---
title: "A rectangular wire loop of length \\(L\\), width \\(w\\), and total resistance \\(R\\) is placed in a uniform magnetic field of magnitude \\(B\\) that points in the \\(+z\\)-direction. The loop is rotated about the \\(y\\)-axis with a constant angular speed \\(\\omega\\). The \\(y\\)-axis passes directly through the center of the loop, bisecting the two sides of length \\(w\\). At time \\(t = 0\\), the loop lies in the \\(xy\\)-plane such that the area vector of the loop is parallel to the magnetic field.  Express all algebraic answers in terms of \\(L\\), \\(w\\), \\(R\\), \\(B\\), \\(\\omega\\), \\(t\\), and physical constants, as appropriate."
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url: "https://nerd-notes.com/ubq/117818/"
date_modified: "2026-08-04T07:58:49+00:00"
---

# A rectangular wire loop of length \(L\), width \(w\), and total resistance \(R\) is placed in a uniform magnetic field of magnitude \(B\) that points in the \(+z\)-direction. The loop is rotated about the \(y\)-axis with a constant angular speed \(\omega\). The \(y\)-axis passes directly through the center of the loop, bisecting the two sides of length \(w\). At time \(t = 0\), the loop lies in the \(xy\)-plane such that the area vector of the loop is parallel to the magnetic field.

Express all algebraic answers in terms of \(L\), \(w\), \(R\), \(B\), \(\omega\), \(t\), and physical constants, as appropriate.

A rectangular wire loop of length \(L\), width \(w\), and total resistance \(R\) is placed in a uniform magnetic field of magnitude \(B\) that points in the \(+z\)-direction. The loop is rotated about the \(y\)-axis with a constant angular speed \(\omega\). The \(y\)-axis passes directly through the center of the loop, bisecting the two sides of length \(w\). At time \(t = 0\), the loop lies in the \(xy\)-plane such that the area vector of the loop is parallel to the magnetic field.

Express all algebraic answers in terms of \(L\), \(w\), \(R\), \(B\), \(\omega\), \(t\), and physical constants, as appropriate.

![A 3D Cartesian coordinate system is shown with mutually perpendicular x, y, and z axes intersecting at an origin. A single rectangular wire loop lies flat within the horizontal xy-plane. The loop has two sides of length '\(w\)' that are parallel to the x-axis, and two sides of length '\(L\)' that are parallel to the y-axis. The y-axis passes directly through the geometric center of the loop, bisecting the two sides of length '\(w\)'. A curved arrow labeled '\(\omega\)' forms a three-quarter circle around the positive y-axis, indicating the direction of continuous rotation. Four equally spaced straight vertical arrows point directly upward in the +z-direction, representing a uniform magnetic field. One of these vertical arrows is labeled with the vector symbol '\(\vec{B}\)'. No other labels, lines, text, variables, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830329-Me1P0R.jpg)

**Part a)** **Derive** an expression for the magnetic flux \(\Phi_B\) through the loop as a function of time \(t\). *(2 points)*

**Part b)** **Derive** an expression for the induced emf \(\varepsilon\) in the loop as a function of time \(t\). *(2 points)*

**Part c)** **Determine** an expression for the instantaneous electrical power \(P\) dissipated by the loop as a function of time \(t\). *(2 points)*

**Part d)** Using the principle of conservation of energy, **derive** an expression for the magnitude of the external torque \(\tau\) required to keep the loop rotating at the constant angular speed \(\omega\) as a function of time \(t\). *(2 points)*

**Part e)** The loop is now rotated at a new constant angular speed of \(2\omega\). **Indicate** whether the maximum external torque required to maintain this rotation is greater than, less than, or equal to the maximum external torque required to maintain the original angular speed \(\omega\). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117818/*