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title: "The electric potential in a region of space, expressed in two-dimensional polar coordinates \\((r, \\theta)\\), is given by \\(V(r, \\theta) = \\dfrac{V_0 R^2 \\cos\\theta}{r^2}\\), where \\(V_0\\) and \\(R\\) are positive constants. The radial component \\(E_r\\) and angular component \\(E_\\theta\\) of the electric field are related to the potential by \\(E_r = -\\dfrac{\\partial V}{\\partial r}\\) and \\(E_\\theta = -\\dfrac{1}{r}\\dfrac{\\partial V}{\\partial \\theta}\\). Which of the following correctly gives the expressions for \\(E_r\\) and \\(E_\\theta\\)?"
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url: "https://nerd-notes.com/ubq/118088/"
date_modified: "2026-08-04T08:05:03+00:00"
---

# The electric potential in a region of space, expressed in two-dimensional polar coordinates \((r, \theta)\), is given by \(V(r, \theta) = \dfrac{V_0 R^2 \cos\theta}{r^2}\), where \(V_0\) and \(R\) are positive constants. The radial component \(E_r\) and angular component \(E_\theta\) of the electric field are related to the potential by \(E_r = -\dfrac{\partial V}{\partial r}\) and \(E_\theta = -\dfrac{1}{r}\dfrac{\partial V}{\partial \theta}\). Which of the following correctly gives the expressions for \(E_r\) and \(E_\theta\)?

The electric potential in a region of space, expressed in two-dimensional polar coordinates \((r, \theta)\), is given by \(V(r, \theta) = \dfrac{V_0 R^2 \cos\theta}{r^2}\), where \(V_0\) and \(R\) are positive constants. The radial component \(E_r\) and angular component \(E_\theta\) of the electric field are related to the potential by \(E_r = -\dfrac{\partial V}{\partial r}\) and \(E_\theta = -\dfrac{1}{r}\dfrac{\partial V}{\partial \theta}\). Which of the following correctly gives the expressions for \(E_r\) and \(E_\theta\)?

![A two-dimensional polar coordinate system with a horizontal polar axis extending to the right from an origin labeled O. A point labeled P is positioned at distance r from O at an angle \theta measured counterclockwise from the horizontal axis. A dashed line segment of length r connects O to P. At point P, a unit vector arrow labeled \hat{r} points radially outward along the extension of line OP. A second unit vector arrow labeled \hat{\theta} starts at P and points perpendicular to \hat{r} in the direction of increasing \theta (counterclockwise). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830703-pNI5U1.jpg)

- **A.** \(E_r = -\dfrac{2 V_0 R^2 \cos\theta}{r^3}\) and \(E_\theta = -\dfrac{V_0 R^2 \sin\theta}{r^3}\)
- **B.** \(E_r = \dfrac{V_0 R^2 \cos\theta}{r^3}\) and \(E_\theta = \dfrac{V_0 R^2 \sin\theta}{r^3}\)
- **C.** \(E_r = \dfrac{2 V_0 R^2 \cos\theta}{r^3}\) and \(E_\theta = -\dfrac{V_0 R^2 \sin\theta}{r^3}\)
- **D.** \(E_r = \dfrac{2 V_0 R^2 \cos\theta}{r^3}\) and \(E_\theta = \dfrac{V_0 R^2 \sin\theta}{r^3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118088/*