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title: "A cylindrical region of radius \\(R\\) contains a spatially uniform magnetic field directed out of the page with magnitude changing at a constant rate \\(\\dfrac{dB}{dt} = \\beta > 0\\). Three closed paths, each enclosing the same area \\(A\\), lie entirely within the region: Path 1 is a circle centered on the magnetic field axis, Path 2 is a circle centered a distance \\(d\\) (where \\(0 < d < R/2\\)) from the axis, and Path 3 is an equilateral triangle. If \\(\\mathcal{E}_1\\), \\(\\mathcal{E}_2\\), and \\(\\mathcal{E}_3\\) are the magnitudes of the induced electromotive force around paths 1, 2, and 3, and \\(E_{\\text{max},1}\\) and \\(E_{\\text{max},2}\\) are the maximum magnitudes of the induced electric field along paths 1 and 2, which of the following correctly compares these quantities?"
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url: "https://nerd-notes.com/ubq/121345/"
date_modified: "2026-08-23T04:59:52+00:00"
---

# A cylindrical region of radius \(R\) contains a spatially uniform magnetic field directed out of the page with magnitude changing at a constant rate \(\dfrac{dB}{dt} = \beta > 0\). Three closed paths, each enclosing the same area \(A\), lie entirely within the region: Path 1 is a circle centered on the magnetic field axis, Path 2 is a circle centered a distance \(d\) (where \(0 < d < R/2\)) from the axis, and Path 3 is an equilateral triangle. If \(\mathcal{E}_1\), \(\mathcal{E}_2\), and \(\mathcal{E}_3\) are the magnitudes of the induced electromotive force around paths 1, 2, and 3, and \(E_{\text{max},1}\) and \(E_{\text{max},2}\) are the maximum magnitudes of the induced electric field along paths 1 and 2, which of the following correctly compares these quantities?

A cylindrical region of radius \(R\) contains a spatially uniform magnetic field directed out of the page with magnitude changing at a constant rate \(\dfrac{dB}{dt} = \beta > 0\). Three closed paths, each enclosing the same area \(A\), lie entirely within the region: Path 1 is a circle centered on the magnetic field axis, Path 2 is a circle centered a distance \(d\) (where \(0 < d < R/2\)) from the axis, and Path 3 is an equilateral triangle. If \(\mathcal{E}_1\), \(\mathcal{E}_2\), and \(\mathcal{E}_3\) are the magnitudes of the induced electromotive force around paths 1, 2, and 3, and \(E_{\text{max},1}\) and \(E_{\text{max},2}\) are the maximum magnitudes of the induced electric field along paths 1 and 2, which of the following correctly compares these quantities?

![A large outer circle of radius \(R\) is centered at origin point \(O\). Small dots with encircling rings are uniformly distributed within the large circle to represent a magnetic field \(\vec{B}\) directed perpendicular to the page. Path 1 is shown as a solid circle of radius \(r_0\) centered directly at \(O\). Path 2 is a solid circle of identical radius \(r_0\) centered at point \(C_2\), which is shifted horizontally to the right of \(O\) by a dashed horizontal line segment labeled \(d\). Path 3 is a solid equilateral triangle located entirely within the bottom-left quadrant of the large outer circle. A dashed radius line extends from \(O\) to the outer boundary labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461192-f22gj4.jpg)

- **A.** \(\mathcal{E}_1 > \mathcal{E}_2 = \mathcal{E}_3\) and \(E_{\text{max},1} = E_{\text{max},2}\)
- **B.** \(\mathcal{E}_1 = \mathcal{E}_2 = \mathcal{E}_3\) and \(E_{\text{max},1} < E_{\text{max},2}\)
- **C.** \(\mathcal{E}_1 = \mathcal{E}_2 = \mathcal{E}_3\) and \(E_{\text{max},1} = E_{\text{max},2}\)
- **D.** \(\mathcal{E}_1 = \mathcal{E}_2 > \mathcal{E}_3\) and \(E_{\text{max},1} < E_{\text{max},2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121345/*