Equal distances simplify the center potential of an arc
You will be able to: Find arc-center potential and distinguish fixed total charge from fixed density comparisons.
Can two differently shaped arcs have the same center potential?
Move a fixed amount of charge around a circle without changing its distance from the center. Each charge piece still contributes k dq/R there, so the center potential stays kQ/R even though the center field can change.
A useful starting point: A ring can have maximum potential where its field is zero →
Words and symbols before equations
- Arc angle β
- The angular extent of the charged arc, in radians.
- Arc length Rβ
- Length of a circular arc of radius R.
- Linear density λ
- Charge per arc length; Q = λRβ for uniform density.
- Fixed-Q comparison
- A comparison that redistributes the same total charge rather than adding more.
What this picture assumes
Positive uniform circular arc from 0 to β, potential evaluated at its center with zero at infinity. One control fixes λ and the other fixes Q according to the selected mode; inactive parameter values do not affect the result.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Fixed λ: Q=3.142 nC; λ=1 nC/m. Center V=28.27 V.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
At the circle center, all elements are at distance R. Therefore V_center = ∫k dq/R = kQ/R, for zero at infinity and a finite arc. No angular projection is needed.
If uniform λ is held fixed, Q = λRβ, so V_center = kλβ. Extending the arc adds charge and increases V for positive λ. If instead Q is held fixed, extending the arc lowers λ but leaves center V unchanged.
A partial arc can produce a nonzero center field even when another arrangement with the same Q and R has the same center potential. Knowing V at one point is not enough to find its derivatives around that point.
A worked example, step by step
An upper semicircle has R = 1 m and λ = +1 nC/m. Find Q and the center potential.
- A semicircle has β = π radians and length πR.
- Q = λπR = π nC.
- V = kQ/R = 9π = 28.27 V.
- A quarter-circle at the same λ and R has half the charge and half the center potential.
State what is held fixed. Extending an arc at fixed λ changes Q; extending it at fixed Q does not.
Does equal center potential require equal center field?
Compare with an explanation
No. The field depends on the directions of contributions and nearby potential variation.
Predict. Change one thing. Explain.
Switch between fixed density and fixed total charge, then change β. Explain why the potential changes in one comparison and stays fixed in the other.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Fixed λ: Q=3.142 nC; λ=1 nC/m. Center V=28.27 V.
Positive uniform circular arc from 0 to β, potential evaluated at its center with zero at infinity. One control fixes λ and the other fixes Q according to the selected mode; inactive parameter values do not affect the result.
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 arc of radius 2 m carries total +4 nC. Find its center V. Then spread the same charge over a full ring of radius 2 m and compare center V and E.
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Compare with the answer and four-point rubric
- 1 point: The arc potential is V = 9(4)/2 = 18 V.
- 1 point: The full ring has the same Q and R, so its center V remains 18 V.
- 1 point: A uniformly charged full ring has zero center field.
- 1 point: A partial arc generally has nonzero center field because directional cancellation is incomplete.
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 1When is V = kQ/R valid at the center?
When every source element is the same distance R from that point.
RECALL 2Does the potential sum need cosφ?
No; it is scalar.
RECALL 3What changes when an arc grows at fixed λ?
Both length and total charge grow.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Equal distances simplify the center potential of an arc
- V_center = kQ/R for charges all at radius R.
- Uniform arc: Q = λRβ, so V_center = kλβ.
- Use radians for β in Rβ and integrals.
Remember: State what is held fixed. Extending an arc at fixed λ changes Q; extending it at fixed Q does not.
Conditions: Positive uniform circular arc from 0 to β, potential evaluated at its center with zero at infinity. One control fixes λ and the other fixes Q according to the selected mode; inactive parameter values do not affect the result.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 2 (official Unit 9) · Objectives 9.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.2, objectives 9.2.A. 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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