---
title: "A real battery with electromotive force \\(\\mathcal{E}\\) and internal resistance \\(r\\) has terminal voltage \\(V_t\\). The battery is connected to an external network where a fixed resistor of resistance \\(R\\) is in series with the positive terminal, followed by a junction that splits into two parallel branches: one branch containing a fixed shunt resistor of resistance \\(R\\) returning to the negative terminal, and the other branch containing a fixed resistor of resistance \\(R\\) in series with an adjustable load resistor \\(R_L\\) that also returns to the negative terminal. Which of the following expressions correctly gives the battery’s terminal voltage \\(V_t\\) in the limit \\(R_L \\to 0\\) and in the limit \\(R_L \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/124912/"
date_modified: "2026-09-28T14:11:55+00:00"
---

# A real battery with electromotive force \(\mathcal{E}\) and internal resistance \(r\) has terminal voltage \(V_t\). The battery is connected to an external network where a fixed resistor of resistance \(R\) is in series with the positive terminal, followed by a junction that splits into two parallel branches: one branch containing a fixed shunt resistor of resistance \(R\) returning to the negative terminal, and the other branch containing a fixed resistor of resistance \(R\) in series with an adjustable load resistor \(R_L\) that also returns to the negative terminal. Which of the following expressions correctly gives the battery’s terminal voltage \(V_t\) in the limit \(R_L \to 0\) and in the limit \(R_L \to \infty\)?

A real battery with electromotive force \(\mathcal{E}\) and internal resistance \(r\) has terminal voltage \(V_t\). The battery is connected to an external network where a fixed resistor of resistance \(R\) is in series with the positive terminal, followed by a junction that splits into two parallel branches: one branch containing a fixed shunt resistor of resistance \(R\) returning to the negative terminal, and the other branch containing a fixed resistor of resistance \(R\) in series with an adjustable load resistor \(R_L\) that also returns to the negative terminal. Which of the following expressions correctly gives the battery's terminal voltage \(V_t\) in the limit \(R_L \to 0\) and in the limit \(R_L \to \infty\)?

![Schematic of a rectangular circuit loop. On the left vertical segment is a battery enclosed in a dashed box, containing a DC source symbol labeled \(\mathcal{E}\) with the longer line on top, in series with a resistor symbol labeled \(r\) below it. Small terminal dots labeled A and B sit on the top and bottom wires just outside the dashed box. From terminal A, the top wire extends rightward through a resistor symbol labeled R. Past this resistor, a junction node connects to a vertical branch containing a resistor symbol labeled R that connects to the bottom return wire. From the same junction node, the top wire continues rightward through another resistor symbol labeled R to an adjustable load resistor symbol labeled \(R_L\) on the right vertical branch, which returns to the bottom wire. The bottom wire runs straight back to terminal B. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604715-BTV2Zz.jpg)

- **A.** In the limit \(R_L \to 0\), \(V_t = 0\); in the limit \(R_L \to \infty\), \(V_t = \mathcal{E}\)
- **B.** In the limit \(R_L \to 0\), \(V_t = \dfrac{R}{r + R}\mathcal{E}\); in the limit \(R_L \to \infty\), \(V_t = \dfrac{2R}{r + 2R}\mathcal{E}\)
- **C.** In the limit \(R_L \to 0\), \(V_t = \dfrac{3R}{2r + 3R}\mathcal{E}\); in the limit \(R_L \to \infty\), \(V_t = \dfrac{2R}{r + 2R}\mathcal{E}\)
- **D.** In the limit \(R_L \to 0\), \(V_t = \dfrac{3R}{2r + 3R}\mathcal{E}\); in the limit \(R_L \to \infty\), \(V_t = \mathcal{E}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124912/*