---
title: "A region of space has an electric potential that varies with position in the \\(xy\\)-plane. A contour map of the equipotential lines is shown in Figure 1. The potential \\(V\\) as a function of position is given by the expression \\( V(x, y) = V_0 – \\alpha x^2 + \\beta y \\), where \\(V_0 = 12 \\text{ V}\\), \\(\\alpha = 1.0 \\text{ V/m}^2\\), and \\(\\beta = 2.0 \\text{ V/m}\\). The coordinates \\(x\\) and \\(y\\) are in meters."
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url: "https://nerd-notes.com/ubq/117862/"
date_modified: "2026-08-04T07:59:53+00:00"
---

# A region of space has an electric potential that varies with position in the \(xy\)-plane. A contour map of the equipotential lines is shown in Figure 1. The potential \(V\) as a function of position is given by the expression \( V(x, y) = V_0 – \alpha x^2 + \beta y \), where \(V_0 = 12 \text{ V}\), \(\alpha = 1.0 \text{ V/m}^2\), and \(\beta = 2.0 \text{ V/m}\). The coordinates \(x\) and \(y\) are in meters.

A region of space has an electric potential that varies with position in the \(xy\)-plane. A contour map of the equipotential lines is shown in Figure 1. The potential \(V\) as a function of position is given by the expression \( V(x, y) = V_0 - \alpha x^2 + \beta y \), where \(V_0 = 12 \text{ V}\), \(\alpha = 1.0 \text{ V/m}^2\), and \(\beta = 2.0 \text{ V/m}\). The coordinates \(x\) and \(y\) are in meters.

![A contour map of equipotential lines on an xy-coordinate grid. The grid has a horizontal x-axis ranging from x = -4 m to 4 m and a vertical y-axis ranging from y = -3 m to 6 m. Several solid curved lines, which are parabolas opening upward, represent the equipotential lines. The lines are symmetrically centered around the y-axis (x=0). The lowest parabola has its vertex at (0, 0) and is labeled '12 V'. Parabola vertices going up the y-axis are at y = 1 m labeled '14 V', y = 2 m labeled '16 V', and y = 3 m labeled '18 V'. Another parabola opens upward with its vertex below the x-axis at (0, -2 m), crossing the x-axis at x = -2 m and x = 2 m, labeled '8 V'. A distinct black dot is placed at the coordinate (2, 2) and is labeled 'P'. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830392-lCbdaM.jpg)

**Part a)** Point \(P\) is located at \((x = 2 \text{ m}, y = 2 \text{ m})\) on the \(12 \text{ V}\) equipotential line.

**Part b)** **Derive** an expression for the electric field vector \(\vec{E}\) in terms of \(x\), \(y\), \(\alpha\), \(\beta\), and unit vectors \(\hat{i}\) and \(\hat{j}\).

**Part c)** On the axes provided, **sketch** a graph of the \(x\)-component of the electric field, \(E_x\), and the \(y\)-component of the electric field, \(E_y\), as a function of \(x\) for \(0 \le x \le 4 \text{ m}\). **Label** the vertical axes with appropriate numerical values to make the graphs quantitative.

**Part d)** A proton is released from rest at point \(P\).

**Part e)** An electron is injected into the region at the origin \((0, 0)\) with an initial velocity \(\vec{v}_0 = v_0 \hat{i}\). **Indicate** whether the electron will ever cross the \(y\)-axis (\(x = 0\)) again after being injected. - [ ] Yes - [ ] No **Justify** your answer using the properties of the electric field and the resulting force on the electron.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117862/*