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title: "An infinite ladder network of capacitors is constructed using identical capacitors, each of capacitance \\(C\\). As shown in the diagram, each stage consists of one capacitor in series along the top line, followed by one capacitor connected in parallel across the output line. Let \\(C_N\\) denote the equivalent capacitance across terminals \\(A\\) and \\(B\\) for a network truncated after \\(N\\) stages, and let \\(C_{\\text{eq}}\\) denote the equivalent capacitance of the infinite network as \\(N \\to \\infty\\). Which choice correctly identifies the expression for \\(C_{\\text{eq}}\\) and describes how \\(C_N\\) changes as the number of stages \\(N\\) increases?"
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url: "https://nerd-notes.com/ubq/118320/"
date_modified: "2026-08-04T08:09:00+00:00"
---

# An infinite ladder network of capacitors is constructed using identical capacitors, each of capacitance \(C\). As shown in the diagram, each stage consists of one capacitor in series along the top line, followed by one capacitor connected in parallel across the output line. Let \(C_N\) denote the equivalent capacitance across terminals \(A\) and \(B\) for a network truncated after \(N\) stages, and let \(C_{\text{eq}}\) denote the equivalent capacitance of the infinite network as \(N \to \infty\). Which choice correctly identifies the expression for \(C_{\text{eq}}\) and describes how \(C_N\) changes as the number of stages \(N\) increases?

An infinite ladder network of capacitors is constructed using identical capacitors, each of capacitance \(C\). As shown in the diagram, each stage consists of one capacitor in series along the top line, followed by one capacitor connected in parallel across the output line. Let \(C_N\) denote the equivalent capacitance across terminals \(A\) and \(B\) for a network truncated after \(N\) stages, and let \(C_{\text{eq}}\) denote the equivalent capacitance of the infinite network as \(N \to \infty\). Which choice correctly identifies the expression for \(C_{\text{eq}}\) and describes how \(C_N\) changes as the number of stages \(N\) increases?

![A circuit diagram showing an infinite ladder network of capacitors connected to input terminals A and B on the left. Terminal A is on the top horizontal wire and Terminal B is on the bottom horizontal wire. Stage 1 consists of a capacitor labeled C on the top horizontal wire, followed by a vertical branch containing a capacitor labeled C connecting the top and bottom wires. Stage 2 repeats this exact configuration with a top horizontal capacitor labeled C and a vertical capacitor labeled C. Stage 3 repeats the same configuration. To the right of Stage 3, three horizontal dashed dots indicate the network continues infinitely to the right. No other labels, lines, or components appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830939-rz1LTD.jpg)

- **A.** \(C_{\text{eq}} = \dfrac{\sqrt{5}+1}{2} C\) ; \(C_N\) monotonically decreases toward \(C_{\text{eq}}\)
- **B.** \(C_{\text{eq}} = \dfrac{\sqrt{5}-1}{2} C\) ; \(C_N\) monotonically increases toward \(C_{\text{eq}}\)
- **C.** \(C_{\text{eq}} = \dfrac{\sqrt{5}-1}{2} C\) ; \(C_N\) monotonically decreases toward \(C_{\text{eq}}\)
- **D.** \(C_{\text{eq}} = \dfrac{\sqrt{3}-1}{2} C\) ; \(C_N\) monotonically increases toward \(C_{\text{eq}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118320/*