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Related Concept Videos

Singularity Functions for Shear01:26

Singularity Functions for Shear

In structural analysis, singularity functions are crucial in simplifying the representation of shear forces in beams under discontinuous loading. These functions describe discontinuous variations in shear force across a beam with varying loads by using a single mathematical expression, regardless of the complexity of the loading conditions. The singularity functions are derived from creating a free-body diagram of the beam and then making conceptual cuts at specific points to examine the shear...
Deflection of a Beam01:19

Deflection of a Beam

Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
Singularity functions, described in an earlier lesson, are powerful mathematical tools that represent discontinuities within a function commonly encountered in structural loading...
Singularity Functions for Bending Moment01:18

Singularity Functions for Bending Moment

Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a...
Gauss's Law: Spherical Symmetry01:26

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 uniform...
Gauss's Law: Cylindrical Symmetry01:20

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 Law01:07

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.

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The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
12:14

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Published on: August 12, 2013

Wigner distribution function for Gaussian-Schell beams in complex matrix optical systems.

D Dragoman

    Applied Optics
    |November 6, 2010
    PubMed
    Summary
    This summary is machine-generated.

    A transformation law for the Wigner distribution function applies to Gaussian-Schell beam propagation in specific complex matrix optical systems. This allows for defining invariant quantities, similar to real matrix systems.

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    Area of Science:

    • Optics
    • Mathematical Physics

    Background:

    • Gaussian-Schell beams are widely used in optical systems.
    • Complex matrix optical systems present challenges for beam propagation analysis.

    Purpose of the Study:

    • To investigate the applicability of the ABCD transformation law to Gaussian-Schell beam propagation in complex matrix optical systems.
    • To identify invariant quantities for such propagation.

    Main Methods:

    • Analysis of Gaussian-Schell beam propagation.
    • Application of the Wigner distribution function.
    • Investigation of matrix optical system transformations.

    Main Results:

    • The ABCD transformation law for the Wigner distribution function holds in particular cases for Gaussian-Schell beams.
    • Invariant quantities analogous to the real matrix case can be defined.

    Conclusions:

    • The study extends the applicability of the ABCD law to complex matrix optical systems under specific conditions.
    • New invariant quantities are proposed for Gaussian-Schell beam propagation analysis.