A logarithmic plot reveals the RC timescale
You will be able to: Linearize discharge data and account for a voltmeter’s finite resistance.
How can a discharge trace reveal an unknown resistance?
A voltmeter watching a capacitor discharge is also a path that can discharge it. Its input resistance may be part of the measured RC time constant, especially with high-resistance circuits.
A useful starting point: Preserve capacitor voltage across a switch change →
Words and symbols before equations
- Normalized voltage
- The dimensionless ratio V(t)/V₀.
- Logarithmic plot
- Here ln(V/V₀) plotted against time.
- Effective resistance R_eff
- Resistance seen by the capacitor, including parallel meter loading.
What this picture assumes
Ideal simulated discharge measurements at multiples of τ, V₀ = 10 V. Meter resistance is parallel with R. No measurement noise is simulated; real fits require uncertainty and residual checks.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- R_eff = 2 kΩ; τ = 2 s; log slope = -0.5 s⁻¹. Meter 6 kΩ is parallel with resistor 3 kΩ. All points are simulated.
- 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 ideal discharge, V/V₀ = e^(−t/τ). Taking a natural logarithm gives ln(V/V₀)=−t/τ, a line of slope −1/τ.
If a finite voltmeter R_m is across a discharge resistor R, then R_eff = RR_m/(R+R_m), and τ = R_eff C. Neglecting the meter can underestimate R.
Use repeated positive-voltage measurements, keep the circuit configuration fixed, and inspect fit residuals. A logarithm cannot be taken at zero or negative voltage; readings near instrument noise need special care. Displayed points are ideal simulated data, not actual measurements.
A worked example, step by step
A discharge log plot has slope −0.5 s⁻¹ and C = 1 mF. What effective resistance is measured?
- τ = −1/slope = 2 s.
- R_eff = τ/C = 2/0.001 = 2000 Ω.
- Thus the measured effective resistance is 2 kΩ.
- If a 6 kΩ meter is in parallel, 1/R = 1/2−1/6 in kΩ⁻¹, giving R = 3 kΩ.
The fitted resistance includes all discharge paths. Do not take logarithms of dimensional voltage or nonpositive readings.
Why divide by V₀ before taking a logarithm?
Compare with an explanation
It creates a dimensionless ratio and sets the ideal intercept to zero.
Predict. Change one thing. Explain.
Increase meter resistance and watch τ approach RC. Compare the simulated voltage samples with the straight logarithmic plot.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
R_eff = 2 kΩ; τ = 2 s; log slope = -0.5 s⁻¹. Meter 6 kΩ is parallel with resistor 3 kΩ. All points are simulated.
Ideal simulated discharge measurements at multiples of τ, V₀ = 10 V. Meter resistance is parallel with R. No measurement noise is simulated; real fits require uncertainty and residual checks.
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 questionPlan a discharge measurement of an unknown R using known C. Include the plotted variables and one instrument effect.
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Compare with the answer and four-point rubric
- 1 point: Measure V₀ and a sequence of positive V(t) values with timestamps.
- 1 point: Plot ln(V/V₀) against t and fit the slope.
- 1 point: Infer τ = −1/slope and R_eff = τ/C.
- 1 point: Account for meter resistance in parallel and avoid noise-dominated near-zero readings.
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 are the log-plot slope units?
s⁻¹.
RECALL 2Are these points measured data?
No, they are ideal simulated points.
RECALL 3How can you check the exponential assumption?
Inspect deviations from a straight log plot and compare repeated measurements.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A logarithmic plot reveals the RC timescale
- ln(V/V₀) = −t/τ.
- τ = −1/slope.
- R_eff = R parallel R_m.
Remember: The fitted resistance includes all discharge paths. Do not take logarithms of dimensional voltage or nonpositive readings.
Conditions: Ideal simulated discharge measurements at multiples of τ, V₀ = 10 V. Meter resistance is parallel with R. No measurement noise is simulated; real fits require uncertainty and residual checks.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 4 (official Unit 11) · Objectives 11.8.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 11.8, objectives 11.8.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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