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title: "A conducting rod of mass \\(m\\) and length \\(\\ell\\) slides along horizontal, frictionless conducting rails connected by a resistor of resistance \\(R\\). A uniform magnetic field of magnitude \\(B\\) is directed perpendicular to the plane of the rails. An external agent applies a force parallel to the rails such that the position of the rod as a function of time \\(t\\) is given by \\(x(t) = C t^2\\), where \\(C\\) is a positive constant and \\(x(0) = 0\\). Which of the following expressions gives the total mechanical work done by the external force on the rod from \\(t = 0\\) to \\(t = T\\)?"
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url: "https://nerd-notes.com/ubq/118749/"
date_modified: "2026-08-04T08:14:07+00:00"
---

# A conducting rod of mass \(m\) and length \(\ell\) slides along horizontal, frictionless conducting rails connected by a resistor of resistance \(R\). A uniform magnetic field of magnitude \(B\) is directed perpendicular to the plane of the rails. An external agent applies a force parallel to the rails such that the position of the rod as a function of time \(t\) is given by \(x(t) = C t^2\), where \(C\) is a positive constant and \(x(0) = 0\). Which of the following expressions gives the total mechanical work done by the external force on the rod from \(t = 0\) to \(t = T\)?

A conducting rod of mass \(m\) and length \(\ell\) slides along horizontal, frictionless conducting rails connected by a resistor of resistance \(R\). A uniform magnetic field of magnitude \(B\) is directed perpendicular to the plane of the rails. An external agent applies a force parallel to the rails such that the position of the rod as a function of time \(t\) is given by \(x(t) = C t^2\), where \(C\) is a positive constant and \(x(0) = 0\). Which of the following expressions gives the total mechanical work done by the external force on the rod from \(t = 0\) to \(t = T\)?

![A horizontal rectangular circuit in the xy-plane consisting of two parallel horizontal conducting rails extending to the right, connected on the left by a vertical resistor labeled R. A vertical conducting rod of length \ell and mass m spans between the rails, positioned at position x. A uniform magnetic field, indicated by a grid of small x symbols labeled \vec{B}, points into the page perpendicular to the plane of the rails. An arrow pointing to the right from the midpoint of the rod represents an external force \vec{F}_{\text{ext}}. The x-axis is indicated parallel to the rails with x = 0 at the left resistor end. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831247-WdlYXF.jpg)

- **A.** \(\dfrac{4 B^2 \ell^2 C^2 T^3}{3 R}\)
- **B.** \(2 m C^2 T^2 + \dfrac{4 B^2 \ell^2 C^2 T^2}{R}\)
- **C.** \(2 m C^2 T^2 + \dfrac{B^2 \ell^2 C^2 T^3}{3 R}\)
- **D.** \(2 m C^2 T^2 + \dfrac{4 B^2 \ell^2 C^2 T^3}{3 R}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118749/*