Add current through concentric area strips
You will be able to: Integrate a nonuniform axial current density over a circular cross-section.
What if current density is larger near the wire’s rim?
A circular cross-section may carry more current near its rim than at its center. Using the center density for the whole area would miss both the changing density and the larger outer annuli.
A useful starting point: Slow carrier drift can produce a substantial current →
Words and symbols before equations
- Radius r
- Distance from the wire axis; outer radius is a.
- Annulus
- A thin ring of area dA = 2πr dr.
- Current-density profile
- Specified dependence of axial J on position.
What this picture assumes
Prescribed axial profile J(r) = J₀[1+β(r/a)²]. Circular cross-section; annular area is 2πr dr. Optional 3D is a schematic wire segment with a labeled radius; wire length is arbitrary and the camera changes no current.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- I = 6.283 A; center J = 1, rim J = 3, area-average J = 2 (all in 10⁶ A/m²).
- 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 axial flow through a perpendicular disk, I = ∫J dA. A radial profile permits thin annuli, each contributing dI = J(r)2πr dr.
For J(r) = J₀[1+β(r/a)²], integrate from 0 to a: I = πa²J₀(1+β/2). The area-weighted average is J₀(1+β/2), not simply the center value.
The supplied profile is a mathematical model, not a claim about every DC wire. Controls use β ≥ 0. The optional spatial view shows the cross-section perpendicular to the wire axis; rotating it cannot change the current.
A worked example, step by step
A wire of radius 1 mm has J₀ = 10⁶ A/m² and β = 2. Find total current.
- Use dA = 2πr dr.
- Integrate J₀[1+2(r/a)²]2πr from 0 to a.
- I = π(10⁻³)²(10⁶)(1+2/2) = 2π A ≈ 6.283 A.
- The uniform-center estimate π A would underestimate by a factor of two.
Do not integrate J with dr alone: the area factor 2πr is essential.
Why does a thin outer ring matter more than an equal-width inner ring?
Compare with an explanation
It has a larger circumference and therefore more area.
Predict. Change one thing. Explain.
Change β at fixed radius and center density. Compare center, rim and average density. Then change radius and explain the area scaling.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
I = 6.283 A; center J = 1, rim J = 3, area-average J = 2 (all in 10⁶ A/m²).
Optional 3D view: current through a cross-section
The circular end face is perpendicular to the wire axis. Length is schematic; use the stated radius to calculate area. This is not a particle-speed animation.
Prescribed axial profile J(r) = J₀[1+β(r/a)²]. Circular cross-section; annular area is 2πr dr. Optional 3D is a schematic wire segment with a labeled radius; wire length is arbitrary and the camera changes no current.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant current, circuit topology, charge or energy conservation, or RC 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 current for the stated profile and predict the effect of doubling radius while keeping J₀ and β fixed.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Write I = 2πJ₀∫₀ᵃ[r+βr³/a²]dr.
- 1 point: Integrate to 2πJ₀[a²/2+βa²/4].
- 1 point: Simplify to πa²J₀(1+β/2).
- 1 point: Doubling a multiplies current by four.
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 density belongs in I = J_avg A?
The area-weighted average.
RECALL 2Does camera rotation alter physical area?
No. It changes only the projection.
RECALL 3Why include a normal direction?
Current counts flow through the oriented cross-section.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Add current through concentric area strips
- I = ∫J·dA.
- Circular annulus: dA = 2πr dr.
- For this profile: I = πa²J₀(1+β/2).
Remember: Do not integrate J with dr alone: the area factor 2πr is essential.
Conditions: Prescribed axial profile J(r) = J₀[1+β(r/a)²]. Circular cross-section; annular area is 2πr dr. Optional 3D is a schematic wire segment with a labeled radius; wire length is arbitrary and the camera changes no current.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 4 (official Unit 11) · Objectives 11.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 11.1, objectives 11.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 4: Electric Circuits, numbered Unit 11 in the official combined Physics C sequence. Topics 11.1–11.8 retain their official identifiers. Models include signed charge flow, prescribed current-density and resistivity integrals, DC resistor networks, nonideal batteries and meters, capacitor combinations and finite-resistance RC transients. Circuit schematics use conventional-current references and explicit node connectivity. Initial capacitor voltage and positive time constants are stated. Unequal ideal sources are never directly wired in parallel. The optional spatial wire view supplements a complete 2D current-density explanation. 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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