---
title: "A rigid circular wire loop of radius \\(a\\) and total resistance \\(R\\) lies fixed in a plane perpendicular to a spatially uniform magnetic field. The magnitude of the magnetic field changes with time \\(t\\) according to \\(B(t) = B_0 \\left(1 + \\dfrac{t}{\\tau}\\right)^3\\), where \\(B_0\\) and \\(\\tau\\) are positive constants. Which of the following expressions represents the total electric charge \\(Q\\) that flows past a point in the loop between \\(t = 0\\) and \\(t = \\tau\\)?"
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/118773/"
date_modified: "2026-08-04T08:14:35+00:00"
---

# A rigid circular wire loop of radius \(a\) and total resistance \(R\) lies fixed in a plane perpendicular to a spatially uniform magnetic field. The magnitude of the magnetic field changes with time \(t\) according to \(B(t) = B_0 \left(1 + \dfrac{t}{\tau}\right)^3\), where \(B_0\) and \(\tau\) are positive constants. Which of the following expressions represents the total electric charge \(Q\) that flows past a point in the loop between \(t = 0\) and \(t = \tau\)?

A rigid circular wire loop of radius \(a\) and total resistance \(R\) lies fixed in a plane perpendicular to a spatially uniform magnetic field. The magnitude of the magnetic field changes with time \(t\) according to \(B(t) = B_0 \left(1 + \dfrac{t}{\tau}\right)^3\), where \(B_0\) and \(\tau\) are positive constants. Which of the following expressions represents the total electric charge \(Q\) that flows past a point in the loop between \(t = 0\) and \(t = \tau\)?

![A circular wire loop is drawn in the plane of the page. A straight dashed line segment extends from the geometric center of the circular loop to its top perimeter, with an arrow head at the perimeter labeled a. Near the upper-left of the loop, text reads R, indicating the total loop resistance. Multiple evenly spaced magnetic field indicators, each drawn as a small circle containing an X symbol to represent a vector directed into the page, are distributed inside and outside the circular loop. Beside the loop, a single arrow pointing into the page is labeled B(t). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831275-rSS5DE.jpg)

- **A.** \(\dfrac{3\pi a^2 B_0}{R}\)
- **B.** \(\dfrac{4\pi a^2 B_0}{R}\)
- **C.** \(\dfrac{7\pi a^2 B_0}{R}\)
- **D.** \(\dfrac{8\pi a^2 B_0}{R}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118773/*