Read the axes before calling a slope resistance
You will be able to: Distinguish resistance, conductance and a changing nonlinear response.
Why is the slope of an I-versus-V graph not R?
A resistor may pass 1 A at 2 V and 2 A at 4 V. Those points lie on a straight line through the origin. A heated filament may not follow the same proportional pattern.
A useful starting point: Add the resistance of thin slices →
Words and symbols before equations
- Ohmic element
- An element modeled with constant R over the stated operating range.
- Conductance
- The reciprocal 1/R, measured in siemens.
- Secant ratio V/I
- Resistance at an operating point; distinct from a local differential slope in a nonlinear model.
What this picture assumes
Illustrative constitutive relation V = R₀I+bI³, not measured bulb data. b = 0 is ohmic. At I = 0 the readout uses the limiting V/I value, never 0/0.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- I = 1 A; V = 3 V; V/I = 3 Ω; local dV/dI = 5 Ω.
- 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 ohmic resistor, V = IR. A V-versus-I graph has slope R; an I-versus-V graph has slope 1/R. Axis order matters.
The comparison model uses V = R₀I + bI³ with b ≥ 0. For nonzero I, V/I = R₀+bI², while dV/dI = R₀+3bI². These coincide only when b = 0 or at the zero-current limit.
The nonlinear curve is an illustrative constitutive model, not a fitted lightbulb. Real heating can change resistance; the ohmic case assumes fixed material conditions within its valid range.
A worked example, step by step
An I-versus-V line rises by 2 A over 8 V. Find R.
- Read the vertical axis as I and horizontal as V.
- Slope = 2/8 = 0.25 A/V.
- This slope is conductance, so R = 1/0.25 = 4 Ω.
- At 12 V the ohmic prediction is I = 3 A.
Resistance is not the slope of every electrical graph; distinguish V/I from dV/dI for nonlinear elements.
At I = 0, should V/I be evaluated as 0/0?
Compare with an explanation
No. Use the limiting ratio R₀ or the local slope, explicitly labeled.
Predict. Change one thing. Explain.
Set b = 0 to see an ohmic line. Increase b and compare the selected-point V/I with the local slope dV/dI.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
I = 1 A; V = 3 V; V/I = 3 Ω; local dV/dI = 5 Ω.
Illustrative constitutive relation V = R₀I+bI³, not measured bulb data. b = 0 is ohmic. At I = 0 the readout uses the limiting V/I value, never 0/0.
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 questionFor V = 3I+2I³, calculate V, V/I and dV/dI at I = 2 A. Explain why the last two differ.
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Compare with the answer and four-point rubric
- 1 point: V = 6+16 = 22 V.
- 1 point: V/I = 11 Ω.
- 1 point: dV/dI = 3+6I² = 27 Ω.
- 1 point: The curve is nonlinear, so a secant through the origin and a local tangent have different slopes.
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 R constant here?
In the b = 0 ohmic model.
RECALL 2What is conductance?
1/R, the slope of an ohmic I-versus-V graph.
RECALL 3What makes a curved graph significant?
The response is not proportional over that range.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Read the axes before calling a slope resistance
- Ohmic: V = IR.
- Slope of I(V) = 1/R.
- Illustrative nonlinear model: V = R₀I+bI³.
Remember: Resistance is not the slope of every electrical graph; distinguish V/I from dV/dI for nonlinear elements.
Conditions: Illustrative constitutive relation V = R₀I+bI³, not measured bulb data. b = 0 is ohmic. At I = 0 the readout uses the limiting V/I value, never 0/0.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 4 (official Unit 11) · Objectives 11.3.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 11.3, objectives 11.3.B. 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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