---
title: "A circuit consists of two loops that share a central branch. The left branch contains an ideal battery of emf \\(\\mathcal{E}\\) and a resistor of resistance \\(R\\). The right branch contains an identical ideal battery of emf \\(\\mathcal{E}\\) and a resistor of resistance \\(R\\), with both batteries oriented to drive current toward the upper junction. The central branch contains a single resistor of resistance \\(R\\). What is the magnitude of the potential difference across the central resistor?"
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url: "https://nerd-notes.com/ubq/118383/"
date_modified: "2026-08-04T08:09:48+00:00"
---

# A circuit consists of two loops that share a central branch. The left branch contains an ideal battery of emf \(\mathcal{E}\) and a resistor of resistance \(R\). The right branch contains an identical ideal battery of emf \(\mathcal{E}\) and a resistor of resistance \(R\), with both batteries oriented to drive current toward the upper junction. The central branch contains a single resistor of resistance \(R\). What is the magnitude of the potential difference across the central resistor?

A circuit consists of two loops that share a central branch. The left branch contains an ideal battery of emf \(\mathcal{E}\) and a resistor of resistance \(R\). The right branch contains an identical ideal battery of emf \(\mathcal{E}\) and a resistor of resistance \(R\), with both batteries oriented to drive current toward the upper junction. The central branch contains a single resistor of resistance \(R\). What is the magnitude of the potential difference across the central resistor?

![A circuit schematic with two rectangular loops sharing a central vertical branch. The outer frame forms a wide rectangle. The left vertical branch contains an ideal battery labeled \(\mathcal{E}\) with its longer positive terminal plate on top, and a resistor labeled \(R\) in series below it. The central vertical branch contains a single resistor labeled \(R\). The right vertical branch contains an identical ideal battery labeled \(\mathcal{E}\) with its longer positive terminal plate on top, and a resistor labeled \(R\) in series below it. Horizontal wire lines connect the top endpoints of all three vertical branches together at a top junction node, and horizontal wire lines connect the bottom endpoints together at a bottom junction node. No other labels, lines, text, or components appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830988-uGkv5Q.jpg)

- **A.** \(\dfrac{1}{3}\mathcal{E}\)
- **B.** \(\dfrac{2}{3}\mathcal{E}\)
- **C.** \(\mathcal{E}\)
- **D.** \(\dfrac{4}{3}\mathcal{E}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118383/*