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

# Capacitor discharge, energy one eighth

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2070 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-eighth
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

A capacitor of capacitance C is discharging through a resistor of resistance R. After how many time constants is the stored energy 1/8 of its initial value?

## Short answer

During discharge, stored energy U = U₀ e^(−2t/RC). Setting U = U₀/8 gives e^(−2t/RC) = 1/8 ⇒ t = (ln 8)/2 RC ≈ 1.04 time constants.

## Answer

During discharge the charge falls as q = q₀ e^(−t/RC), so the stored energy U = q²/2C falls as

**U = U₀ e^(−2t/RC)**

Let t = x RC, where x is the number of time constants. We need U/U₀ = 1/8.

```text
e^(−2x) = 1/8
  2x    = ln 8 = 2.079
   x    = 1.04
```

**Answer: after about 1.04 time constants (t ≈ 1.04 RC).**

Note that the energy decays twice as fast as the charge: it falls to 1/e² of its initial value after 1 time constant.

## Check yourself

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

1. During discharge through R, the stored energy varies as:
   A. U = U₀ e^(−t/RC)
   B. U = U₀ e^(−2t/RC)
   C. U = U₀ e^(−t/2RC)
   D. U = U₀(1 − e^(−2t/RC))
   Answer: B. U = q²/2C and q = q₀e^(−t/RC), so the exponent doubles. (Key point: Energy decay)

2. For U/U₀ = 1/8 with t = x·RC, the equation is:
   A. e^(−2x) = 1/8
   B. e^(−x) = 1/8
   C. e^(−x) = 1/√8
   D. 1 − e^(−2x) = 1/8
   Answer: A. Substitute into U/U₀ = e^(−2x). (Key point: Setting up)

3. The number of time constants is:
   A. about 2.08
   B. about 0.52
   C. about 1.04
   D. about 8
   Answer: C. x = ln 8 / 2 = 2.079/2 ≈ 1.04. (Key point: Answer)

4. How does the decay rate of energy compare with that of charge?
   A. They decay at the same rate
   B. Energy decays half as fast
   C. Energy does not decay
   D. Energy decays twice as fast as charge
   Answer: D. Energy goes as q², so after one time constant it is 1/e² of the initial value. (Key point: Energy vs charge)

## Close variants set in other exams

- [Capacitor discharge, energy one fourth](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acapacitor-discharge-energy-quarter) — 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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