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title: "A large circular wire loop of radius \\(R\\) lies flat on a horizontal table and carries a time-dependent current \\(I(t) = ct\\) in the counterclockwise direction, where \\(c\\) is a positive constant. A small circular conducting loop of radius \\(r\\) (\\(r \\ll R\\)) is placed in the same plane, concentric with the large loop. As viewed from above, which of the following correctly identifies the direction of the induced current in the small loop and the proportionality of the induced electromotive force \\(\\mathcal{E}\\) in terms of the radii \\(r\\) and \\(R\\)?"
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url: "https://nerd-notes.com/ubq/124816/"
date_modified: "2026-09-28T14:11:30+00:00"
---

# A large circular wire loop of radius \(R\) lies flat on a horizontal table and carries a time-dependent current \(I(t) = ct\) in the counterclockwise direction, where \(c\) is a positive constant. A small circular conducting loop of radius \(r\) (\(r \ll R\)) is placed in the same plane, concentric with the large loop. As viewed from above, which of the following correctly identifies the direction of the induced current in the small loop and the proportionality of the induced electromotive force \(\mathcal{E}\) in terms of the radii \(r\) and \(R\)?

A large circular wire loop of radius \(R\) lies flat on a horizontal table and carries a time-dependent current \(I(t) = ct\) in the counterclockwise direction, where \(c\) is a positive constant. A small circular conducting loop of radius \(r\) (\(r \ll R\)) is placed in the same plane, concentric with the large loop. As viewed from above, which of the following correctly identifies the direction of the induced current in the small loop and the proportionality of the induced electromotive force \(\mathcal{E}\) in terms of the radii \(r\) and \(R\)?

![A top-down view showing two concentric circles drawn with solid black lines on a white background. The outer circle has radius \(R\), and the inner circle is much smaller with radius \(r\). From the common center of the circles, a straight solid arrow extends toward the upper-right at a 45-degree angle to the perimeter of the inner circle, labeled \(r\). A second straight solid arrow extends from the common center toward the upper-left at a 135-degree angle to the perimeter of the outer circle, labeled \(R\). On the perimeter of the outer circle, exactly two curved arrows point counterclockwise, indicating the direction of current labeled \(I(t)\). The interior of both circles is unshaded white. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604690-UKvHVi.jpg)

- **A.** Direction: Counterclockwise ; Proportionality: \(\mathcal{E} \propto \dfrac{r^2}{R^2}\)
- **B.** Direction: Clockwise ; Proportionality: \(\mathcal{E} \propto \dfrac{r^2}{R}\)
- **C.** Direction: Counterclockwise ; Proportionality: \(\mathcal{E} \propto \dfrac{r^2}{R}\)
- **D.** Direction: Clockwise ; Proportionality: \(\mathcal{E} \propto \dfrac{r^2}{R^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124816/*