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Related Concept Videos

Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

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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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Spherical Coordinates01:23

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Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
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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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Electric Field of a Non Uniformly Charged Sphere01:22

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Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
Consider a non-uniformly charged sphere, for which the density of charge depends only on the distance from a point in space and not on the direction. Such a sphere has a spherically symmetrical charge distribution. Here, the electric...
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Theorem of Pappus01:24

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The Theorem of Pappus, also known as the Pappus–Guldinus Theorem, provides a geometric method for determining the volume and surface area of solids generated by the revolution of a plane region or a plane curve about an external axis. The theorem consists of two related statements. The first addresses the volume of solids formed by rotating plane areas, while the second addresses the surface area generated by rotating plane curves. Both results depend on the location of the centroid,...
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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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Generalized Poincaré sphere.

Zhi-Cheng Ren, Ling-Jun Kong, Si-Min Li

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    |October 20, 2015
    PubMed
    Summary
    This summary is machine-generated.

    Researchers developed a generalized Poincaré sphere (G sphere) unifying vector field descriptors. This new geometric model incorporates spin angular momentum (SAM) and orbital angular momentum (OAM) for broader applications.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • The standard Poincaré sphere is limited in describing complex vector fields.
    • Existing models struggle to unify descriptors for diverse vector field properties.

    Purpose of the Study:

    • To introduce a generalized Poincaré sphere (G sphere) and generalized Stokes parameters (G parameters) for unifying vector field descriptors.
    • To extend the geometric representation beyond the standard Poincaré sphere by incorporating higher dimensions.

    Main Methods:

    • Constructing the G sphere by extending Jones vector bases to general vector bases.
    • Incorporating continuously changeable ellipticity (spin angular momentum, SAM) and higher-dimensional orbital angular momentum (OAM).
    • Defining spherical shells where poles represent orthogonal vector bases with varying SAM and OAM.

    Main Results:

    • The G sphere provides a unified geometric representation for various vector fields.
    • Higher-order Poincaré spheres are identified as specific spherical shells within the G sphere.
    • A flexible scheme is presented for generating all vector fields representable on the G sphere.

    Conclusions:

    • The G sphere offers a comprehensive framework for understanding and manipulating vector fields.
    • This generalization unifies concepts related to SAM and OAM in a single geometric model.
    • The proposed scheme enables the generation of a wide range of vector fields with tailored properties.