Test capacitance with a straight-line graph
You will be able to: Design a parallel-plate capacitance investigation and interpret a fitted slope.
Which graph reveals the plate-area dependence?
A group keeps plate separation fixed and measures capacitance for several overlapping areas. If the ideal model applies, capacitance increases linearly with area. The graph can reveal both the expected slope and an unwanted constant stray capacitance.
A useful starting point: Predict a charged particle’s path between plates →
Words and symbols before equations
- Controlled gap d
- Plate separation held fixed and parallel during the area trial.
- Slope ΔC/ΔA
- Capacitance change per area change, in F/m².
- Stray capacitance
- Extra capacitance from wires, surroundings or apparatus.
- Synthetic data
- Calculated illustrative values, not real measurements.
What this picture assumes
Synthetic vacuum-plate data at areas 0.005–0.020 m² with fixed selected gap. Added stray capacitance is constant for every point. ε₀=8.85×10⁻¹² F/m. No measurement noise, edge effects or changing lead geometry; these are not measured results.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Slope=8.85e-9 F/m²; intercept=20 pF. Orange intercept is a fitted extension of the model, not an additional physical zero-area measurement. Data are synthetic.
- 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 fixed gap and dielectric, C=(κ ε₀/d)A. A C-versus-A graph has slope κ ε₀/d. For a separation experiment at fixed A, plot C against 1/d to obtain a straight line instead of plotting against d.
Measure overlap area and separation, keep plates parallel, use low-voltage capacitance equipment and repeat measurements. Minimize lead movement and surrounding conductors. If an approximately constant stray capacitance is present, the fitted intercept is nonzero while the ideal slope can remain unchanged.
A slope mismatch or curvature may indicate changing gap, nonparallel plates, edge effects or variable stray capacitance. Do not force the fit through the origin merely because the ideal formula has no intercept. The explorer’s exact points cannot demonstrate experimental precision.
A worked example, step by step
A C-versus-A fit has slope 8.85×10⁻⁹ F/m² in air approximated as κ=1. Infer d. A constant 20 pF intercept is also observed.
- Slope=ε₀/d, so d=ε₀/slope.
- d=(8.85×10⁻¹²)/(8.85×10⁻⁹)=0.001 m.
- The 20 pF intercept is consistent with approximately constant stray capacitance.
- Subtracting a justified offset can help; forcing the raw fit through zero would bias the inferred slope.
A nonzero intercept is information, not something to delete automatically. Keep units on slopes.
Does a constant stray capacitance change the ideal slope?
Compare with an explanation
Not in this simple additive model; it shifts the intercept.
Predict. Change one thing. Explain.
Change gap and an added constant stray capacitance. Compare the slope and intercept of the synthetic C-versus-A graph. Explain why only one changes when a constant offset is added.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Slope=8.85e-9 F/m²; intercept=20 pF. Orange intercept is a fitted extension of the model, not an additional physical zero-area measurement. Data are synthetic.
Synthetic vacuum-plate data at areas 0.005–0.020 m² with fixed selected gap. Added stray capacitance is constant for every point. ε₀=8.85×10⁻¹² F/m. No measurement noise, edge effects or changing lead geometry; these are not measured results.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant charge, field or energy 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 questionA C-versus-A graph has slope 4.425×10⁻⁹ F/m² for vacuum plates. (a) Infer gap. (b) State a controlled variable. (c) Predict the effect of a constant 10 pF stray capacitance. (d) Name one cause of nonlinear data.
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Compare with the answer and four-point rubric
- 1 point: d=8.85×10⁻¹²/(4.425×10⁻⁹)=0.002 m.
- 1 point: Gap, plate parallelism or dielectric must be controlled during the area trial.
- 1 point: The vertical intercept increases by 10 pF.
- 1 point: Changing separation, substantial edge effects or variable stray capacitance.
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 1Useful area-trial graph?
C vertically versus overlap area A horizontally.
RECALL 2What does its slope measure?
κ ε₀/d under the ideal plate model.
RECALL 3Why use several measurements?
To estimate a trend and reveal scatter, offsets or curvature.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Test capacitance with a straight-line graph
- Fixed d,κ: slope of C versus A = κ ε₀/d.
- Fixed A,κ: C versus 1/d is linear.
- Measured C may include stray capacitance.
Remember: A nonzero intercept is information, not something to delete automatically. Keep units on slopes.
Conditions: Synthetic vacuum-plate data at areas 0.005–0.020 m² with fixed selected gap. Added stray capacitance is constant for every point. ε₀=8.85×10⁻¹² F/m. No measurement noise, edge effects or changing lead geometry; these are not measured results.
Refresh Kid · AP Physics 2 Unit 2 (official Unit 10) · Objectives 10.6.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.6, objectives 10.6.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Refresh Kid calls this the second AP Physics 2 unit; College Board numbers it Unit 10; the first unit in this course is official Unit 9. 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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