---
title: "A cylindrical conductor of outer radius \\(R\\) carries a pulse of current directed parallel to its central axis.  The conduction electron density varies with distance \\(r\\) from the central axis according to \\(n(r) = n_0 \\left(\\dfrac{r}{R}\\right)\\), and all electrons move with a time-dependent drift speed \\(v_d(t) = v_0 \\left(1 – \\dfrac{t^2}{T^2}\\right)\\) for \\(0 \\le t \\le T\\), where \\(n_0\\), \\(v_0\\), and \\(T\\) are positive constants and \\(e\\) is the elementary charge.  Which of the following expressions correctly represents the total charge \\(Q\\) that passes through the cross section of the conductor in the region \\(\\dfrac{R}{2} \\le r \\le R\\) during the time interval from \\(t = 0\\) to \\(t = T\\)?"
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url: "https://nerd-notes.com/ubq/121586/"
date_modified: "2026-08-23T05:24:10+00:00"
---

# A cylindrical conductor of outer radius \(R\) carries a pulse of current directed parallel to its central axis.

The conduction electron density varies with distance \(r\) from the central axis according to \(n(r) = n_0 \left(\dfrac{r}{R}\right)\), and all electrons move with a time-dependent drift speed \(v_d(t) = v_0 \left(1 – \dfrac{t^2}{T^2}\right)\) for \(0 \le t \le T\), where \(n_0\), \(v_0\), and \(T\) are positive constants and \(e\) is the elementary charge.

Which of the following expressions correctly represents the total charge \(Q\) that passes through the cross section of the conductor in the region \(\dfrac{R}{2} \le r \le R\) during the time interval from \(t = 0\) to \(t = T\)?

A cylindrical conductor of outer radius \(R\) carries a pulse of current directed parallel to its central axis.

The conduction electron density varies with distance \(r\) from the central axis according to \(n(r) = n_0 \left(\dfrac{r}{R}\right)\), and all electrons move with a time-dependent drift speed \(v_d(t) = v_0 \left(1 - \dfrac{t^2}{T^2}\right)\) for \(0 \le t \le T\), where \(n_0\), \(v_0\), and \(T\) are positive constants and \(e\) is the elementary charge.

Which of the following expressions correctly represents the total charge \(Q\) that passes through the cross section of the conductor in the region \(\dfrac{R}{2} \le r \le R\) during the time interval from \(t = 0\) to \(t = T\)?

![A perspective view of a horizontal cylindrical conductor of radius \(R\) extending along a horizontal dashed central axis. The circular front face shows a concentric dashed inner circle of radius \(\dfrac{R}{2}\) centered on the axis. The annular region between the inner circle of radius \(\dfrac{R}{2}\) and the outer circle of radius \(R\) is shaded with light gray fill. A solid radial line segment extends from the center to the inner boundary labeled \(\dfrac{R}{2}\), and another solid radial line segment extends from the center to the outer perimeter labeled \(R\). A variable radial distance arrow from the center to an arbitrary radius within the annulus is labeled \(r\). Three parallel horizontal arrows point to the right along the length of the cylinder, each labeled \(v_d\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462650-GjLUop.jpg)

- **A.** \(Q = 2\pi e n_0 v_0 \int_0^T \int_{R/2}^R \left(\dfrac{r}{R}\right) \left(1 - \dfrac{t^2}{T^2}\right) dr\,dt\)
- **B.** \(Q = 2\pi e n_0 v_0 \int_0^T \int_{R/2}^R \left(\dfrac{r^2}{R}\right) \left(1 - \dfrac{t^2}{T^2}\right) dr\,dt\)
- **C.** \(Q = \pi e n_0 v_0 \int_0^T \int_{R/2}^R \left(\dfrac{r^2}{R}\right) \left(1 - \dfrac{t^2}{T^2}\right) dr\,dt\)
- **D.** \(Q = 2\pi e n_0 v_0 \int_0^T \int_0^R \left(\dfrac{r^2}{R}\right) \left(1 - \dfrac{t^2}{T^2}\right) dr\,dt\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121586/*