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Related Concept Videos

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.
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 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: Planar Symmetry01:27

Gauss's Law: Planar Symmetry

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...
Plane Electromagnetic Waves I01:30

Plane Electromagnetic Waves I

The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of Maxwell's equations. As Maxwell's equations hold in free space, the predicted electromagnetic waves do not require a medium for their propagation. An EM wave comprises an electric field, defined as the force per charge on a stationary charge, and a magnetic field, which is the force per charge on a moving charge.
The EM field is assumed to be a...
Distribution of Stresses in a Narrow Rectangular Beam01:11

Distribution of Stresses in a Narrow Rectangular Beam

In studying beam stress distribution, examining an elemental section is essential. To determine the average shearing stress on this face, the calculated shear is divided by the surface area. Importantly, shearing stresses on the beam's transverse and horizontal planes mirror each other, indicating a consistent stress distribution along the upper region of the beam. Notably, shearing stresses are absent at the beam's upper and lower surfaces due to the absence of applied forces in these areas.

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Related Experiment Video

Updated: Jun 12, 2026

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

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

Published on: August 12, 2013

Nonparaxial Gaussian beams.

S Nemoto

    Applied Optics
    |June 22, 2010
    PubMed
    Summary
    This summary is machine-generated.

    Researchers established a lower bound for Gaussian beam waist size, ensuring the paraxial approximation remains valid. Beams smaller than this bound are nonparaxial Gaussian beams, where standard approximations fail.

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    Area of Science:

    • Optics and Photonics
    • Electromagnetism

    Background:

    • Gaussian beams are fundamental in optics, often analyzed using the paraxial approximation.
    • The paraxial approximation assumes beam width is much larger than the wavelength of light.

    Purpose of the Study:

    • To define the lower bound for Gaussian beam waist size where the paraxial approximation is valid.
    • To analyze nonparaxial Gaussian beams and the effectiveness of first-order corrections.

    Main Methods:

    • Theoretical analysis of Gaussian beam propagation.
    • Derivation of conditions for paraxial approximation validity.
    • Investigation of first-order correction effectiveness for nonparaxial beams.

    Main Results:

    • A specific lower bound for Gaussian beam waist size was determined.
    • Nonparaxial Gaussian beams were characterized for waist sizes below this bound.
    • The range of effectiveness for first-order corrections was clarified.

    Conclusions:

    • The paraxial approximation is valid only for Gaussian beams with waist sizes exceeding a defined lower bound.
    • First-order corrections are ineffective when the paraxial approximation completely fails for nonparaxial Gaussian beams.