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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 a...

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: 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

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.

Stokes' Law

Stokes' Law

Viscous forces, like friction, are intermolecular forces that resist the relative motion of molecules over each other. When a solid body moves through a liquid, viscous forces drag it in the opposite direction. The force's magnitude depends on the solid's shape and size, as well as its speed and the liquid's coefficient of viscosity, density and temperature.
The expression for the force on a solid spherical object in a fluid is called Stokes' law. Stokes' law is valid only...

Motion Of A Charged Particle In A Magnetic Field

Motion Of A Charged Particle In A Magnetic Field

A charged particle experiences a force when moving through a magnetic field. Consider the field to be uniform and the charged particle to move perpendicular to it. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion. Since the magnetic force is perpendicular to the direction of motion, a charged particle follows a curved path. The particle continues to follow this curved path until it forms a complete circle. Another way to look at this is that the...

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Related Experiment Video

Updated: Apr 3, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

Light trajectory in Bessel-Gauss vortex beams.

Chen Xie, Remo Giust, Vytautas Jukna

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|September 15, 2015

Summary

This summary is machine-generated.

Bessel-Gauss vortex beams transition from expanding rings to a diffraction-free state. Using hollow input beams can eliminate this transition regime for better beam control.

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Published on: August 12, 2013

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Area of Science:

Optics and Photonics
Laser Physics

Background:

Bessel-Gauss vortex beams exhibit unique propagation characteristics.
Understanding their early-stage dynamics is crucial for applications.

Purpose of the Study:

To investigate the transition regime in Bessel-Gauss vortex beam propagation.
To characterize the beam structure and identify methods to control its dynamics.

Main Methods:

Utilized the eikonal equation to model beam structure.
Employed analytical, numerical, and experimental approaches.
Investigated the effect of hollow input beams.

Main Results:

Observed a progressive lateral expansion of the main intensity ring during the transition regime.
Characterized the beam structure using hyperboloids with variable waists, forming a tapered tubular caustic.
Demonstrated excellent agreement between analytical, numerical, and experimental results.
Showed that hollow input beams eliminate the transition regime.

Conclusions:

The early propagation of Bessel-Gauss vortex beams involves a distinct transition regime.
The eikonal equation effectively describes the beam's hyperbolic structure.
Hollow input beams offer a method to bypass the transition, leading to a more direct diffraction-free propagation.