---
title: "A point charge \\(q\\) is located on the axis of symmetry of an open paraboloid surface, at a distance \\(d\\) directly below the center of its circular opening of radius \\(R\\). The surface is open at the top rim and closed at the bottom vertex. Which of the following expressions represents the magnitude of the electric flux \\(\\Phi_E\\) through the open paraboloid surface?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/117952/"
date_modified: "2026-08-04T08:02:41+00:00"
---

# A point charge \(q\) is located on the axis of symmetry of an open paraboloid surface, at a distance \(d\) directly below the center of its circular opening of radius \(R\). The surface is open at the top rim and closed at the bottom vertex. Which of the following expressions represents the magnitude of the electric flux \(\Phi_E\) through the open paraboloid surface?

A point charge \(q\) is located on the axis of symmetry of an open paraboloid surface, at a distance \(d\) directly below the center of its circular opening of radius \(R\). The surface is open at the top rim and closed at the bottom vertex. Which of the following expressions represents the magnitude of the electric flux \(\Phi_E\) through the open paraboloid surface?

![An open paraboloid surface extending upward, drawn as a smooth U-shaped curve rotated about a vertical central dashed line representing the axis of symmetry. The top of the paraboloid is a flat horizontal ellipse representing a circular opening of radius R, with a horizontal dashed line from the central axis to the right edge labeled R. Below the vertex of the paraboloid, on the vertical dashed axis of symmetry, a small black circle represents a point charge labeled q. A vertical dimension line with arrowheads indicates a distance d from charge q up to the plane of the circular top opening. Exactly three electric field vectors with arrowheads originate from charge q and pass outward through the curved sides of the paraboloid surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830560-doXDJJ.jpg)

- **A.** \(\dfrac{q}{\varepsilon_0} \left( 1 - \dfrac{d}{\sqrt{R^2 + d^2}} \right)\)
- **B.** \(\dfrac{q}{2\varepsilon_0} \left( \dfrac{d}{\sqrt{R^2 + d^2}} \right)\)
- **C.** \(\dfrac{q}{2\varepsilon_0} \left( 1 - \dfrac{d}{\sqrt{R^2 + d^2}} \right)\)
- **D.** \(\dfrac{q}{2\varepsilon_0} \left( 1 - \dfrac{R}{\sqrt{R^2 + d^2}} \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117952/*