---
title: "A circuit consists of two loops connected between two nodes, \\(a\\) and \\(b\\). The left branch contains an ideal battery of EMF \\(\\mathcal{E}_1\\) in series with a resistor of resistance \\(R\\), with its positive terminal directed toward node \\(a\\). The central branch contains a single resistor of resistance \\(R\\). The right branch contains an ideal battery of EMF \\(\\mathcal{E}_2\\) in series with a resistor of resistance \\(2R\\), with its positive terminal also directed toward node \\(a\\). In terms of \\(\\mathcal{E}_1\\) and \\(\\mathcal{E}_2\\), what is the electric potential difference \\(V_a – V_b\\) across the central branch?"
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url: "https://nerd-notes.com/ubq/121571/"
date_modified: "2026-08-23T05:24:03+00:00"
---

# A circuit consists of two loops connected between two nodes, \(a\) and \(b\). The left branch contains an ideal battery of EMF \(\mathcal{E}_1\) in series with a resistor of resistance \(R\), with its positive terminal directed toward node \(a\). The central branch contains a single resistor of resistance \(R\). The right branch contains an ideal battery of EMF \(\mathcal{E}_2\) in series with a resistor of resistance \(2R\), with its positive terminal also directed toward node \(a\). In terms of \(\mathcal{E}_1\) and \(\mathcal{E}_2\), what is the electric potential difference \(V_a – V_b\) across the central branch?

A circuit consists of two loops connected between two nodes, \(a\) and \(b\). The left branch contains an ideal battery of EMF \(\mathcal{E}_1\) in series with a resistor of resistance \(R\), with its positive terminal directed toward node \(a\). The central branch contains a single resistor of resistance \(R\). The right branch contains an ideal battery of EMF \(\mathcal{E}_2\) in series with a resistor of resistance \(2R\), with its positive terminal also directed toward node \(a\). In terms of \(\mathcal{E}_1\) and \(\mathcal{E}_2\), what is the electric potential difference \(V_a - V_b\) across the central branch?

![A circuit schematic consisting of two adjacent rectangular loops sharing a single central vertical branch. The top horizontal wire connects a solid circular dot labeled a to the top of the left vertical branch and the top of the right vertical branch. The bottom horizontal wire connects a solid circular dot labeled b to the bottom of the left vertical branch and the bottom of the right vertical branch. The left vertical branch contains an ideal battery labeled \(\mathcal{E}_1\) with its longer positive plate at the top and shorter negative plate below it, in series with a resistor labeled R. The central vertical branch contains a single resistor labeled R between node a and node b. The right vertical branch contains an ideal battery labeled \(\mathcal{E}_2\) with its longer positive plate at the top and shorter negative plate below it, in series with a resistor labeled 2R. No other labels, lines, text, or components appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/circuit-diagram-1-1787462643-cejkwZ.jpg)

- **A.** \(\dfrac{\mathcal{E}_1 + \mathcal{E}_2}{3}\)
- **B.** \(\dfrac{2\mathcal{E}_1 + \mathcal{E}_2}{5}\)
- **C.** \(\dfrac{\mathcal{E}_1 + 2\mathcal{E}_2}{5}\)
- **D.** \(\dfrac{2\mathcal{E}_1 - \mathcal{E}_2}{5}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121571/*