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AQA's Required Practical 9 (RP9) investigates the discharge of a capacitor through a resistor to determine the time constant τ = RC and extract the capacitance C of an unknown capacitor. This is one of the 12 required practicals examined in Paper 3 of the AQA 7408 specification; it is a substantial exam-able experiment and a quantitative test of the exponential-decay model developed in Lesson 5.
Spec mapping (AQA 7408 §3.7.4, Required Practical 9): This lesson covers the experimental procedure for charging a capacitor via a DC supply through a known resistor, the transition from charging to discharging via a switch, the measurement of V across the capacitor as a function of time t, the transformation V → ln V to linearise the exponential decay, the graphical extraction of the time constant τ = RC from the gradient of ln V vs t, the propagation of uncertainties from R-tolerance, voltmeter loading and timer resolution, and the CPAC competencies assessed (precise measurement, recording, analysis, evaluation). Refer to the official AQA 7408 specification document for the precise wording of CPAC criteria and required-practical specification.
Synoptic links — where this resurfaces in 7408:
- §3.7.4 Capacitance / charge-discharge (Lessons 4–5 here) — the underlying theory V = V₀ e^(−t/RC) is the entire mathematical content of the practical.
- §3.5 Electricity — Ohm's law, Kirchhoff's voltage law and voltmeter loading are the foundational tools used to design and critique the experimental setup.
- §3.1 Measurements and their errors — RP9 is a showcase of A-Level error analysis: timer-resolution uncertainty, voltmeter input impedance, resistor tolerance, and how each propagates to C.
- §3.8.1.2 Radioactive decay — the same N = N₀ e^(−λt) form is examined in RP12 (radioactive-decay simulation); both linearise via natural log and extract a time constant from gradient. Examined synoptically.
- §3.6.1 Simple harmonic motion / RP7 — RP9 (first-order exponential) and RP7 (SHM) jointly anchor the time-dependent practical work in the spec.
- §3.4 Mechanics — the data-logging and graph-fitting skills developed here transfer directly to terminal-velocity and impulse experiments.
A capacitor of capacitance C, initially charged to voltage V₀, discharged through a resistor R, has its instantaneous voltage given by the first-order linear ODE:
dV/dt = −V/(RC)
with solution:
V(t) = V₀ e^(−t/τ), τ = RC
Taking natural logs of both sides linearises the equation:
ln V = ln V₀ − t/τ
This is of the form y = mx + c with:
A plot of ln V vs t therefore yields a straight line whose gradient gives 1/(RC) and whose intercept independently checks the initial voltage. If R is known, C is extracted from C = −1/(m × R).
A minimal RP9 setup uses:
| Component | Specification | Role |
|---|---|---|
| Capacitor (unknown) | electrolytic, 100 μF – 10 mF, polarised | the device under test |
| Resistor | metal-film, 1% tolerance, 10–470 kΩ | sets τ via RC |
| DC supply | 0–12 V variable, low-noise | charges the capacitor |
| Voltmeter | digital, R_input ≥ 10 MΩ | reads V_C(t) |
| Stopwatch or data logger | resolution ≤ 0.1 s | records t |
| SPDT switch | low-resistance, mid-position-break | toggles charge ↔ discharge |
| Connecting leads | low-resistance, short | minimise lead inductance |
For the higher-precision variant, a data logger sampling at ~10 Hz replaces the stopwatch and voltmeter, eliminating timer-reaction-time uncertainty and producing 50–100 data points instead of 6–10. AQA accepts either approach for the Required Practical, but data-logger setups are now standard in well-equipped sixth-form labs.
graph LR
A["DC supply<br/>(charge position)"] --> B["SPDT switch"]
B --> C["Capacitor C<br/>under test"]
C --> D["Voltmeter<br/>R_V ≥ 10 MΩ"]
B -.->|discharge position| E["Resistor R<br/>known, 1% tolerance"]
E --> C
style A fill:#3b82f6,color:#fff
style C fill:#27ae60,color:#fff
style E fill:#f59e0b,color:#fff
The SPDT switch with a mid-position break ensures that at the moment of switching from charge to discharge, the supply is disconnected before the resistor is connected — preventing a brief direct short of the supply through R during the transition.
Step 1 — Charge the capacitor. With the switch in the charge position, connect the capacitor to the DC supply for at least 5τ (a few seconds to a few minutes, depending on RC). The capacitor's voltage rises along V(t) = V₀(1 − e^(−t/RC)) and asymptotes to V₀ at the supply voltage.
Step 2 — Confirm equilibrium. Read the voltage. It should equal the supply voltage to within the meter's tolerance. If it doesn't, allow more charging time.
Step 3 — Switch to discharge. Flick the switch to the discharge position. Simultaneously start the stopwatch and read the initial voltage V₀ (this should still equal the supply voltage at t = 0).
Step 4 — Record V vs t. At pre-arranged time intervals (typically Δt = τ/3 or similar to give roughly 8–12 data points spread over 0 ≤ t ≤ 3τ), record (t, V). Stop when V has fallen to a few percent of V₀ — beyond this, the voltmeter resolution dominates the uncertainty.
Step 5 — Repeat for replicate trials. Re-charge the capacitor and repeat at least three times. Average the V(t) values at each t. This averages out one-off measurement errors and tightens the gradient uncertainty.
Step 6 — Tabulate. Make a clear table of (t, V_mean, V_std, ln V_mean), with units and decimal places consistent with the meter resolution.
Step 7 — Plot. Plot ln V_mean against t on graph paper or computer software. Draw the line of best fit through all points (use a least-squares regression where available, or judge by eye). The line should be straight to within scatter; visible curvature indicates non-ideal behaviour.
Step 8 — Extract τ. Calculate the gradient m of the best-fit line by reading two well-separated points on the line (not necessarily data points). Then τ = −1/m, and C = τ/R.
Step 9 — Calculate uncertainty. Estimate the gradient uncertainty Δm from the spread of points around the line of best fit (or from the standard error of regression in software). Propagate via Δτ/τ = Δm/m, then ΔC/C = √((Δτ/τ)² + (ΔR/R)²).
Step 10 — Compare with marked value. If the capacitor has a marked nominal capacitance (with tolerance, often ±20% for electrolytic types), compare your measured C ± ΔC against the marked range. Discuss any disagreement.
A student measures the discharge of an unknown capacitor through R = 100 kΩ (1% tolerance). The capacitor was charged to V₀ = 8.0 V.
| t / s | V (trial 1) / V | V (trial 2) / V | V (trial 3) / V | V_mean / V | ln(V_mean) |
|---|---|---|---|---|---|
| 0 | 8.00 | 7.99 | 8.01 | 8.00 | 2.079 |
| 5 | 5.91 | 5.94 | 5.89 | 5.91 | 1.777 |
| 10 | 4.38 | 4.35 | 4.40 | 4.38 | 1.477 |
| 15 | 3.24 | 3.21 | 3.23 | 3.23 | 1.172 |
| 20 | 2.39 | 2.40 | 2.36 | 2.38 | 0.867 |
| 25 | 1.77 | 1.75 | 1.79 | 1.77 | 0.571 |
| 30 | 1.31 | 1.30 | 1.32 | 1.31 | 0.270 |
Gradient extraction: from a least-squares fit to (t, ln V_mean), m = (0.270 − 2.079) / (30 − 0) = −1.809/30 = −0.0603 s⁻¹ (with successive-difference verification giving consistent values −0.0604, −0.0600, −0.0610, −0.0610, −0.0592, −0.0602 — confirming linearity).
τ = −1/m = 1/0.0603 = 16.58 s
C = τ/R = 16.58 / (1.00 × 10⁵) = 1.66 × 10⁻⁴ F = 166 μF
Uncertainty analysis. Suppose the gradient uncertainty from the spread of points is Δm = ±0.0008 s⁻¹ (~1.3%). The resistor is 1% tolerance, so ΔR/R = 0.01.
ΔC/C = √((Δm/m)² + (ΔR/R)²) = √(0.013² + 0.010²) = √(0.000169 + 0.000100) = √0.000269 = 0.0164 ≈ 1.6%
ΔC = 1.6% × 166 = 2.7 μF
Reported result: C = 166 ± 3 μF, consistent with a nominal "150 μF ± 20%" marking (which spans 120–180 μF), suggesting the capacitor is at the upper end of its marked tolerance.
| Source | Magnitude (typical) | Mitigation |
|---|---|---|
| Timer resolution / reaction time (manual) | ±0.2 s | Use data logger; record t at intervals long enough that 0.2 s is < 5% of the interval |
| Voltmeter resolution | 1 in 999 ≈ 0.1% | Negligible for most readings; matters at low V where signal/quantisation is poor |
| Voltmeter input impedance | R_V ~ 10 MΩ | Choose R ≪ R_V (e.g. R = 100 kΩ ≪ 10 MΩ); meter loads circuit by R/R_V ≈ 1% |
| Resistor tolerance | ±1% (metal-film); ±5% (carbon) | Use precision metal-film; calibrate against known R if precision is critical |
| Capacitor dielectric leakage | ~0.1%/s for electrolytic | Use plastic-film or ceramic for high-precision work; minimise total experiment time |
| Initial voltage drift before t=0 | ~1% if switching slow | Practise rapid switching; double-blind start with helper calling t=0 |
| Best-fit line judgement | ~1–3% (graphical); <1% (least-squares) | Use software least-squares fit; report standard error of regression |
Most contributions can be made negligible by careful design. The dominant residual error in a well-conducted RP9 is typically the resistor tolerance (1%) plus the best-fit gradient uncertainty (1–3%), giving a final C uncertainty of ~2–4%.
The Common Practical Assessment Criteria assessed by RP9 include:
Specimen question modelled on the AQA 7408 Paper 3 (Practical Skills) format.
A student investigates the capacitance of an unmarked electrolytic capacitor by discharging it through a fixed resistor R = (47 ± 1) kΩ. The capacitor is initially charged to V₀ = 5.0 V; the student records V across the capacitor every 2.0 s using a manual stopwatch and digital voltmeter. Data are shown below:
| t / s | V / V |
|---|---|
| 0 | 5.00 |
| 2 | 3.95 |
| 4 | 3.13 |
| 6 | 2.47 |
| 8 | 1.96 |
| 10 | 1.55 |
| 12 | 1.22 |
(a) Design: State one systematic error in this measurement procedure and explain its effect on the final value of C. Suggest one experimental improvement to reduce it. (3 marks) (b) Analysis: Using a linearised plot, calculate the value of C with an estimate of the percentage uncertainty in C. (4 marks) (c) Evaluation: The student claims their measured C agrees with the marked nominal value of "330 μF ± 20%". Comment, with reference to your answer in (b), on whether this claim is justified. (2 marks)
| Part | Marks | Assessment Objective | What is being tested |
|---|---|---|---|
| (a) | 3 | AO3 (evaluation) | Identification + explanation + targeted improvement |
| (b) | 4 | AO2 (application) + AO3 (analysis) | Linearisation, gradient + C, % uncertainty |
| (c) | 2 | AO3 (evaluation) | Comparison within combined uncertainty bounds |
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