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title: "A planar direct current circuit consists of four parallel vertical branches connected across common top and bottom horizontal rails of negligible resistance, forming three adjacent meshes. The branch currents are defined as \\(I_1\\) (directed upward in the far-left branch), \\(I_2\\) (directed downward in the second branch), \\(I_3\\) (directed downward in the third branch), and \\(I_4\\) (directed downward in the far-right branch). Both batteries \\(\\mathcal{E}_1\\) and \\(\\mathcal{E}_2\\) have their positive terminals oriented toward the top rail. Which of the following correctly pairs the loop equation for the clockwise traversal of the outer perimeter loop with the maximum number of linearly independent loop equations needed to fully determine all branch currents?"
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url: "https://nerd-notes.com/ubq/121609/"
date_modified: "2026-08-23T05:24:23+00:00"
---

# A planar direct current circuit consists of four parallel vertical branches connected across common top and bottom horizontal rails of negligible resistance, forming three adjacent meshes. The branch currents are defined as \(I_1\) (directed upward in the far-left branch), \(I_2\) (directed downward in the second branch), \(I_3\) (directed downward in the third branch), and \(I_4\) (directed downward in the far-right branch). Both batteries \(\mathcal{E}_1\) and \(\mathcal{E}_2\) have their positive terminals oriented toward the top rail. Which of the following correctly pairs the loop equation for the clockwise traversal of the outer perimeter loop with the maximum number of linearly independent loop equations needed to fully determine all branch currents?

A planar direct current circuit consists of four parallel vertical branches connected across common top and bottom horizontal rails of negligible resistance, forming three adjacent meshes. The branch currents are defined as \(I_1\) (directed upward in the far-left branch), \(I_2\) (directed downward in the second branch), \(I_3\) (directed downward in the third branch), and \(I_4\) (directed downward in the far-right branch). Both batteries \(\mathcal{E}_1\) and \(\mathcal{E}_2\) have their positive terminals oriented toward the top rail. Which of the following correctly pairs the loop equation for the clockwise traversal of the outer perimeter loop with the maximum number of linearly independent loop equations needed to fully determine all branch currents?

![A planar circuit schematic drawn in clean black lines on a white background with two horizontal rails and four vertical branches forming three adjacent rectangular meshes. The top rail connects the tops of all four branches; the bottom rail connects the bottoms. The far-left branch contains an ideal battery labeled \(\mathcal{E}_1\) with its longer positive line on top and shorter negative line on bottom, in series with a resistor labeled \(R_1\); an arrow labeled \(I_1\) points upward along this branch. The second branch contains a resistor labeled \(R_2\) with an arrow labeled \(I_2\) pointing downward. The third branch contains a resistor labeled \(R_3\) with an arrow labeled \(I_3\) pointing downward. The far-right branch contains a resistor labeled \(R_4\) above an ideal battery labeled \(\mathcal{E}_2\) with its longer positive line on top and shorter negative line on bottom; an arrow labeled \(I_4\) pointing downward along this branch. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462663-mhi063.jpg)

- **A.** Outer loop equation: \(\mathcal{E}_1 - I_1 R_1 - I_4 R_4 - \mathcal{E}_2 = 0\) Number of independent loop equations: \(3\)
- **B.** Outer loop equation: \(\mathcal{E}_1 - I_1 R_1 - I_4 R_4 - \mathcal{E}_2 = 0\) Number of independent loop equations: \(4\)
- **C.** Outer loop equation: \(\mathcal{E}_1 - I_1 R_1 - I_2 R_2 - I_3 R_3 - I_4 R_4 - \mathcal{E}_2 = 0\) Number of independent loop equations: \(3\)
- **D.** Outer loop equation: \(\mathcal{E}_1 - I_1 R_1 + I_4 R_4 + \mathcal{E}_2 = 0\) Number of independent loop equations: \(4\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121609/*