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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 has...

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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,...

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Gauss's Law: Planar Symmetry

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Related Experiment Video

Updated: Apr 21, 2026

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Published on: February 27, 2016

Vector spherical quasi-Gaussian vortex beams.

¹Los Alamos National Laboratory, MS D429, Los Alamos, New Mexico 87545, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 30, 2014

Summary

Researchers developed high-order quasi-Gaussian (qG) vortex beams for describing focused electromagnetic beams. This new model offers an exact solution satisfying Maxwell

Area of Science:

Electromagnetism and Optics
Mathematical Physics

Background:

Describing focused electromagnetic beams, particularly vortex beams, requires advanced mathematical models.
Existing methods for higher-order beams often involve complex corrections or intensive computations.

Purpose of the Study:

To derive model equations for efficiently computing radiation profiles of tightly spherically focused higher-order electromagnetic vortex beams.
To introduce a new analytical solution for these beams that exactly satisfies fundamental electromagnetic equations.

Main Methods:

Vectorial analysis utilizing the complex-source-point method.
Derivation of closed-form expressions for high-order quasi-Gaussian (qG) vortex beams.

Main Results:

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Last Updated: Apr 21, 2026

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Published on: February 27, 2016

High-speed Particle Image Velocimetry Near Surfaces

High-speed Particle Image Velocimetry Near Surfaces

Published on: June 24, 2013

Preparation of Free-Surface Hyperbolic Water Vortices

Preparation of Free-Surface Hyperbolic Water Vortices

Published on: July 28, 2023

A novel high-order quasi-Gaussian (qG) vortex beam solution was developed, exactly satisfying the vector Helmholtz and Maxwell's equations.

The solution rigorously describes strongly focused or divergent vortex wave fields without higher-order corrections or intensive numerical methods.

Properties of these beams were illustrated using closed-form expressions and computational results, emphasizing beam waist and polarization schemes.

Conclusions:

The high-order qG vortex beam model provides an exact and computationally efficient method for analyzing focused electromagnetic vortex beams.
This approach simplifies the study of complex beam characteristics, such as degree, order, Rayleigh range, and polarization.