=0; uniform solid cylinder uses V(R)=0 and ignores the line-reference control. Cylinder total charge per length is λ. A thin line is singular at r=0; positive-radius controls exclude that location.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 2 (official Unit 9) · Objectives 9.2.A, 9.2.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.2, objectives 9.2.A, 9.2.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 2: Electric Potential, numbered Unit 9 in the official combined Physics C sequence. Topics 9.1–9.3 retain their official identifiers. Models assume electrostatic fields and state whether source charges are fixed or free. Potential integrals use the specified rods, ring, arc and infinite line or cylinder. Infinite-line examples use a finite reference radius, not zero potential at infinity. The optional 3D equipotential surface uses height to represent volts, not a particle trajectory. 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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