---
title: "A circuit contains an inductor of inductance \\(L\\) and a resistor of resistance \\(R\\) connected in a single closed loop. At time \\(t = 0\\), the current in the loop is \\(I_0\\), and no external voltage source is connected. The current in the circuit decays according to \\(I(t) = I_0 e^{-tR/L}\\). What is the total thermal energy dissipated in the resistor between \\(t = 0\\) and time \\(t = \\tau\\), where \\(\\tau = \\dfrac{L}{R}\\) is the time constant of the circuit?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/118701/"
date_modified: "2026-08-04T08:13:48+00:00"
---

# A circuit contains an inductor of inductance \(L\) and a resistor of resistance \(R\) connected in a single closed loop. At time \(t = 0\), the current in the loop is \(I_0\), and no external voltage source is connected. The current in the circuit decays according to \(I(t) = I_0 e^{-tR/L}\). What is the total thermal energy dissipated in the resistor between \(t = 0\) and time \(t = \tau\), where \(\tau = \dfrac{L}{R}\) is the time constant of the circuit?

A circuit contains an inductor of inductance \(L\) and a resistor of resistance \(R\) connected in a single closed loop. At time \(t = 0\), the current in the loop is \(I_0\), and no external voltage source is connected. The current in the circuit decays according to \(I(t) = I_0 e^{-tR/L}\). What is the total thermal energy dissipated in the resistor between \(t = 0\) and time \(t = \tau\), where \(\tau = \dfrac{L}{R}\) is the time constant of the circuit?

![A simple single-loop circuit schematic drawn with black lines on a white background. The loop is rectangular with two vertical branches and two horizontal wires connecting them at top and bottom. The left vertical branch contains an inductor symbol drawn as four rounded coils labeled \(L\). The right vertical branch contains a resistor symbol drawn as a zigzag line labeled \(R\). Near the top horizontal wire, a horizontal arrow pointing left is labeled \(I_0\). No other labels, lines, text, or symbols appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831228-dNpSNv.jpg)

- **A.** \(\dfrac{1}{2} L I_0^2 \left(1 - e^{-1}\right)\)
- **B.** \(L I_0^2 \left(1 - e^{-2}\right)\)
- **C.** \(\dfrac{1}{2} L I_0^2 \left(1 - e^{-2}\right)\)
- **D.** \(\dfrac{1}{2} L I_0^2 e^{-2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118701/*