---
title: "A point charge \\(+q\\) is located at a distance \\(d\\) above an infinite, grounded conducting plane lying in the \\(xy\\)-plane. To analyze the electrostatic field in the region above the plane, the grounded conductor is replaced by an equivalent image charge \\(-q\\) positioned at a distance \\(d\\) directly below the plane at position \\((0, 0, -d)\\). The electric field magnitude just outside the surface of a conductor in electrostatic equilibrium is related to the surface charge density by \\(E = \\dfrac{|\\sigma|}{\\varepsilon_0}\\). Which of the following expressions gives the magnitude of the induced surface charge density \\(|\\sigma(r)|\\) on the conducting plane as a function of the radial distance \\(r\\) from the point directly beneath the charge?"
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url: "https://nerd-notes.com/ubq/118008/"
date_modified: "2026-08-04T08:02:52+00:00"
---

# A point charge \(+q\) is located at a distance \(d\) above an infinite, grounded conducting plane lying in the \(xy\)-plane. To analyze the electrostatic field in the region above the plane, the grounded conductor is replaced by an equivalent image charge \(-q\) positioned at a distance \(d\) directly below the plane at position \((0, 0, -d)\). The electric field magnitude just outside the surface of a conductor in electrostatic equilibrium is related to the surface charge density by \(E = \dfrac{|\sigma|}{\varepsilon_0}\). Which of the following expressions gives the magnitude of the induced surface charge density \(|\sigma(r)|\) on the conducting plane as a function of the radial distance \(r\) from the point directly beneath the charge?

A point charge \(+q\) is located at a distance \(d\) above an infinite, grounded conducting plane lying in the \(xy\)-plane. To analyze the electrostatic field in the region above the plane, the grounded conductor is replaced by an equivalent image charge \(-q\) positioned at a distance \(d\) directly below the plane at position \((0, 0, -d)\). The electric field magnitude just outside the surface of a conductor in electrostatic equilibrium is related to the surface charge density by \(E = \dfrac{|\sigma|}{\varepsilon_0}\). Which of the following expressions gives the magnitude of the induced surface charge density \(|\sigma(r)|\) on the conducting plane as a function of the radial distance \(r\) from the point directly beneath the charge?

![A side-view diagram showing a horizontal line representing a conducting plane along the x-axis. A point labeled +q is located on the vertical z-axis at height d above the origin (0,0). Directly below the origin on the z-axis, at distance d, a dashed point labeled -q is shown. A point P is marked on the horizontal line at distance r to the right of the origin. Two straight dashed lines connect +q to P and -q to P, each of length R equal to the square root of r squared plus d squared. At point P, two downward-angled vector arrows of equal length represent electric field components pointing into the lower region, forming an angle theta with the horizontal line. A vertical dashed z-axis line passes through +q, the origin, and -q. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830572-42J3E1.jpg)

- **A.** \(\dfrac{qd}{4\pi (r^2 + d^2)^{3/2}}\)
- **B.** \(\dfrac{qd}{2\pi (r^2 + d^2)^{3/2}}\)
- **C.** \(\dfrac{qr}{2\pi (r^2 + d^2)^{3/2}}\)
- **D.** \(\dfrac{qd^2}{2\pi (r^2 + d^2)^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118008/*