---
title: "Capacitor discharge, energy one fourth"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:capacitor-discharge-energy-quarter"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/capacitor-discharge-energy-quarter
site: BCT Engineering
---

# Capacitor discharge, energy one fourth

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2068 Chaitra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/capacitor-discharge-energy-quarter
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

A capacitor of capacitance 'C' is discharged through a resistor of resistance 'R'. After how many time constants is the energy stored becomes one fourth of initial value?

## Short answer

Energy U = U_0 e^(−2t/RC). Set U/U_0 = 1/4 gives e^(−2t/RC) = 1/4 → t/RC = (ln4)/2 = ln2 ≈ 0.693. So after 0.693 time constants.

## Answer

Charge on a discharging capacitor: q = q₀ e^(−t/RC). Stored energy U = q²/2C, so

**U = U₀ e^(−2t/RC)**

Let t = x RC and U/U₀ = 1/4.

```text
e^(−2x) = 1/4
  2x    = ln 4 = 1.386
   x    = 0.693 = ln 2
```

**Answer: after 0.693 time constants (t = RC ln 2 ≈ 0.69 RC).**

This is the same time at which the charge falls to 1/2 of its initial value, because (1/2)² = 1/4.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Stored energy during discharge falls as:
   A. U = U₀e^(−2t/RC)
   B. U = U₀e^(−t/RC)
   C. U = U₀e^(−t/2RC)
   D. U = U₀/(1+t/RC)
   Answer: A. U ∝ q². (Key point: Energy decay)

2. Setting e^(−2x) = 1/4 gives x:
   A. ln 4 ≈ 1.386
   B. ln 2 ≈ 0.693
   C. 2 ln 2
   D. 0.25
   Answer: B. 2x = ln 4, so x = ln 4/2 = ln 2. (Key point: Solve)

3. Why is this the same time at which charge falls to half?
   A. Because energy is proportional to q
   B. Because the time constant doubles
   C. Because energy goes as q², and (1/2)² = 1/4
   D. Because charge falls linearly
   Answer: C. q = q₀/2 gives U = U₀/4. (Key point: Link to charge)

## Close variants set in other exams

- [Capacitor discharge, energy one eighth](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acapacitor-discharge-energy-eighth) — Engineering Physics, chapter 8, asked 2×

## Related in Engineering Physics

- [Energy stored in coil, time constants](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acoil-energy-stored-time-constants) — Engineering Physics, chapter 8
- [Energy stored in parallel capacitors](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aparallel-capacitors-energy-stored) — Engineering Physics, chapter 7

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