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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...
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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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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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Gauss's Law in Dielectrics01:17

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Consider a polar dielectric placed in an external field. In such a dielectric, opposite charges on adjacent dipoles neutralize each other, such that the net charge within the dielectric is zero. When a polar dielectric is inserted in between the capacitor plates, an electric field is generated due to the presence of net charges near the edge of the dielectric and the metal plates interface. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral. An...
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Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...
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A spherical capacitor consists of two concentric conducting spherical shells of radii R1 (inner shell) and R2 (outer shell). The shells have equal and opposite charges of +Q and −Q, respectively. For an isolated conducting spherical capacitor, the radius of the outer shell can be considered to be infinite.
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Difference of two Gaussian Schell-model cross-spectral densities.

Franco Gori, Massimo Santarsiero

    Optics Letters
    |May 3, 2014
    PubMed
    Summary

    Researchers explored the difference between Gaussian Schell-model cross-spectral densities (CSDs). They identified conditions for a valid CSD difference, crucial for optical coherence theory and wave propagation studies.

    Area of Science:

    • Optics and Photonics
    • Wave Propagation
    • Statistical Optics

    Background:

    • Gaussian Schell-model beams are widely used in optical coherence theory.
    • Cross-spectral densities (CSDs) describe the spectral coherence properties of light fields.
    • Understanding the properties of differences between CSDs is essential for analyzing complex light fields.

    Purpose of the Study:

    • To determine the conditions under which the difference of two Gaussian Schell-model cross-spectral densities (CSDs) results in a valid CSD.
    • To derive criteria for ensuring the non-negative definiteness of the resulting CSD.
    • To investigate conditions for shape-invariant CSD differences during propagation.

    Main Methods:

    • Mathematical analysis of Gaussian Schell-model CSDs.

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  • Derivation of conditions for non-negative definiteness.
  • Investigation of propagation invariance properties.

    • Main Results:

    • A sufficient condition for the non-negative definiteness of the difference of two Gaussian Schell-model CSDs was derived.
    • A necessary and sufficient condition was established for CSD differences that maintain their shape during propagation.
    • The study provides a framework for constructing valid CSDs from differences.

    Conclusions:

    • The difference of two Gaussian Schell-model CSDs can represent a valid CSD under specific mathematical conditions.
    • The findings contribute to a deeper understanding of coherence properties and light field analysis.
    • This research offers practical criteria for optical engineers and physicists working with partially coherent light sources.