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title: "A circular region of radius \\(R\\) in the \\(xy\\)-plane contains a time-dependent, non-uniform magnetic field directed perpendicular to the plane. Inside the region (\\(r \\le R\\)), the magnitude of the magnetic field is given by \\(B(r,t) = C t \\left(\\dfrac{r}{R}\\right)^2\\), where \\(C\\) is a positive constant and \\(r\\) is the radial distance from the central axis. Outside the region (\\(r > R\\)), the magnetic field is zero. Which of the following expressions gives the magnitude of the induced electric field \\(E(r)\\) as a function of distance \\(r\\) from the central axis inside the region (\\(r < R\\))?"
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url: "https://nerd-notes.com/ubq/118756/"
date_modified: "2026-08-04T08:14:09+00:00"
---

# A circular region of radius \(R\) in the \(xy\)-plane contains a time-dependent, non-uniform magnetic field directed perpendicular to the plane. Inside the region (\(r \le R\)), the magnitude of the magnetic field is given by \(B(r,t) = C t \left(\dfrac{r}{R}\right)^2\), where \(C\) is a positive constant and \(r\) is the radial distance from the central axis. Outside the region (\(r > R\)), the magnetic field is zero. Which of the following expressions gives the magnitude of the induced electric field \(E(r)\) as a function of distance \(r\) from the central axis inside the region (\(r < R\))?

A circular region of radius \(R\) in the \(xy\)-plane contains a time-dependent, non-uniform magnetic field directed perpendicular to the plane. Inside the region (\(r \le R\)), the magnitude of the magnetic field is given by \(B(r,t) = C t \left(\dfrac{r}{R}\right)^2\), where \(C\) is a positive constant and \(r\) is the radial distance from the central axis. Outside the region (\(r > R\)), the magnetic field is zero. Which of the following expressions gives the magnitude of the induced electric field \(E(r)\) as a function of distance \(r\) from the central axis inside the region (\(r < R\))?

![A top-down view of a circular region of radius R centered at the origin of an xy-coordinate plane. Inside the circular boundary of radius R, small dots surrounded by circles indicate a magnetic field directed out of the page. A concentric dashed circular path of radius r is drawn inside the region, where r is less than R. At four evenly spaced points along the dashed path, four arrows labeled E of equal length point tangentially in the counterclockwise direction. A solid straight line arrow labeled R extends from the origin to the outer boundary at an angle of 45 degrees. A solid straight line arrow labeled r extends from the origin to the dashed circle at an angle of 135 degrees. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831249-NrfSGL.jpg)

- **A.** \(\dfrac{C r^3}{2 R^2}\)
- **B.** \(\dfrac{C r^3}{6 R^2}\)
- **C.** \(\dfrac{C r^3}{8 R^2}\)
- **D.** \(\dfrac{C r^3}{4 R^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118756/*