Separate material resistivity from object resistance
You will be able to: Use R = ρL/A with consistent units and explicit temperature conditions.
Why is a long thin wire more resistive?
Two wires made from the same material can oppose current differently. A longer wire gives a longer resistive path; a thicker wire provides more conducting area.
A useful starting point: Follow connections before following the drawing →
Words and symbols before equations
- Resistance R
- Voltage-to-current ratio for a specified object, in ohms Ω.
- Resistivity ρ
- Material property, in Ω·m, which can depend on temperature.
- Length L and area A
- Path length and perpendicular cross-sectional area in SI units.
What this picture assumes
Uniform area and fixed resistivity at a specified temperature. No heating feedback. The wire sketch is schematic; displayed length and area are numerical inputs.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- R = 0.04 Ω from ρL/A. Length, material and cross-sectional area are independent inputs.
- 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 uniform material and cross-section, E = ρJ, ΔV = EL and I = JA combine to give R = ΔV/I = ρL/A.
Doubling length doubles resistance. Doubling radius quadruples area and therefore divides resistance by four. The area unit mm² must be converted to 10⁻⁶ m².
The model holds resistivity fixed. Real metal resistivity generally changes with temperature, so heating can invalidate a fixed-R approximation. Geometry and material are separate controls.
A worked example, step by step
A 2 m wire has A = 1 mm² and ρ = 2×10⁻⁸ Ω·m. Find R.
- Convert area to 10⁻⁶ m².
- Use R = ρL/A.
- R = (2×10⁻⁸)(2)/(10⁻⁶) = 0.040 Ω.
- Doubling length at fixed area and material would give 0.080 Ω.
Resistivity belongs to material under stated conditions; resistance also depends on geometry.
If both L and A double, does R change?
Compare with an explanation
No. The ratio L/A stays fixed.
Predict. Change one thing. Explain.
Vary length, area and resistivity separately. Predict multiplicative changes before reading R. Temperature is held fixed through the chosen ρ.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
R = 0.04 Ω from ρL/A. Length, material and cross-sectional area are independent inputs.
Uniform area and fixed resistivity at a specified temperature. No heating feedback. The wire sketch is schematic; displayed length and area are numerical inputs.
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 questionWire B has three times the length and twice the radius of wire A, made from the same material at the same temperature. Find R_B/R_A.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Use the ratio of ρL/A.
- 1 point: The common resistivity cancels.
- 1 point: The area ratio is 2² = 4.
- 1 point: R_B/R_A = 3/4.
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 must be uniform for ρL/A?
Cross-sectional area and resistivity along the path.
RECALL 2Why does larger area lower R?
It permits more parallel carrier flow.
RECALL 3Why specify temperature?
Material resistivity may change as temperature changes.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Separate material resistivity from object resistance
- R = ρL/A for uniform material and area.
- A_circle = πa².
- 1 mm² = 10⁻⁶ m².
Remember: Resistivity belongs to material under stated conditions; resistance also depends on geometry.
Conditions: Uniform area and fixed resistivity at a specified temperature. No heating feedback. The wire sketch is schematic; displayed length and area are numerical inputs.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 4 (official Unit 11) · Objectives 11.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 11.3, objectives 11.3.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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