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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...
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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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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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Generalized modified hollow vortex Gaussian beam.

Hassan T Al-Ahsab, Mingjian Cheng, Ibrahim G H Loqman

    Journal of the Optical Society of America. A, Optics, Image Science, and Vision
    |August 12, 2025
    PubMed
    Summary

    Researchers introduced a new modified hollow vortex Gaussian beam (MHVGB). This novel vortex beam offers enhanced control over optical properties and shows promise for applications in optical manipulation and particle trapping.

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    Area of Science:

    • Optics and Photonics
    • Beam Propagation
    • Laser Physics

    Background:

    • Vortex beams are of significant interest due to their unique propagation characteristics and diverse applications.
    • Existing hollow beam models have limitations in controllability.

    Purpose of the Study:

    • To introduce and analyze a new modified hollow vortex Gaussian beam (MHVGB).
    • To investigate the propagation dynamics and focusing properties of MHVGBs.
    • To explore the potential applications of MHVGBs.

    Main Methods:

    • Derivation of an analytical expression for MHVGB propagation through ABCD optical systems using the Collins integral formula and paraxial approximation.
    • Comparison of MHVGB propagation with special cases like hollow Gaussian beams.
    • Experimental validation of the theoretical model.
    • Investigation of tight focusing of radially polarized MHVGBs.

    Main Results:

    • An analytical expression for MHVGB propagation was derived.
    • The modification parameter provides enhanced control over hollow beam properties compared to existing models.
    • Experimental results validated the theoretical predictions for MHVGB propagation.
    • Formation of flat-topped beams was observed during tight focusing of radially polarized MHVGBs.

    Conclusions:

    • The modified hollow vortex Gaussian beam (MHVGB) is a novel and controllable vortex beam model.
    • MHVGBs exhibit unique propagation and focusing behaviors.
    • This new beam model holds potential for applications in optical manipulation and particle trapping.