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Gauss's Law: Cylindrical Symmetry01:20

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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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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...
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Gauss's Law: Planar Symmetry01:27

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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...
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The moment-area method is an analytical tool used in structural engineering to determine the slope and deflection of beams under various loads. Consider a cantilever with a concentrated load and moment at the free end. The first step is constructing a free-body diagram to calculate the reactions at the fixed end. Next, the bending moment diagram is plotted to visualize how the bending moment varies along the beam's length, focusing on points where the bending moment equals zero.
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Related Experiment Video

Updated: Mar 21, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
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Noncoaxial Bessel-Gauss beams.

Chaohong Huang, Yishu Zheng, Hanqing Li

    Journal of the Optical Society of America. A, Optics, Image Science, and Vision
    |May 4, 2016
    PubMed
    Summary

    Researchers introduced novel noncoaxial Gauss-truncated Bessel beams, which exhibit asymmetrical intensity and unique vortex behavior during propagation. These beams offer new possibilities for optical manipulation and vortex generation.

    Area of Science:

    • Optics and Photonics
    • Quantum Optics

    Background:

    • Bessel beams are known for their non-diffracting properties.
    • Bessel-Gauss beams combine Bessel and Gaussian beam characteristics.
    • Controlling beam propagation and vortex dynamics is crucial in optical applications.

    Purpose of the Study:

    • To introduce and characterize a new family of noncoaxial Gauss-truncated Bessel beams.
    • To analyze the propagation dynamics and angular spectra of these novel beams.
    • To investigate the orbital angular momentum and vortex behavior of the proposed beams.

    Main Methods:

    • Multiplying conventional symmetrical Bessel beams by a noncoaxial Gauss function.
    • Deriving closed-form solutions for angular spectra and paraxial propagation.
    • Analyzing the intensity distributions and vortex properties during propagation.

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    Main Results:

    • The proposed beams exhibit asymmetrical intensity distributions.
    • These beams carry the same orbital angular momentum per photon as Bessel-Gauss beams.
    • Noncoaxial Bessel-Gauss beams demonstrate intensity rotation and transverse vortex shifts during propagation.
    • Paired unit vortices with opposite signs can be generated depending on beam parameters.

    Conclusions:

    • Noncoaxial Gauss-truncated Bessel beams represent a new class of optical beams with unique propagation characteristics.
    • These beams offer potential for controlled vortex generation and manipulation in optical systems.
    • The derived analytical solutions facilitate further theoretical and experimental investigations.