---
title: "A circuit consists of a single closed loop containing an ideal battery of electromotive force \\(\\mathcal{E}_1\\), a rechargeable ideal battery of electromotive force \\(\\mathcal{E}_2\\) (where \\(\\mathcal{E}_1 > \\mathcal{E}_2\\)), and a resistor of resistance \\(R\\). The two batteries are oriented in opposition such that their positive terminals face each other along the top wire, resulting in a steady counterclockwise current \\(I\\) through the loop.  Which of the following statements correctly identifies the rate at which battery 1 converts chemical energy into electrical energy and provides a valid justification based on the conservation of energy?"
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url: "https://nerd-notes.com/ubq/121555/"
date_modified: "2026-08-23T05:24:00+00:00"
---

# A circuit consists of a single closed loop containing an ideal battery of electromotive force \(\mathcal{E}_1\), a rechargeable ideal battery of electromotive force \(\mathcal{E}_2\) (where \(\mathcal{E}_1 > \mathcal{E}_2\)), and a resistor of resistance \(R\). The two batteries are oriented in opposition such that their positive terminals face each other along the top wire, resulting in a steady counterclockwise current \(I\) through the loop.

Which of the following statements correctly identifies the rate at which battery 1 converts chemical energy into electrical energy and provides a valid justification based on the conservation of energy?

A circuit consists of a single closed loop containing an ideal battery of electromotive force \(\mathcal{E}_1\), a rechargeable ideal battery of electromotive force \(\mathcal{E}_2\) (where \(\mathcal{E}_1 > \mathcal{E}_2\)), and a resistor of resistance \(R\). The two batteries are oriented in opposition such that their positive terminals face each other along the top wire, resulting in a steady counterclockwise current \(I\) through the loop.

Which of the following statements correctly identifies the rate at which battery 1 converts chemical energy into electrical energy and provides a valid justification based on the conservation of energy?

![A rectangular schematic circuit diagram with four branches. The left vertical branch contains an ideal battery labeled \(\mathcal{E}_1\), with the long line (positive terminal) at the top and the short thick line (negative terminal) at the bottom. The right vertical branch contains an ideal battery labeled \(\mathcal{E}_2\), with the long line (positive terminal) at the top and the short thick line (negative terminal) at the bottom. The top horizontal branch contains a zig-zag resistor symbol labeled \(R\) at its center. The bottom branch is a straight solid wire. A curved arrow labeled \(I\) is drawn just inside the loop pointing in the counterclockwise direction. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462640-X3N7kt.jpg)

- **A.** The rate is equal to \((\mathcal{E}_1 - \mathcal{E}_2)I\), because only the net electromotive force of the loop supplies electrical power to the circuit components.
- **B.** The rate is equal to \((\mathcal{E}_1 + \mathcal{E}_2)I\), because both batteries contribute energy to maintain the current \(I\) through the total resistance of the loop.
- **C.** The rate is equal to \(\mathcal{E}_1 I\), because conservation of energy requires that the power delivered by battery 1 equals the rate at which battery 2 absorbs chemical energy (\(\mathcal{E}_2 I\)) plus the rate of thermal dissipation in the resistor (\(I^2 R\)).
- **D.** The rate is equal to \(I^2 R\), because all the electrical energy delivered by battery 1 is fully dissipated as thermal energy in the resistor.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121555/*