Calculate the capacitor time constant
Use τ = RC to determine the discharge timescale and interpret the fraction remaining after one time constant.
Section 19 · Capacitance
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 19. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use charge, potential difference, current, resistance, graph area, energy and SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the time constant for a discharging capacitor?
What you will learn
- Use τ = RC
- Express resistance and capacitance in SI units
- Interpret the time constant on an exponential discharge graph
Resistance and capacitance set the discharge timescale
For a capacitor discharging through a resistor, the time constant is τ = RC. Use resistance in ohms and capacitance in farads to obtain τ in seconds because Ω F = s.
A larger R or C gives a larger time constant and slower discharge. A smaller value gives a faster response. The time constant describes the scale of the decay rather than a time at which the capacitor becomes exactly empty.
One time constant leaves about 37 percent
For exponential discharge, after t = τ the charge, voltage and current magnitude are multiplied by e⁻¹, about 0.368 of their initial values. After a further time constant, the remaining fraction is multiplied by 0.368 again.
On a graph, τ can be estimated from the time at which the decaying quantity reaches about 37 percent of its initial magnitude. Check that the graph begins at t = 0 with the stated initial value.
Pause and explain
After one time constant, approximately what fraction of the initial voltage remains?
Worked example
A 4.7 μF capacitor discharges through a 2.0 MΩ resistor. Calculate τ.
Show the worked solution
- Convert 4.7 μF to 4.7 × 10⁻⁶ F and 2.0 MΩ to 2.0 × 10⁶ Ω.
- Use τ = RC.
- State seconds.
Answer τ = 2.0 × 10⁶ × 4.7 × 10⁻⁶ = 9.4 s.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find τ for R = 100 kΩ and C = 22 μF.
Give me a hint
Convert both prefixes, then multiply.
Compare my reasoning
- R = 1.0 × 10⁵ Ω.
- C = 22 × 10⁻⁶ F.
- τ = RC = 2.2 s.
τ = 2.2 s.
Look for these in your work
- Resistance converted
- Capacitance converted
- Product in seconds
A capacitor has τ = 5.0 s and initial voltage 12 V. Estimate its voltage after 5.0 s.
Give me a hint
After one time constant, 0.368 of the initial value remains.
Compare my reasoning
- t = τ.
- V = V₀e⁻¹ ≈ 12 × 0.368.
- Evaluate in volts.
V ≈ 4.4 V.
Look for these in your work
- One-time-constant condition identified
- Remaining fraction applied
- Voltage estimated
A discharge graph reaches 3.7 V from an initial 10 V after 8.0 s. Estimate the time constant.
Give me a hint
Compare the remaining fraction with e⁻¹.
Compare my reasoning
- 3.7/10 = 0.37 of the initial voltage.
- This is approximately e⁻¹.
- The elapsed time is therefore about one time constant.
τ ≈ 8.0 s.
Look for these in your work
- Remaining fraction calculated
- 37 percent criterion used
- Timescale estimated
Common mistakes
- Using kilohms and microfarads without considering the powers of ten
- Using R/C instead of RC
- Treating one time constant as complete discharge
- Saying 37 percent has decayed instead of 37 percent remains
What can you explain now?
What is the time constant for a discharging capacitor?
Compare with the explanation
τ = RC
The resistance–capacitance product sets the timescale.
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Make it stick
What does the time constant tell you about a capacitor's discharge?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.