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title: "A circuit consists of three vertical branches connected in parallel between an upper node labeled \\(X\\) and a lower grounded rail labeled \\(Z\\) (where \\(V_Z = 0\\text{ V}\\)). The left branch contains a real battery with EMF \\(\\mathcal{E}_1 = 36\\text{ V}\\) and internal resistance \\(r_1 = 1\\text{ }\\Omega\\) (positive terminal facing upward) in series with an external resistor \\(R_1 = 3\\text{ }\\Omega\\), with node \\(W\\) located at the positive terminal between \\(r_1\\) and \\(R_1\\). The middle branch contains a single resistor \\(R_2 = 4\\text{ }\\Omega\\). The right branch contains a real battery with EMF \\(\\mathcal{E}_2 = 6\\text{ V}\\) and internal resistance \\(r_2 = 1\\text{ }\\Omega\\) (positive terminal facing upward) in series with an external resistor \\(R_3 = 1\\text{ }\\Omega\\), with node \\(Y\\) located at the positive terminal between \\(r_2\\) and \\(R_3\\). Which of the following correctly ranks the electric potentials \\(V_W\\), \\(V_X\\), \\(V_Y\\), and \\(V_Z\\) at the labeled nodes?"
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url: "https://nerd-notes.com/ubq/124886/"
date_modified: "2026-09-28T14:11:49+00:00"
---

# A circuit consists of three vertical branches connected in parallel between an upper node labeled \(X\) and a lower grounded rail labeled \(Z\) (where \(V_Z = 0\text{ V}\)). The left branch contains a real battery with EMF \(\mathcal{E}_1 = 36\text{ V}\) and internal resistance \(r_1 = 1\text{ }\Omega\) (positive terminal facing upward) in series with an external resistor \(R_1 = 3\text{ }\Omega\), with node \(W\) located at the positive terminal between \(r_1\) and \(R_1\). The middle branch contains a single resistor \(R_2 = 4\text{ }\Omega\). The right branch contains a real battery with EMF \(\mathcal{E}_2 = 6\text{ V}\) and internal resistance \(r_2 = 1\text{ }\Omega\) (positive terminal facing upward) in series with an external resistor \(R_3 = 1\text{ }\Omega\), with node \(Y\) located at the positive terminal between \(r_2\) and \(R_3\). Which of the following correctly ranks the electric potentials \(V_W\), \(V_X\), \(V_Y\), and \(V_Z\) at the labeled nodes?

A circuit consists of three vertical branches connected in parallel between an upper node labeled \(X\) and a lower grounded rail labeled \(Z\) (where \(V_Z = 0\text{ V}\)). The left branch contains a real battery with EMF \(\mathcal{E}_1 = 36\text{ V}\) and internal resistance \(r_1 = 1\text{ }\Omega\) (positive terminal facing upward) in series with an external resistor \(R_1 = 3\text{ }\Omega\), with node \(W\) located at the positive terminal between \(r_1\) and \(R_1\). The middle branch contains a single resistor \(R_2 = 4\text{ }\Omega\). The right branch contains a real battery with EMF \(\mathcal{E}_2 = 6\text{ V}\) and internal resistance \(r_2 = 1\text{ }\Omega\) (positive terminal facing upward) in series with an external resistor \(R_3 = 1\text{ }\Omega\), with node \(Y\) located at the positive terminal between \(r_2\) and \(R_3\). Which of the following correctly ranks the electric potentials \(V_W\), \(V_X\), \(V_Y\), and \(V_Z\) at the labeled nodes?

![A circuit schematic with three parallel vertical branches connected between a horizontal top wire and a horizontal bottom wire. The bottom wire has a ground symbol at its center and is labeled with a solid dot as node Z. The top wire has a solid dot at the center junction labeled as node X. The left vertical branch contains, from bottom to top, a battery symbol labeled \(\mathcal{E}_1\) with a longer upper line, a resistor zigzag labeled \(r_1\), a solid dot labeled as node W, and a resistor zigzag labeled \(R_1\) meeting the top wire. The middle vertical branch contains a single resistor zigzag labeled \(R_2\) connecting the top wire at node X to the bottom wire at node Z. The right vertical branch contains, from bottom to top, a battery symbol labeled \(\mathcal{E}_2\) with a longer upper line, a resistor zigzag labeled \(r_2\), a solid dot labeled as node Y, and a resistor zigzag labeled \(R_3\) meeting the top wire. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604709-4SbCew.jpg)

- **A.** \(V_W > V_X > V_Y > V_Z\)
- **B.** \(V_W > V_Y > V_X > V_Z\)
- **C.** \(V_X > V_W > V_Y > V_Z\)
- **D.** \(V_W > V_X > V_Z > V_Y\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124886/*