---
title: "A thin circular wire of radius \\(R\\) lies in the \\(xy\\)-plane centered at the origin. A steady current flows counterclockwise around the wire such that the current magnitude as a function of the azimuthal angle \\(\\theta\\) (measured counterclockwise from the positive \\(x\\)-axis) is given by \\(I(\\theta) = I_0 \\cos^2\\theta\\), where \\(I_0\\) is a positive constant. Which of the following expressions represents the correct integral setup using the Biot–Savart law to determine the magnetic field vector \\(\\vec{B}\\) at the origin?"
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url: "https://nerd-notes.com/ubq/125073/"
date_modified: "2026-09-28T14:16:43+00:00"
---

# A thin circular wire of radius \(R\) lies in the \(xy\)-plane centered at the origin. A steady current flows counterclockwise around the wire such that the current magnitude as a function of the azimuthal angle \(\theta\) (measured counterclockwise from the positive \(x\)-axis) is given by \(I(\theta) = I_0 \cos^2\theta\), where \(I_0\) is a positive constant. Which of the following expressions represents the correct integral setup using the Biot–Savart law to determine the magnetic field vector \(\vec{B}\) at the origin?

A thin circular wire of radius \(R\) lies in the \(xy\)-plane centered at the origin. A steady current flows counterclockwise around the wire such that the current magnitude as a function of the azimuthal angle \(\theta\) (measured counterclockwise from the positive \(x\)-axis) is given by \(I(\theta) = I_0 \cos^2\theta\), where \(I_0\) is a positive constant. Which of the following expressions represents the correct integral setup using the Biot–Savart law to determine the magnetic field vector \(\vec{B}\) at the origin?

![A set of perpendicular horizontal and vertical axes intersecting at a central origin point, labeled x on the right end and y on the upper end. A single solid circle of radius R is centered at the origin in the xy-plane. A dashed straight line segment extends from the origin to a point on the circle in the first quadrant, making an angle labeled \theta with the positive x-axis. At this point on the circle, a short solid arrow tangent to the circle points counterclockwise, labeled d\vec{\ell}. A curved arrow along the circumference indicates the counterclockwise current flow labeled I(\theta). A dot at the center marks the origin. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790605003-OPsP3k.jpg)

- **A.** \(\vec{B} = \dfrac{\mu_0 I_0}{4\pi R} \left(\int_0^{2\pi} \cos^2\theta\,d\theta\right) \hat{k}\)
- **B.** \(\vec{B} = \dfrac{\mu_0 I_0}{2R} \left(\int_0^{2\pi} \cos^2\theta\,d\theta\right) \hat{k}\)
- **C.** \(\vec{B} = \dfrac{\mu_0 I_0}{4\pi R} \left(\int_0^{2\pi} \cos^2\theta \sin\theta\,d\theta\right) \hat{k}\)
- **D.** \(\vec{B} = \dfrac{\mu_0 I_0}{4\pi R} \left(\int_0^{\pi} \cos^2\theta\,d\theta\right) \hat{k}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/125073/*