---
title: "In the circuit shown, three parallel branches connect a central node P to a common lower conductor held at ground potential (\\(V = 0\\text{ V}\\)).  – The left branch contains an ideal battery of EMF \\(\\mathcal{E}_1\\) in series with a resistor of resistance \\(R_1\\), with the positive terminal connected toward node P. – The middle branch contains an ideal battery of EMF \\(\\mathcal{E}_2\\) in series with a resistor of resistance \\(R_2\\), with the negative terminal connected toward node P. – The right branch contains an ideal current source that delivers a constant current \\(I_0\\) into node P.  Which of the following is a correct expression for the electric potential \\(V_P\\) at node P in terms of the given quantities?"
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url: "https://nerd-notes.com/ubq/121599/"
date_modified: "2026-08-23T05:24:17+00:00"
---

# In the circuit shown, three parallel branches connect a central node P to a common lower conductor held at ground potential (\(V = 0\text{ V}\)).

– The left branch contains an ideal battery of EMF \(\mathcal{E}_1\) in series with a resistor of resistance \(R_1\), with the positive terminal connected toward node P.
– The middle branch contains an ideal battery of EMF \(\mathcal{E}_2\) in series with a resistor of resistance \(R_2\), with the negative terminal connected toward node P.
– The right branch contains an ideal current source that delivers a constant current \(I_0\) into node P.

Which of the following is a correct expression for the electric potential \(V_P\) at node P in terms of the given quantities?

In the circuit shown, three parallel branches connect a central node P to a common lower conductor held at ground potential (\(V = 0\text{ V}\)).

- The left branch contains an ideal battery of EMF \(\mathcal{E}_1\) in series with a resistor of resistance \(R_1\), with the positive terminal connected toward node P.
- The middle branch contains an ideal battery of EMF \(\mathcal{E}_2\) in series with a resistor of resistance \(R_2\), with the negative terminal connected toward node P.
- The right branch contains an ideal current source that delivers a constant current \(I_0\) into node P.

Which of the following is a correct expression for the electric potential \(V_P\) at node P in terms of the given quantities?

![A circuit schematic consisting of three parallel vertical branches connected between a top horizontal wire and a bottom horizontal wire. The top wire contains a marked node labeled P. The bottom wire has a standard three-bar ground symbol attached beneath its center. The leftmost vertical branch contains, in order from top to bottom, an ideal battery labeled \mathcal{E}_1 (oriented with its longer positive plate on top and shorter negative plate on the bottom) in series with a resistor labeled R_1. The middle vertical branch contains, in order from top to bottom, an ideal battery labeled \mathcal{E}_2 (oriented with its shorter negative plate on top and longer positive plate on the bottom) in series with a resistor labeled R_2. The rightmost vertical branch contains a circle representing an ideal current source with a single arrow inside pointing vertically upward toward node P, labeled I_0. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462657-MbAvju.jpg)

- **A.** \(V_P = \dfrac{\mathcal{E}_1 R_1 - \mathcal{E}_2 R_2 + I_0 R_1 R_2}{R_1 + R_2}\)
- **B.** \(V_P = \dfrac{\mathcal{E}_1 R_2 - \mathcal{E}_2 R_1 - I_0 R_1 R_2}{R_1 + R_2}\)
- **C.** \(V_P = \dfrac{\mathcal{E}_1 R_2 + \mathcal{E}_2 R_1 + I_0 R_1 R_2}{R_1 + R_2}\)
- **D.** \(V_P = \dfrac{\mathcal{E}_1 R_2 - \mathcal{E}_2 R_1 + I_0 R_1 R_2}{R_1 + R_2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121599/*