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title: "The electric potential in a region of two-dimensional space is given by the function \\(V(x, y) = \\alpha x^2 – \\beta y^2\\), where \\(\\alpha\\) and \\(\\beta\\) are positive constants. Which of the following correctly gives the electric field vector \\(\\vec{E}\\) at a point \\((x_0, y_0)\\) located in the first quadrant (where \\(x_0 > 0\\) and \\(y_0 > 0\\)), and identifies the quadrant of the \\(xy\\)-plane toward which \\(\\vec{E}\\) points?"
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url: "https://nerd-notes.com/ubq/118098/"
date_modified: "2026-08-04T08:05:06+00:00"
---

# The electric potential in a region of two-dimensional space is given by the function \(V(x, y) = \alpha x^2 – \beta y^2\), where \(\alpha\) and \(\beta\) are positive constants. Which of the following correctly gives the electric field vector \(\vec{E}\) at a point \((x_0, y_0)\) located in the first quadrant (where \(x_0 > 0\) and \(y_0 > 0\)), and identifies the quadrant of the \(xy\)-plane toward which \(\vec{E}\) points?

The electric potential in a region of two-dimensional space is given by the function \(V(x, y) = \alpha x^2 - \beta y^2\), where \(\alpha\) and \(\beta\) are positive constants. Which of the following correctly gives the electric field vector \(\vec{E}\) at a point \((x_0, y_0)\) located in the first quadrant (where \(x_0 > 0\) and \(y_0 > 0\)), and identifies the quadrant of the \(xy\)-plane toward which \(\vec{E}\) points?

![A Cartesian coordinate plane with a horizontal axis labeled x and a vertical axis labeled y intersecting at an origin labeled O. Arrowheads are on both ends of both axes. The four quadrants are labeled in text: Quadrant I in the top-right region, Quadrant II in the top-left region, Quadrant III in the bottom-left region, and Quadrant IV in the bottom-right region. A single solid black dot is located in the top-right region at positive x and positive y, accompanied by the text label (x_0, y_0). No field vectors, equipotential lines, curves, or other markings appear on the axes.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830706-POovaE.jpg)

- **A.** \(\vec{E} = 2\alpha x_0 \hat{i} + 2\beta y_0 \hat{j}\) ; Quadrant I
- **B.** \(\vec{E} = 2\alpha x_0 \hat{i} - 2\beta y_0 \hat{j}\) ; Quadrant IV
- **C.** \(\vec{E} = -2\alpha x_0 \hat{i} + 2\beta y_0 \hat{j}\) ; Quadrant II
- **D.** \(\vec{E} = -2\alpha x_0 \hat{i} - 2\beta y_0 \hat{j}\) ; Quadrant III

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118098/*