---
title: "An open hemispherical surface of radius \\(R\\) is bounded by a circular rim in the \\(xy\\)-plane centered at the origin, with the curved dome extending into the region \\(z \\ge 0\\). A non-uniform magnetic field directed everywhere parallel to the \\(z\\)-axis is given in cylindrical coordinates by \\(\\vec{B} = B_0\\left(\\dfrac{r}{R}\\right)^2 \\hat{k}\\), where \\(r = \\sqrt{x^2 + y^2}\\) is the distance from the \\(z\\)-axis and \\(B_0\\) is a positive constant. Which of the following expressions is equal to the total magnetic flux \\(\\Phi_B\\) passing outward through the curved hemispherical surface?"
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url: "https://nerd-notes.com/ubq/121307/"
date_modified: "2026-08-23T04:59:39+00:00"
---

# An open hemispherical surface of radius \(R\) is bounded by a circular rim in the \(xy\)-plane centered at the origin, with the curved dome extending into the region \(z \ge 0\). A non-uniform magnetic field directed everywhere parallel to the \(z\)-axis is given in cylindrical coordinates by \(\vec{B} = B_0\left(\dfrac{r}{R}\right)^2 \hat{k}\), where \(r = \sqrt{x^2 + y^2}\) is the distance from the \(z\)-axis and \(B_0\) is a positive constant. Which of the following expressions is equal to the total magnetic flux \(\Phi_B\) passing outward through the curved hemispherical surface?

An open hemispherical surface of radius \(R\) is bounded by a circular rim in the \(xy\)-plane centered at the origin, with the curved dome extending into the region \(z \ge 0\). A non-uniform magnetic field directed everywhere parallel to the \(z\)-axis is given in cylindrical coordinates by \(\vec{B} = B_0\left(\dfrac{r}{R}\right)^2 \hat{k}\), where \(r = \sqrt{x^2 + y^2}\) is the distance from the \(z\)-axis and \(B_0\) is a positive constant. Which of the following expressions is equal to the total magnetic flux \(\Phi_B\) passing outward through the curved hemispherical surface?

![A three-dimensional coordinate system with a vertical z-axis labeled z, a horizontal x-axis extending to the right labeled x, and an axis extending forward-left labeled y. Centered at the origin in the xy-plane is a flat circular base of radius R drawn as an ellipse with a solid boundary. Rising above the xy-plane into the positive z region is a smooth hemispherical dome of radius R, rendered with a light gray shaded surface. Three upward vertical arrows of increasing length with distance from the z-axis indicate the magnetic field \vec{B}, each oriented parallel to the z-axis in the positive z-direction and labeled \vec{B}. A dashed line extends radially from the origin to the circular rim labeled R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461179-675Hr5.jpg)

- **A.** \(\dfrac{1}{4}\pi B_0 R^2\)
- **B.** \(\dfrac{1}{3}\pi B_0 R^2\)
- **C.** \(\dfrac{1}{2}\pi B_0 R^2\)
- **D.** \(\dfrac{2}{3}\pi B_0 R^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121307/*