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title: "A flat, square conducting loop with side length \\(a\\), total resistance \\(R\\), and moment of inertia \\(I\\) about its central symmetry axis rotates in a region containing a uniform magnetic field \\(\\vec{B}\\). The rotation axis lies in the plane of the loop and is perpendicular to \\(\\vec{B}\\). At an instant when the normal to the loop makes an angle \\(\\theta\\) with \\(\\vec{B}\\), the loop has an instantaneous angular velocity \\(\\omega = \\dfrac{d\\theta}{dt}\\). Which of the following differential equations correctly describes the rate of change of the loop’s angular velocity \\(\\dfrac{d\\omega}{dt}\\) due to electromagnetic damping?"
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url: "https://nerd-notes.com/ubq/118746/"
date_modified: "2026-08-04T08:14:06+00:00"
---

# A flat, square conducting loop with side length \(a\), total resistance \(R\), and moment of inertia \(I\) about its central symmetry axis rotates in a region containing a uniform magnetic field \(\vec{B}\). The rotation axis lies in the plane of the loop and is perpendicular to \(\vec{B}\). At an instant when the normal to the loop makes an angle \(\theta\) with \(\vec{B}\), the loop has an instantaneous angular velocity \(\omega = \dfrac{d\theta}{dt}\). Which of the following differential equations correctly describes the rate of change of the loop’s angular velocity \(\dfrac{d\omega}{dt}\) due to electromagnetic damping?

A flat, square conducting loop with side length \(a\), total resistance \(R\), and moment of inertia \(I\) about its central symmetry axis rotates in a region containing a uniform magnetic field \(\vec{B}\). The rotation axis lies in the plane of the loop and is perpendicular to \(\vec{B}\). At an instant when the normal to the loop makes an angle \(\theta\) with \(\vec{B}\), the loop has an instantaneous angular velocity \(\omega = \dfrac{d\theta}{dt}\). Which of the following differential equations correctly describes the rate of change of the loop's angular velocity \(\dfrac{d\omega}{dt}\) due to electromagnetic damping?

![A 3D perspective drawing of a flat square loop of side length a in the xy-plane region. A dashed central axis of rotation passes vertically through the midpoints of the top and bottom sides of the square loop. A uniform magnetic field represented by three horizontal parallel arrows pointing to the right labeled vector B. A unit normal vector labeled hat n extends perpendicularly from the center of the square loop surface. The angle between vector B and hat n is labeled theta. A curved arrow around the vertical rotation axis indicates angular velocity omega. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831246-UfBfQx.jpg)

- **A.** \(\dfrac{d\omega}{dt} = -\dfrac{B^2 a^4 \cos^2\theta}{I R} \omega\)
- **B.** \(\dfrac{d\omega}{dt} = -\dfrac{B^2 a^4 \sin\theta \cos\theta}{I R} \omega\)
- **C.** \(\dfrac{d\omega}{dt} = -\dfrac{B^2 a^4 \sin^2\theta}{I R} \omega\)
- **D.** \(\dfrac{d\omega}{dt} = -\dfrac{B^2 a^4 \sin\theta}{I R} \omega\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118746/*