---
title: "Two separate circuits, Circuit 1 and Circuit 2, each consist of a conducting bar of length \\(\\ell\\) sliding horizontally to the right at a constant speed \\(v\\) along a pair of frictionless, parallel conducting rails separated by distance \\(\\ell\\). Both circuits are located in identical, uniform magnetic fields \\(\\vec{B}\\) directed perpendicular to the plane of the rails, and the rails have negligible resistance. Circuit 1 contains a load resistor of resistance \\(R\\), while Circuit 2 contains a load resistor of resistance \\(2R\\). Which row in the table correctly compares the induced electromotive force \\(\\mathcal{E}_2\\), the magnitude of the magnetic braking force \\(F_{B,2}\\), and the rate of electrical energy dissipation \\(P_2\\) in Circuit 2 to the corresponding quantities in Circuit 1?  | Row | Induced EMF \\(\\mathcal{E}_2\\) | Magnetic Force \\(F_{B,2}\\) | Dissipated Power \\(P_2\\) | | :— | :— | :— | :— | | A | \\(\\mathcal{E}_1\\) | \\(\\dfrac{1}{2}F_{B,1}\\) | \\(\\dfrac{1}{2}P_1\\) | | B | \\(\\dfrac{1}{2}\\mathcal{E}_1\\) | \\(\\dfrac{1}{2}F_{B,1}\\) | \\(\\dfrac{1}{4}P_1\\) | | C | \\(\\mathcal{E}_1\\) | \\(2F_{B,1}\\) | \\(2P_1\\) | | D | \\(\\mathcal{E}_1\\) | \\(\\dfrac{1}{2}F_{B,1}\\) | \\(\\dfrac{1}{4}P_1\\) |"
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url: "https://nerd-notes.com/ubq/121343/"
date_modified: "2026-08-23T04:59:49+00:00"
---

# Two separate circuits, Circuit 1 and Circuit 2, each consist of a conducting bar of length \(\ell\) sliding horizontally to the right at a constant speed \(v\) along a pair of frictionless, parallel conducting rails separated by distance \(\ell\). Both circuits are located in identical, uniform magnetic fields \(\vec{B}\) directed perpendicular to the plane of the rails, and the rails have negligible resistance. Circuit 1 contains a load resistor of resistance \(R\), while Circuit 2 contains a load resistor of resistance \(2R\). Which row in the table correctly compares the induced electromotive force \(\mathcal{E}_2\), the magnitude of the magnetic braking force \(F_{B,2}\), and the rate of electrical energy dissipation \(P_2\) in Circuit 2 to the corresponding quantities in Circuit 1?

| Row | Induced EMF \(\mathcal{E}_2\) | Magnetic Force \(F_{B,2}\) | Dissipated Power \(P_2\) |
| :— | :— | :— | :— |
| A | \(\mathcal{E}_1\) | \(\dfrac{1}{2}F_{B,1}\) | \(\dfrac{1}{2}P_1\) |
| B | \(\dfrac{1}{2}\mathcal{E}_1\) | \(\dfrac{1}{2}F_{B,1}\) | \(\dfrac{1}{4}P_1\) |
| C | \(\mathcal{E}_1\) | \(2F_{B,1}\) | \(2P_1\) |
| D | \(\mathcal{E}_1\) | \(\dfrac{1}{2}F_{B,1}\) | \(\dfrac{1}{4}P_1\) |

Two separate circuits, Circuit 1 and Circuit 2, each consist of a conducting bar of length \(\ell\) sliding horizontally to the right at a constant speed \(v\) along a pair of frictionless, parallel conducting rails separated by distance \(\ell\). Both circuits are located in identical, uniform magnetic fields \(\vec{B}\) directed perpendicular to the plane of the rails, and the rails have negligible resistance. Circuit 1 contains a load resistor of resistance \(R\), while Circuit 2 contains a load resistor of resistance \(2R\). Which row in the table correctly compares the induced electromotive force \(\mathcal{E}_2\), the magnitude of the magnetic braking force \(F_{B,2}\), and the rate of electrical energy dissipation \(P_2\) in Circuit 2 to the corresponding quantities in Circuit 1?

| Row | Induced EMF \(\mathcal{E}_2\) | Magnetic Force \(F_{B,2}\) | Dissipated Power \(P_2\) |
| :--- | :--- | :--- | :--- |
| A | \(\mathcal{E}_1\) | \(\dfrac{1}{2}F_{B,1}\) | \(\dfrac{1}{2}P_1\) |
| B | \(\dfrac{1}{2}\mathcal{E}_1\) | \(\dfrac{1}{2}F_{B,1}\) | \(\dfrac{1}{4}P_1\) |
| C | \(\mathcal{E}_1\) | \(2F_{B,1}\) | \(2P_1\) |
| D | \(\mathcal{E}_1\) | \(\dfrac{1}{2}F_{B,1}\) | \(\dfrac{1}{4}P_1\) |

![A schematic showing two horizontal parallel lines representing conducting rails. On the far left, a vertical zigzag resistor symbol labeled \(R\) connects the top and bottom rails. A vertical solid line segment representing a conducting crossbar spans between the rails at the center of the diagram. A horizontal arrow pointing to the right originates from the midpoint of the crossbar and is labeled \(v\). A vertical double-headed arrow between the two rails is labeled \(\ell\). An array of six \(\times\) symbols arranged in two horizontal rows of three is distributed across the region enclosed by the rails, and one label \(\vec{B}\) is positioned near the top right of the array. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461189-dpKHQP.jpg)

- **A.** \(\mathcal{E}_2 = \mathcal{E}_1\), \(F_{B,2} = \dfrac{1}{2}F_{B,1}\), and \(P_2 = \dfrac{1}{2}P_1\)
- **B.** \(\mathcal{E}_2 = \dfrac{1}{2}\mathcal{E}_1\), \(F_{B,2} = \dfrac{1}{2}F_{B,1}\), and \(P_2 = \dfrac{1}{4}P_1\)
- **C.** \(\mathcal{E}_2 = \mathcal{E}_1\), \(F_{B,2} = 2F_{B,1}\), and \(P_2 = 2P_1\)
- **D.** \(\mathcal{E}_2 = \mathcal{E}_1\), \(F_{B,2} = \dfrac{1}{2}F_{B,1}\), and \(P_2 = \dfrac{1}{4}P_1\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121343/*