Use the capacitor discharge equation
Apply the exponential equation to voltage, charge or current during a capacitor's discharge through a resistor.
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 sign of the exponent in a passive discharge equation?
What you will learn
- Use x = x₀e⁻ᵗ/ᴿᶜ for discharge
- Select x as voltage, charge or current magnitude
- Calculate the remaining quantity at a stated time
One exponential form describes each decaying quantity
For a capacitor discharging through a resistor, x = x₀e⁻ᵗ/ᴿᶜ, where x can represent charge Q, potential difference V or current magnitude I at time t; x₀ is its initial value. Since τ = RC, the same relation is x = x₀e⁻ᵗ/τ.
The ratio x/x₀ = e⁻ᵗ/ᴿᶜ is dimensionless. Use t, R and C in compatible SI units so that t/(RC) has no unit. The equation predicts a smooth decrease towards zero.
Solve for a fraction before substituting
To find the amount remaining, calculate the exponential factor first, then multiply by x₀. For example, after 2τ the remaining fraction is e⁻² ≈ 0.135. This fraction applies separately to Q, V and current magnitude in the simple circuit model.
If solving for elapsed time, rearrange x/x₀ = e⁻ᵗ/τ using natural logarithms: t = −τ ln(x/x₀). The ratio must lie between zero and one during a passive discharge.
Pause and explain
A quantity has x₀ = 50 units and t/τ = 1. What remains?
Worked example
An 80 μC charge discharges with R = 100 kΩ and C = 10 μF. Find the charge remaining after 2.0 s.
Show the worked solution
- τ = RC = 100 × 10³ × 10 × 10⁻⁶ = 1.0 s.
- Q = Q₀e⁻ᵗ/τ = 80e⁻² μC.
- Evaluate the remaining charge.
Answer Q ≈ 80 × 0.135 = 10.8 μC.
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Test this topic →From guided practice to a new situation
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.
A 120 V capacitor voltage has time constant 4.0 s. Find the voltage after 8.0 s.
Give me a hint
Calculate t/τ, then multiply the initial voltage by the exponential fraction.
Compare my reasoning
- t/τ = 8.0/4.0 = 2.
- V = 120e⁻².
- Evaluate the result in volts.
V ≈ 16.2 V.
Look for these in your work
- Time ratio found
- Negative exponential applied
- Remaining voltage calculated
A current magnitude decreases from 3.0 mA with τ = 0.50 s. Find its magnitude after 0.20 s.
Give me a hint
Use the exponential factor with elapsed time divided by the time constant.
Compare my reasoning
- t/τ = 0.20/0.50 = 0.40.
- I = 3.0e⁻⁰·⁴ mA.
- Evaluate while retaining current-magnitude units.
I ≈ 2.0 mA.
Look for these in your work
- Time ratio calculated
- Exponential equation applied
- Current magnitude stated
A capacitor's voltage falls to 25 percent of its initial value. Express the elapsed time in terms of τ.
Give me a hint
Use the natural logarithm of the remaining voltage fraction to isolate time.
Compare my reasoning
- V/V₀ = 0.25.
- t = −τ ln(0.25).
- Evaluate the logarithm.
t ≈ 1.39τ.
Look for these in your work
- Fraction formed
- Logarithmic rearrangement used
- Time ratio stated
Common mistakes
- Using e⁺ᵗ/ᴿᶜ instead of a negative exponent
- Applying the initial value at the wrong time
- Mixing seconds with a time constant in milliseconds
- Forgetting that current's sign depends on the chosen reference direction
What can you explain now?
What is the sign of the exponent in a passive discharge equation?
Compare with the explanation
Negative
The quantity decreases as time increases.
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Make it stick
How do you use the discharge equation to find the remaining fraction and the elapsed time?
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.