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Related Concept Videos

Electromagnetic Wave Equation01:24

Electromagnetic Wave Equation

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Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete.
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations:...
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Plane Electromagnetic Waves I01:30

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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...
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Plane Electromagnetic Waves II01:29

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Consider a plane wavefront traveling in position x-direction with a constant speed. This wavefront can be utilized to obtain the relationship between electric and magnetic fields with the help of Faraday's law.
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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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Electromagnetic Waves01:30

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James Clerk Maxwell formulated a single theory combining all the electric and magnetic effects scientists knew during that time, calling the phenomena his theory predicted “Electromagnetic waves”. He brought together all the work that had been done by brilliant physicists such as Oersted, Coulomb, Gauss, and Faraday and added his own insights to develop the overarching theory of electromagnetism. Maxwell’s equations, combined with the Lorentz force law, encompass all the laws...
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Electromagnetic waves can travel in the vacuum as well as in matter. For example light, which is an electromagnetic wave, can travel through air, water, or glass.
Consider the electromagnetic wave passing through a dielectric medium. In such a case, Maxwell's equations get modified. In Ampere's law, ε0 , the dielectric permittivity of free space is replaced with ε, the permittivity of dielectric. Also, the vacuum permeability μ0 is replaced by the permeability of the medium, μ.
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Related Experiment Video

Updated: Mar 31, 2026

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Spherical space Bessel-Legendre-Fourier localized modes solver for electromagnetic waves.

Mohammed A Alzahrani, Robert C Gauthier

    Optics Express
    |October 20, 2015
    PubMed
    Summary

    This study presents a novel method for solving Maxwell

    Area of Science:

    • Computational Electromagnetics
    • Mathematical Physics

    Background:

    • Solving Maxwell's equations is crucial for understanding electromagnetic phenomena.
    • Traditional methods can be computationally intensive for complex dielectric structures.

    Purpose of the Study:

    • To develop a compact and accurate numerical method for solving Maxwell's vector wave equations.
    • To enable efficient computation of electromagnetic modes in spherical dielectric configurations.

    Main Methods:

    • Series expansion of field components and dielectric profiles using spherical Bessel functions, Legendre polynomials, and complex exponentials (BLF).
    • Formulation of a compact eigenvalue matrix for Maxwell's equations.
    • Solving the matrix to obtain frequencies and field profiles.

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    Main Results:

    • The BLF basis functions lead to a compact eigenvalue matrix formulation.
    • The method accurately solves for steady-state electromagnetic modes.
    • Numerical results are validated against complementary techniques.

    Conclusions:

    • The developed BLF matrix method offers an accurate and efficient approach for analyzing electromagnetic modes in spherical dielectric systems.
    • This technique is suitable for desktop PC computations, enhancing accessibility.