---
title: "An infinite two-dimensional square grid is constructed with identical resistors, each of resistance \\(R\\), connected at every junction node. A battery supplies a net current \\(I\\) that enters the grid at node \\(A\\) and exits at an adjacent node \\(B\\). Which of the following claims correctly gives the equivalent resistance \\(R_{eq}\\) between nodes \\(A\\) and \\(B\\), along with the correct physical justification?"
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url: "https://nerd-notes.com/ubq/118454/"
date_modified: "2026-08-04T08:10:13+00:00"
---

# An infinite two-dimensional square grid is constructed with identical resistors, each of resistance \(R\), connected at every junction node. A battery supplies a net current \(I\) that enters the grid at node \(A\) and exits at an adjacent node \(B\). Which of the following claims correctly gives the equivalent resistance \(R_{eq}\) between nodes \(A\) and \(B\), along with the correct physical justification?

An infinite two-dimensional square grid is constructed with identical resistors, each of resistance \(R\), connected at every junction node. A battery supplies a net current \(I\) that enters the grid at node \(A\) and exits at an adjacent node \(B\). Which of the following claims correctly gives the equivalent resistance \(R_{eq}\) between nodes \(A\) and \(B\), along with the correct physical justification?

![A schematic diagram showing a portion of an infinite two-dimensional square grid of identical resistors. Each grid segment represents a resistor labeled R. Two adjacent junction nodes near the center are prominently labeled A and B, connected directly by a single horizontal resistor R. An arrow labeled I points into node A from an external current source, and an arrow labeled I points out of node B to an external sink. Dashed lines extend outward from the outer boundaries of the drawn grid in all four directions to indicate that the grid extends infinitely. No other labels, text, or construction lines appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831013-DHGuOY.jpg)

- **A.** \(R_{eq} = \dfrac{R}{2}\), because injecting current \(I\) into node \(A\) sends \(\dfrac{I}{4}\) through the resistor between \(A\) and \(B\), and extracting current \(I\) from node \(B\) draws \(\dfrac{I}{4}\) through the same resistor, giving a net current of \(\dfrac{I}{2}\) through that resistor.
- **B.** \(R_{eq} = \dfrac{R}{4}\), because four identical paths branch symmetrically from node \(A\), so the total current \(I\) splits equally into four parallel paths, each having an equivalent resistance of \(R\).
- **C.** \(R_{eq} = \dfrac{R}{2}\), because the infinite network surrounding the edge between \(A\) and \(B\) acts as a single equivalent resistor of resistance \(R\) connected in parallel with the resistor directly between \(A\) and \(B\).
- **D.** \(R_{eq} = R\), because the current takes the path of least resistance through the single resistor connecting nodes \(A\) and \(B\), while all surrounding indirect paths through the infinite grid present infinite total resistance.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118454/*