---
title: "A circuit contains two ideal batteries with emfs \\(\\varepsilon_1\\) and \\(\\varepsilon_2\\), and three resistors with resistances \\(R_1\\), \\(R_2\\), and \\(R_3\\), arranged in a two-mesh network as shown. Both batteries are oriented with their positive terminals connected toward the upper conductor. In terms of the given quantities, which of the following expressions is equal to the magnitude of the downward current \\(I_2\\) through resistor \\(R_2\\)?"
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url: "https://nerd-notes.com/ubq/120398/"
date_modified: "2026-08-23T04:34:27+00:00"
---

# A circuit contains two ideal batteries with emfs \(\varepsilon_1\) and \(\varepsilon_2\), and three resistors with resistances \(R_1\), \(R_2\), and \(R_3\), arranged in a two-mesh network as shown. Both batteries are oriented with their positive terminals connected toward the upper conductor. In terms of the given quantities, which of the following expressions is equal to the magnitude of the downward current \(I_2\) through resistor \(R_2\)?

A circuit contains two ideal batteries with emfs \(\varepsilon_1\) and \(\varepsilon_2\), and three resistors with resistances \(R_1\), \(R_2\), and \(R_3\), arranged in a two-mesh network as shown. Both batteries are oriented with their positive terminals connected toward the upper conductor. In terms of the given quantities, which of the following expressions is equal to the magnitude of the downward current \(I_2\) through resistor \(R_2\)?

![A rectangular circuit diagram with two adjacent rectangular meshes sharing a central vertical wire. The outer perimeter and central divider form three vertical parallel branches connected by continuous top and bottom horizontal wires. The left vertical branch contains a battery with emf labeled \(\varepsilon_1\) (represented by two parallel horizontal plates with the longer plate on top) above a resistor labeled \(R_1\). The central vertical branch contains a single resistor labeled \(R_2\). The right vertical branch contains a battery with emf labeled \(\varepsilon_2\) (longer plate on top) above a resistor labeled \(R_3\). Solid dots indicate the top and bottom junction nodes. No other labels, lines, text, or arrows appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459667-vO9Er8.jpg)

- **A.** \(\dfrac{\varepsilon_1 R_3 + \varepsilon_2 R_1}{R_1 R_2 + R_2 R_3 + R_3 R_1}\)
- **B.** \(\dfrac{\varepsilon_1 R_3 - \varepsilon_2 R_1}{R_1 R_2 + R_2 R_3 + R_3 R_1}\)
- **C.** \(\dfrac{\varepsilon_1 R_1 + \varepsilon_2 R_3}{R_1 R_2 + R_2 R_3 + R_3 R_1}\)
- **D.** \(\dfrac{\varepsilon_1 R_3 + \varepsilon_2 R_1}{R_2(R_1 + R_3)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120398/*