Test a decay model against readings
A smooth exponential curve is a prediction, not evidence that a physical capacitor follows it. Your task is to estimate a discharge time constant, use it to predict an unused reading, and investigate one reason the estimate might differ from nominal RC.
Use either instructor-supervised measurements or the original synthetic practice data below. State which you used. The invented readings support analysis practice, not a claim that you operated equipment. The browser retains its ideal 6 V, 1000 µF capacitor and selected resistor; it does not ingest readings, fit parameters, or simulate meter loading. Keep your raw-data sheet separately and summarize the calculation in your notebook.
Before opening the data, set resistance to 2 kΩ. Capture discharge predictions at 1, 2 and 3 s. Sketch the voltage curve and predict whether equal one-second intervals remove equal voltages or equal fractions. The ideal predictions are approximately 3.639, 2.207 and 1.339 V. Decide in advance to estimate the time constant from the 1 s and 3 s readings and reserve 2 s as a check.
Optional supervised apparatus investigation
Use an instructor-approved low-voltage RC apparatus: a current-limited 6 V DC supply, a nominal 2 kΩ resistor rated at least 0.25 W, a nominal 1000 µF capacitor rated at least 16 V, suitable leads, a two-position switch that selects charging or source-disconnected discharge, and a voltage logger with known input resistance and sampling interval. A polarized capacitor must have its marked positive terminal on the positive side. Have the instructor verify polarity, switch wiring, source current limit and logger compatibility before energizing. Do not use mains circuits, camera-flash capacitors or salvaged high-voltage equipment.
Measure voltage across the capacitor; do not connect a current-mode meter across it. The switch must remove the source before selecting the resistor-only discharge loop. Disconnect power before changing wiring. Stored charge remains after disconnection: discharge through the approved resistor and verify the residual voltage with the instructor before handling or rewiring; do not short the terminals. These instructions apply only to this small supervised apparatus, not to arbitrary charged capacitors.
- Record component labels and tolerances, logger model, voltage range, accuracy specification, input resistance and sample interval. Measure the isolated resistor only when unpowered and after verified discharge. Do not equate display resolution with accuracy.
- Agree on the switching time marker and sample at least ten times per expected 2 s time constant, so intervals are at most 0.2 s. A slow hand-read display cannot resolve this transient well. Record the actual timestamps and any switch delay instead of inventing exact times.
- Charge through the resistor until voltage is stable, recording the actual initial voltage. Switch to the source-disconnected discharge loop and log for about 10 s. Retain every raw sample; avoid replacing small late readings with zero.
- Repeat three runs with the same initial-voltage procedure. Analyze each separately before comparing their estimates. Repeated runs test repeatability; they are not three independent capacitors or independent calibrations.
- Keep one interior reading out of the parameter estimate. Report its prediction and residual before deciding whether a changed model is warranted. Record any departures from the planned method.
Analyze the supplied readings
These values are invented for this activity. They represent a nominal 2 kΩ / 1000 µF discharge with no claimed hardware provenance. The stipulated bounds for analysis are ±0.02 V per reading and ±0.01 s per timestamp. They are not specifications for your equipment.
| Time (s) | Voltage (V) | Planned use |
|---|
| 0.00 | 6.01 | Initial reading; additional check |
| 1.00 | 3.63 | Parameter estimate |
| 2.00 | 2.22 | Reserved prediction check |
| 3.00 | 1.33 | Parameter estimate |
| 4.00 | 0.82 | Additional prediction check |
A discharge model can be written V(t) = A exp(−t/τ). Here V is voltage, t is time in seconds, A is the fitted voltage at the chosen time origin, and τ (tau) is a positive time constant in seconds. The exponential function exp describes a constant fractional decrease over equal intervals. A constant error in the chosen time origin changes A but not the ratio-based estimate of τ; unequal timestamp errors still matter.
Divide the two endpoint voltages to eliminate A. The result is τ = (t₂ − t₁) / ln(V₁/V₂). The symbol ln means natural logarithm, the inverse of exp. Its argument V₁/V₂ is a dimensionless ratio. For positive decaying readings, V₁ exceeds V₂, so both numerator and denominator are positive. Zero or negative readings cannot be used in this logarithm; readings close to the instrument floor can make it unreliable.
With the selected endpoints, τ = 2 / ln(3.63/1.33) ≈ 1.9919 s. If resistance really is 2000 Ω and other discharge paths are negligible, inferred capacitance is C = τ/R ≈ 0.000996 F = 996 µF. The decay alone identifies a product RC, not R and C separately. Choosing 2000 Ω is an additional assumption unless separately measured.
Predict before judging the fit
Predict the reserved 2 s value from the 1 s anchor: V predicted = 3.63 exp[−(2 − 1)/1.9919] ≈ 2.19725 V. Equivalently, because 2 s is halfway between the fitting times, the predicted voltage is the geometric mean, the square root of 3.63 × 1.33. The residual is observed minus predicted: 2.22 − 2.19725 ≈ +0.02275 V.
A positive residual means the reading lies above the prediction. It does not by itself establish a faulty capacitor. The prediction inherits uncertainty from its endpoint readings and times, so comparing this residual only with the reserved reading’s ±0.02 V bound would be incomplete. The nominal browser predicts 2.20728 V at 2 s; distinguish that nominal prediction from the fitted prediction. Do not silently round 1.9919 s to 2 s and call the browser a fit.
As a simple consistency example, hold the timestamps exact and raise both fitting voltages by 0.01 V, within their stipulated bounds. The midpoint prediction becomes the square root of 3.64 × 1.34, approximately 2.2085 V. That overlaps the reserved reading’s interval 2.20–2.24 V. Thus these bounds permit a compatible curve; this example neither proves the exponential model nor assigns a probability to it. Reusing the check reading to tune the estimate would turn it into fitting evidence, requiring another unused check.
Check the additional prediction
Using the original 1 s anchor and fitted time constant, predict the 4 s voltage before comparing it with 0.82 V. The prediction is approximately 0.805 V and the residual approximately +0.015 V. Both checks are public practice once revealed; they are not a hidden assessment of independent mastery.
Bound the estimate without inventing a confidence level
If each timestamp can vary independently by 0.01 s, the endpoint separation spans 1.98–2.02 s. The largest voltage ratio is 3.65/1.31 and the smallest is 3.61/1.35. Since τ increases with the time separation and decreases with the voltage ratio, conservative endpoint combinations give:
- Lower bound: 1.98 / ln(3.65/1.31) ≈ 1.9323 s.
- Upper bound: 2.02 / ln(3.61/1.35) ≈ 2.0537 s.
These are worst-case bounds under the stipulated independent ranges, not a 95% confidence interval. Correlated calibration or timing errors require a different treatment. If R is known only to be within 1980–2020 Ω, combining these bounds gives capacitance from approximately 957 to 1037 µF, under the same ideal discharge assumptions. A parameter interval alone is not a full voltage prediction interval: the anchor voltage and time constant are related through the same observations.
Ask whether the measuring instrument changes the circuit
The ideal browser treats its voltage reading as nonloading. A real voltage logger with finite input resistance R meter adds a parallel discharge path. With leakage neglected, the effective resistance is R effective = R × R meter / (R + R meter) and the measured decay time constant is R effective × C. The logger reads voltage while also drawing current.
For R = 2000 Ω and C = 0.001 F, a hypothetical 10,000 Ω input gives R effective ≈ 1666.7 Ω and τ ≈ 1.667 s. Ignoring the meter would infer about 833 µF even though the capacitor is 1000 µF. A 10,000,000 Ω input instead gives τ ≈ 1.9996 s, a much smaller effect. These are calculated comparison cases, not modes in the browser or claims about a particular logger.
A short apparent time constant could also reflect a smaller capacitor, a smaller resistor, leakage, or timing errors. One exponential trace cannot distinguish all these causes. Propose a discriminating follow-up: measure the isolated resistor, verify logger input resistance, or repeat with a known different resistor while retaining the same capacitor and logger. A straight logarithmic plot alone does not establish that meter loading is absent; a parallel resistor still gives exponential decay. Changing input resistance may require different apparatus and instructor approval rather than rewiring an energized circuit.
Transfer and report
An unfamiliar discharge gives 4.0 V at 0.5 s and 2.0 V at 1.5 s. A separately established effective discharge resistance is 3000 Ω. Estimate τ and C, and predict voltage at 2.5 s without using the browser’s fixed components. Identify which additional observation would test your model.
Check the discharge measurement transfer
The voltage halves every second, so τ = 1/ln(2) ≈ 1.4427 s. Capacitance is τ/3000 ≈ 0.0004809 F = 481 µF. Another second predicts 1.0 V at 2.5 s. An unused later reading with its uncertainty tests that prediction; another point used in the fit would not serve the same role. The stated resistance must include every relevant discharge path for this capacitance inference.
Submit a circuit sketch with meter placement, data provenance, raw readings with units, an advance prediction, a captured nominal comparison, your fitted time constant, an unused-reading residual, uncertainty assumptions, and one alternative explanation with a discriminating follow-up. Preserve the raw readings even if they disagree. Notebook completion checks that you supplied text; it does not grade experimental reasoning or establish physical laboratory proficiency.
The underlying exponential relations are explained in OpenStax: RC circuits. This procedure, dataset, uncertainty exercise and transfer case are original learning activities.
