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Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.

Gauss's Law: Cylindrical Symmetry

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...

Gauss's Law: Problem-Solving

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area...

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

Gauss's Law: Spherical Symmetry

Gauss's Law: Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half...

Gauss's Law in Dielectrics

Gauss's Law in Dielectrics

Consider a polar dielectric placed in an external field. In such a dielectric, opposite charges on adjacent dipoles neutralize each other, such that the net charge within the dielectric is zero. When a polar dielectric is inserted in between the capacitor plates, an electric field is generated due to the presence of net charges near the edge of the dielectric and the metal plates interface. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral. An...

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Related Experiment Video

Updated: Aug 15, 2025

Fabrication of Gate-tunable Graphene Devices for Scanning Tunneling Microscopy Studies with Coulomb Impurities

Fabrication of Gate-tunable Graphene Devices for Scanning Tunneling Microscopy Studies with Coulomb Impurities

Published on: July 24, 2015

Gaussian Curvature Effects on Graphene Quantum Dots.

Sergio de-la-Huerta-Sainz¹, Angel Ballesteros¹, Nicolás A Cordero^1,2,3

¹Physics Department, Universidad de Burgos, E-09001 Burgos, Spain.

Nanomaterials (Basel, Switzerland)

|January 8, 2023

Summary

Bending hexagonal graphene quantum dots alters their electronic properties. Researchers found a strong link between Gaussian curvature and quantum regeneration times, with unique behavior in hyperboloid shapes possibly indicating a phase transition.

Keywords:

DFT Gaussian curvature graphene phase transition pseudo-magnetic field quantum revival

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Related Experiment Videos

Last Updated: Aug 15, 2025

Fabrication of Gate-tunable Graphene Devices for Scanning Tunneling Microscopy Studies with Coulomb Impurities

Fabrication of Gate-tunable Graphene Devices for Scanning Tunneling Microscopy Studies with Coulomb Impurities

Published on: July 24, 2015

Graphene Enclosure of Chemically Fixed Mammalian Cells for Liquid-Phase Electron Microscopy

Graphene Enclosure of Chemically Fixed Mammalian Cells for Liquid-Phase Electron Microscopy

Published on: September 21, 2020

Nanofabrication of Gate-defined GaAs/AlGaAs Lateral Quantum Dots

Nanofabrication of Gate-defined GaAs/AlGaAs Lateral Quantum Dots

Published on: November 1, 2013

Area of Science:

Condensed Matter Physics
Materials Science
Quantum Mechanics

Background:

Graphene nanostructures display exotic properties when deformed.
Understanding mechanical and electronic responses of bent graphene is crucial.

Purpose of the Study:

Investigate mechanical and electronic properties of bent hexagonal graphene quantum dots.
Explore the impact of Gaussian curvature on quantum regeneration times.

Main Methods:

Employing density functional theory (DFT) for simulations.
Bending graphene quantum dots on spherical, cylindrical, and hyperboloid surfaces.
Analyzing curvature energy and quantum regeneration times (classic and revival).

Main Results:

A strong correlation was found between Gaussian curvature and regeneration times.
The hyperboloid surface showed a special divergence in revival time.
This divergence may be linked to curvature-induced pseudo-magnetic fields and phase transitions.

Conclusions:

Gaussian curvature significantly influences the electronic properties and quantum regeneration of graphene quantum dots.
Hyperboloid curvature may induce phase transitions in these nanostructures.