When a charged capacitor discharges through a resistor, the voltage decays exponentially with time. The time constant τ = RC determines how quickly this happens. By measuring τ for different resistance values, you can verify the linear relationship τ = RC and calculate the capacitance of the capacitor. A data logger makes this investigation particularly powerful, generating rich continuous voltage-time data for each resistance. This is an outstanding IB Diploma Physics HL Internal Assessment topic.

This practical is suitable for IB Diploma Physics HL.

Background Theory

When a capacitor of capacitance C discharges through a resistance R, the voltage V across the capacitor at time t is:

V = V₀ e⁻^(t/RC)

Taking natural logarithms: ln V = ln V₀ − t/RC

A graph of ln V against t is linear with gradient = −1/RC. The time constant τ = RC is the time for V to fall to V₀/e ≈ 0.368 V₀. Since τ = RC, a graph of τ against R should be linear with gradient = C, allowing the capacitance to be determined experimentally.

Variables

• Independent variable (IV): Resistance R (Ω) — e.g. 10k, 20k, 33k, 47k, 68k, 100k Ω using fixed resistors or a decade resistance box
• Dependent variable (DV): Time constant τ (s) — determined from the gradient of a ln V vs t graph using a data logger
• Controlled variables (CV): Same capacitor throughout, same initial charging voltage V₀, same circuit connections, temperature (resistance of components is temperature-dependent)

Equipment

• Electrolytic capacitor (e.g. 470 μF or 1000 μF) — note the polarity
• Fixed resistors or a decade resistance box (10k–100k Ω)
• DC power supply (e.g. 9 V)
• Data logger with voltage probe (e.g. Pasco, Vernier) OR oscilloscope
• Switch (to disconnect power supply and start discharge)
• Connecting wires
• Multimeter (to verify resistance values)

Safety

⚠️ Ensure the electrolytic capacitor is connected with correct polarity — reverse connection can cause it to rupture. Do not exceed the capacitor’s rated voltage. No chemical hazards — no waste disposal required.

Method

1. Connect the circuit: power supply → switch → capacitor, with the resistor in parallel with the capacitor. Connect the data logger voltage probe across the capacitor.
2. Set the data logger to record voltage at intervals of 0.1 s (or shorter for small R values).
3. Close the switch to charge the capacitor fully to V₀. Wait until the voltage is stable.
4. Start the data logger recording, then open the switch to begin discharge through the resistor.
5. Record until V has fallen to less than 0.1 V₀ (at least 2–3 time constants).
6. Export the voltage-time data. Plot ln V against t. Calculate the gradient and determine τ = −1/gradient.
7. Repeat for each resistance value R. Take at least three runs per resistance and calculate a mean τ.

Results Table

R (kΩ)	τ₁ (s)	τ₂ (s)	τ₃ (s)	Mean τ (s)
10
20
33
47
68
100

Analysis

1. For each resistance, plot ln V (y-axis) against t (x-axis) using your data logger output. Draw a best-fit straight line. Calculate the gradient = −1/τ. Record τ and its uncertainty from the spread of the three runs.

2. Plot mean τ (y-axis, s) against R (x-axis, Ω). The relationship τ = RC predicts a straight line through the origin.

3. Calculate the gradient of this graph: gradient = C (in farads).

4. Compare your calculated C to the stated value on the capacitor. Calculate the percentage error.

5. Use worst-case lines on the τ vs R graph to determine δC and express your result as C ± δC.

Discussion Points

• Why does the voltage decay exponentially rather than linearly?
• Why does increasing R increase the time constant? Explain in terms of the discharge current.
• Why does the graph of ln V vs t become linear when the voltage decays exponentially?
• Why might your calculated C differ from the stated value on the capacitor? Consider tolerance ratings (typically ±20% for electrolytic capacitors).
• What would happen to the discharge curve if the temperature of the resistor increased significantly during discharge?

IA Guidance

The use of a data logger gives this IA rich continuous data and allows highly precise determination of τ from each ln V vs t graph, making the uncertainty analysis particularly sophisticated. To score highly:

• Research Design: Justify your resistance range — R too small means τ is too short to measure accurately; R too large means discharge takes too long. Show how you chose your range. Explain why a data logger is superior to manual voltage readings.
• Data Analysis: Include at least two ln V vs t graphs (one for smallest R, one for largest). Show gradient calculations clearly. Plot τ vs R with error bars from the spread of three runs. Use worst-case lines to determine δC.
• Conclusion: State C ± δC and compare to the manufacturer’s stated value. Note that electrolytic capacitors have a tolerance of ±20% — discuss whether your result is consistent with this.
• Evaluation: Discuss the internal resistance of the power supply and connecting wires as a source of systematic error (they add to R, making τ slightly larger than expected). Suggest measuring total circuit resistance with a multimeter for a more accurate R value.

• Practical Physics