Use a graph to separate dielectric behavior from stray capacitance
You will be able to: Design and interpret a capacitance-versus-inverse-thickness experiment with uncertainty.
How can measurements reveal a dielectric constant?
A capacitance meter measures more than the ideal plates: leads and nearby objects can add a small offset. Instead of trusting one reading, compare several thicknesses and fit a straight line.
A useful starting point: The battery connection decides the dielectric response →
Words and symbols before equations
- Inverse thickness 1/d
- Horizontal graph variable; if d is in mm, units are mm⁻¹.
- Slope s
- Change in measured capacitance divided by change in 1/d.
- Stray capacitance C_s
- An approximately constant extra capacitance from leads and surroundings.
- Systematic effect
- A repeated bias, such as an air gap or compressed thickness.
What this picture assumes
Ideal simulated data, not actual experimental measurements. Four known thicknesses: 0.5, 1, 2 and 4 mm. Model assumes full filling, constant area and constant parallel stray capacitance. Real uncertainty, air gaps and leakage require separate evaluation.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Slope = 177 pF·mm; intercept = 5 pF; inferred κ = 2. Simulated readings at d = 0.5, 1, 2, 4 mm: 359, 182, 93.5, 49.25 pF.
- 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 fixed area A and a uniform fully filling dielectric, C_meas = κε₀A/d + C_s. Plot C_meas against 1/d: slope is κε₀A in consistent SI units, and intercept estimates the constant stray term.
In this investigation, C is in pF and d in mm. Then s = κε₀A × 10¹⁵ pF·mm, so κ = s/(ε₀A×10¹⁵). Labeling axes avoids a thousand-fold thickness conversion error.
Keep overlap area and lead arrangement fixed, measure actual thickness, repeat readings and fit several points. Examine residuals rather than forcing a line through zero. An air gap or compressed layers can change the effective capacitance. The displayed data are ideal simulated readings with a stated constant offset, not student measurements or a validated instrument.
A worked example, step by step
For A = 0.010 m², a fitted C-versus-1/d line has slope 177 pF·mm and intercept 5 pF. Estimate κ.
- The expected vacuum slope in these units is ε₀A×10¹⁵ = 88.5 pF·mm.
- κ = 177/88.5 = 2.00.
- The 5 pF intercept represents the model’s constant stray capacitance.
- Use fit uncertainty and area/thickness uncertainties to judge precision; do not subtract an arbitrary offset without evidence.
A nonzero intercept need not imply an incorrect dielectric law. Air gaps and changing lead geometry can invalidate the constant-offset model.
Why avoid forcing the fitted line through the origin?
Compare with an explanation
An approximately constant stray capacitance can create a nonzero intercept and bias an origin-forced slope.
Predict. Change one thing. Explain.
Change the simulated dielectric constant and then the stray offset. Identify which changes slope and which changes intercept. Compare the 0.5, 1, 2 and 4 mm measurements.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Slope = 177 pF·mm; intercept = 5 pF; inferred κ = 2. Simulated readings at d = 0.5, 1, 2, 4 mm: 359, 182, 93.5, 49.25 pF.
Ideal simulated data, not actual experimental measurements. Four known thicknesses: 0.5, 1, 2 and 4 mm. Model assumes full filling, constant area and constant parallel stray capacitance. Real uncertainty, air gaps and leakage require separate evaluation.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant conductor equilibrium, charge conservation, capacitance 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 questionDesign a measurement of κ using plates, a capacitance meter and several dielectric thicknesses. Include a graph, controls and a plausible source of bias.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Keep facing area and lead arrangement fixed; measure actual thickness and repeat readings.
- 1 point: Plot capacitance against inverse thickness and fit a line with an intercept.
- 1 point: Infer κ from slope divided by ε₀A in consistent units, and use the intercept to estimate a constant stray term.
- 1 point: Discuss an air gap, compression or changing surroundings and assess residuals and measurement uncertainty.
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 1Why collect several thicknesses?
To test the functional relationship and fit slope and offset.
RECALL 2What must remain fixed?
Area, material and measurement arrangement apart from thickness.
RECALL 3Are the investigation points real measurements?
No. They are labeled ideal simulated data for learning how a fit works.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Use a graph to separate dielectric behavior from stray capacitance
- C_meas = κε₀A/d + C_s.
- SI plot slope of C versus 1/d is κε₀A.
- For pF and mm⁻¹ axes: κ = s/(ε₀A×10¹⁵).
Remember: A nonzero intercept need not imply an incorrect dielectric law. Air gaps and changing lead geometry can invalidate the constant-offset model.
Conditions: Ideal simulated data, not actual experimental measurements. Four known thicknesses: 0.5, 1, 2 and 4 mm. Model assumes full filling, constant area and constant parallel stray capacitance. Real uncertainty, air gaps and leakage require separate evaluation.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 3 (official Unit 10) · Objectives 10.4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.4, objectives 10.4.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 3: Conductors and Capacitors, numbered Unit 10 in the official combined Physics C sequence. Topics 10.1–10.4 retain their official identifiers. Models state the electrostatic conditions, geometry approximations and whether charge or voltage stays fixed. Capacitor geometries include parallel plates, concentric spheres and long coaxial cylinders. Dielectric comparisons assume a fully filling ideal linear material. The optional 3D plate view uses explicitly different gap and lateral scales to show the small separation. 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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