--- title: "Relaxation time constant from continuity" subject: "Electromagnetism" chapter: 2 key: "semester_3/electromagnetism:relaxation-time-continuity-derive" source: https://www.bctengineer.com/q/semester_3/electromagnetism/relaxation-time-continuity-derive site: BCT Engineering --- # Relaxation time constant from continuity Electromagnetism · Chapter 2: Electric field - Asked in: 2076 Chaitra, 4 marks; 2068 Chaitra, 3+4 marks (asked 2 times) - Video: [2.13 CONTINUITY EQUATION and RELAXATION TIME for IES,GATE](https://www.youtube.com/watch?v=HiWqt--pXpU) — EPOV CHANNEL - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/relaxation-time-continuity-derive - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2 ## Question Define Relaxation Time Constant (RTC) and, using the continuity equation, elaborate the concept of RTC with necessary derivations. ## Short answer Relaxation time constant T_r = ε/σ is the time for charge density inside a conductor to decay to 1/e of its initial value. From continuity ∇·J = −∂ρ_v/∂t and Ohm's law J = σE, Gauss's law gives ∂ρ_v/∂t + (σ/ε)ρ_v = 0, yielding ρ_v = ρ₀ e^{−t/T_r}. ## Answer #### Relaxation time constant When free charge is suddenly placed inside a conducting (or lossy dielectric) material, it does not stay. It decays exponentially as the charge moves to the surface. The **relaxation time constant** T_r = ε/σ is the time in which the volume charge density inside the material falls to 1/e (about 36.8 %) of its initial value. #### Derivation using the continuity equation The point form of the continuity equation expresses charge conservation: ```text ∇ · J = −∂ρ_v/∂t ``` For a conducting medium, Ohm's law in point form gives J = σE, and E = D/ε. Then ```text ∇ · J = σ ∇ · E = (σ/ε) ∇ · D ``` By Gauss's law, ∇ · D = ρ_v. Therefore ```text (σ/ε) ρ_v = −∂ρ_v/∂t ∂ρ_v/∂t + (σ/ε) ρ_v = 0 ``` This is a first-order differential equation. Its solution is ```text ρ_v = ρ₀ e^(−(σ/ε) t) = ρ₀ e^(−t/T_r) where T_r = ε/σ (seconds) ``` Here ρ₀ is the charge density at t = 0. #### Interpretation - The charge density inside a conductor decays exponentially to zero. The charge that was inside appears as surface charge on the boundary of the conductor. - Larger σ means faster decay. At t = T_r, ρ_v = 0.368 ρ₀. - **Good conductor** (copper, σ = 5.8 × 10⁷ S/m, ε ≈ ε₀): T_r = 8.854 × 10⁻¹²/5.8 × 10⁷ ≈ 1.5 × 10⁻¹⁹ s. This is extremely short, so we may say no charge remains inside a conductor. - **Good insulator** (e.g. fused quartz, σ ≈ 10⁻¹⁷ S/m): T_r is of the order of days to years, so charge placed in it stays for a long time. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the relaxation time T_r? A. σ/ε, the time to reach zero charge B. ε/σ, the time for charge density to fall to 1/e of its initial value C. σε, the time to double charge D. 1/ε, the time to remove charge Answer: B. T_r = ε/σ is the time in which the volume charge density decays to 1/e (about 36.8%). (Key point: Definition) 2. Combining the continuity equation, J = σE and Gauss's law gives what equation? A. ∂ρ_v/∂t + (σ/ε)ρ_v = 0 B. ∂ρ_v/∂t = σερ_v C. ∇·J = ρ_v D. ∂ρ_v/∂t - (σ/ε)ρ_v = 0 Answer: A. ∇·J = (σ/ε)ρ_v = -∂ρ_v/∂t, so ∂ρ_v/∂t + (σ/ε)ρ_v = 0. (Key point: Differential equation) 3. What is the solution? A. ρ_v = ρ₀ e^(+t/T_r) B. ρ_v = ρ₀ e^(-t/T_r) C. ρ_v = ρ₀ (1 - t/T_r) D. ρ_v = ρ₀ T_r Answer: B. The first-order equation gives exponential decay with time constant T_r. (Key point: Solution) 4. What happens to charge placed inside a good conductor? A. It stays in the interior B. It decays very quickly and appears as surface charge C. It becomes larger D. It oscillates forever Answer: B. For copper T_r is about 1.5×10⁻¹⁹ s, so interior charge vanishes and moves to the surface. (Key point: Where charge goes) ## Related in Electromagnetism - [Ampere's law vs continuity equation](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aampere-law-conflict-continuity) — Electromagnetism, chapter 3, asked 2× - [Derive continuity equation](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acontinuity-equation-derive-forms) — Electromagnetism, chapter 2, asked 2× ## Related in other subjects - [Relaxation time and mean free path](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Arelaxation-time-mean-free-path) — Electrical Engineering Material, chapter 2, asked 4× - [Damped oscillator relaxation time numerical](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adamped-relaxation-amplitude-energy-fifty) — Engineering Physics, chapter 1, asked 2× - [Maxwell's equations and continuity equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-continuity-equation) — Engineering Physics, chapter 9, asked 2× - [Continuity equation basis and Bernoulli equation](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Acontinuity-and-bernoulli) — Hydro Power, chapter 1