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Related Concept Videos

Hyperbolas01:30

Hyperbolas

540
A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse...
540
Geometry of Hyperbolas01:30

Geometry of Hyperbolas

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A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
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Gauss's Law: Planar Symmetry01:27

Gauss's Law: Planar Symmetry

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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: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

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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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Symmetry in Maxwell's Equations01:28

Symmetry in Maxwell's Equations

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Once the fields have been calculated using Maxwell's four equations, the Lorentz force equation gives the force that the fields exert on a charged particle moving with a certain velocity. The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be...
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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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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Hyperbolic-symmetry vector fields.

Xu-Zhen Gao, Yue Pan, Meng-Qiang Cai

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    |December 25, 2015
    PubMed
    Summary
    This summary is machine-generated.

    Researchers developed a novel hyperbolic coordinate system and hyperbolic-symmetry vector fields. These optical fields were experimentally generated and focused, demonstrating potential for micro-structure fabrication.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • Orthogonal coordinate systems are fundamental in physics and engineering.
    • Vector optical fields with tailored polarization exhibit unique properties for light manipulation.

    Purpose of the Study:

    • To introduce and construct a new hyperbolic coordinate system.
    • To design and experimentally generate hyperbolic-symmetry vector optical fields.
    • To investigate the focusing behavior and micro-fabrication applications of these fields.

    Main Methods:

    • Development of a novel hyperbolic coordinate system.
    • Design and experimental generation of hyperbolic-symmetry vector optical fields.
    • Analysis of tight focusing properties and fabrication of micro-structures using these fields.

    Main Results:

    • Successful construction of the hyperbolic coordinate system.
    • Experimental demonstration of hyperbolic-symmetry vector optical fields.
    • Observation of tight focusing and successful micro-structure fabrication on K9 glass surfaces.

    Conclusions:

    • The novel hyperbolic coordinate system and vector fields offer new possibilities in optical manipulation.
    • The demonstrated micro-fabrication capability highlights the practical applications of these engineered optical fields.