--- title: "Growth and decay of current in LR circuit" subject: "Engineering Physics" chapter: 8 key: "semester_1/engineering_physics:lr-circuit-growth-decay" source: https://www.bctengineer.com/q/semester_1/engineering_physics/lr-circuit-growth-decay site: BCT Engineering --- # Growth and decay of current in LR circuit Engineering Physics · Chapter 8: Electromagnetism - Asked in: 2073 Chaitra, 5 marks; 2071 Chaitra, 5 marks; 2070 Chaitra, 5 marks; 2067 Asoj, 5 marks (asked 4 times) - Video: [growth and decay of current in lr circuit](https://www.youtube.com/watch?v=laDGSqVg36U) — Eduphile - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/engineering_physics/lr-circuit-growth-decay - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8 ## Question Derive the relation for rise (growth) and fall (decay) of current in an LR circuit. Plot graph between current and time for both cases and explain the figure and the inductive time constant. ## Short answer For growth, i = (ε/R)(1 - e^(-t/τ)) with τ = L/R. For decay, i = i0 e^(-t/τ). The time constant τ is the time for current to reach 63.2% of final value in growth or fall to 36.8% in decay; after ~5τ it's within 1% of final. ## Answer Circuit: battery emf ε, resistor R, inductor L in series with a switch. The **inductive time constant** is τ = L/R. ```text +----- R -----+ | | ε =| L (coil) | | +---- S ------+ ``` #### Growth of current (switch to the battery) 1. Kirchhoff's loop rule: ε − iR − L di/dt = 0 2. Rearrange: di/(ε/R − i) = (R/L) dt 3. Integrate from i = 0 at t = 0: −ln[(ε/R − i)/(ε/R)] = Rt/L 4. **i = (ε/R)(1 − e^(−t/τ)) = i₀(1 − e^(−t/τ))**, where i₀ = ε/R #### Decay of current (battery removed, circuit closed through R) 1. Loop rule with ε = 0: iR + L di/dt = 0 2. di/i = −(R/L) dt. Integrate from i = i₀ at t = 0: 3. **i = i₀ e^(−t/τ)** #### Graph ```text i Growth i Decay i0 |- - - - - - - _____ i0 |. | . ' | `. | .' | `. | .' | ` . | .' | ` . . . |.'_____________________ t |_____________________ t 0 tau 2tau 3tau 0 tau 2tau 3tau ``` The growth curve rises towards i₀ and the decay curve falls towards zero; the two are mirror images. #### Explanation - **Time constant τ = L/R** (seconds): the time in which the current grows to 63.2% of i₀ (1 − e⁻¹), or decays to 36.8% (e⁻¹) of i₀. - At t = 0 the inductor opposes any change: growth starts with i = 0 and slope ε/L. - After about 5τ the current is within 1% of its final value. - A large L or small R gives a larger τ, so the current changes more slowly. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The current growth in an LR circuit is A. i = (ε/R) e^(−t/τ) B. i = (ε/R)(1 − e^(−t/τ)) C. i = ε R (1 − e^(−t/τ)) D. i = (ε/L)(1 − e^(−t/τ)) Answer: B. Solving ε − iR − L di/dt = 0 gives i = i0(1 − e^(−t/τ)). (Key point: Growth) 2. The inductive time constant τ equals A. RL B. R/L C. L/R D. 1/(LR) Answer: C. τ = L/R; it is the time to reach 63.2% of i0. (Key point: Time constant) 3. During decay, after t = τ the current is A. 63.2% of i0 B. 36.8% of i0 C. Zero D. 50% of i0 Answer: B. i = i0 e^(−1) = 0.368 i0. (Key point: Decay) 4. A larger L or smaller R makes the current A. Change faster B. Reach a higher final value C. Change more slowly D. Oscillate Answer: C. τ = L/R increases, so the change is slower; the final value ε/R depends only on R. (Key point: Effect of L) ## Related in Engineering Physics - [Rise and fall of current in RC circuit](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Arc-circuit-charge-discharge-current) — Engineering Physics, chapter 8 ## Related in other subjects - [Pure inductive circuit, zero real power](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Apure-inductive-circuit-power-derivation) — Basic Electrical Engineering, chapter 6, asked 2× - [RL circuit current as a function of time](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aapp-rl-circuit-current) — Mathematics I, chapter 4, asked 4× - [Temperature rise-time curve under heating](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Aheating-curve-time-constant) — Electric Machine Design, chapter 2 - [Townsend current growth with secondary ionization](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atownsend-current-growth-secondary) — High Voltage Engineering, chapter 6, asked 2×