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Beams with Symmetric Loadings01:15

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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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Beams with Unsymmetric Loadings01:17

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Analyzing a supported beam under unsymmetrical loadings is essential in structural engineering to understand how beams respond to varied force distributions. This analysis involves calculating the deflection and identifying points where the slope of the beam is zero, which are crucial for ensuring structural stability and functionality.
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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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Gauss's Law: Spherical Symmetry01:26

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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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    We introduce novel pseudo-Schell model (PSM) sources with Dual-Laguerre Gaussian (DLG) coherence. These sources enable tunable partial coherence and controllable self-focusing for light beams and vortex beams.

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    Area of Science:

    • Optics and Photonics
    • Coherence Theory
    • Beam Propagation

    Background:

    • Pseudo-Schell model (PSM) sources are widely used in optical research.
    • Controlling the coherence properties of light is crucial for various applications.
    • Laguerre-Gaussian (LG) beams are known for their unique phase structures.

    Purpose of the Study:

    • To introduce a novel class of pseudo-Schell model (PSM) sources.
    • To investigate the self-focusing properties of light fields generated by these sources.
    • To demonstrate the generation and characteristics of vortex beams with tunable coherence.

    Main Methods:

    • Developing a new class of PSM sources with a Dual-Laguerre Gaussian (DLG) spectral degree of coherence.
    • Analyzing the propagation dynamics of light fields radiated from DLG-PSM sources.
    • Generating and characterizing DLG-PSM vortex beams with arbitrary topological charges.

    Main Results:

    • The DLG-PSM sources exhibit perfect coherence for co-radial points and tunable partial coherence for non-radial points.
    • Light fields from DLG-PSM sources show controllable self-focusing effects, with adjustable focal length and peak intensity.
    • DLG-PSM vortex beams maintain a stable annular profile with a central intensity null and exhibit controllable self-focusing during propagation.

    Conclusions:

    • The novel DLG-PSM sources offer unique control over coherence properties and beam propagation.
    • The ability to tune self-focusing behavior and generate stable vortex beams opens new avenues for optical applications.
    • This research provides a versatile platform for advanced optical beam shaping and manipulation.