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Related Concept Videos

Differential Form of Maxwell's Equations01:17

Differential Form of Maxwell's Equations

James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and Faraday.
Propagation Speed of Electromagnetic Waves01:30

Propagation Speed of Electromagnetic Waves

Electromagnetic waves are consistent with Ampere's law. Assuming there is no conduction current Ampere's law is given as:
Electromagnetic Wave Equation01:24

Electromagnetic Wave Equation

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: What...
Plane Electromagnetic Waves II01:29

Plane Electromagnetic Waves II

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.
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...
Symmetry in Maxwell's Equations01:28

Symmetry in Maxwell's Equations

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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Related Experiment Video

Updated: Jun 23, 2026

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture
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Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis.

S Johnson, J Joannopoulos

    Optics Express
    |May 7, 2009
    PubMed
    Summary

    A new 3D algorithm computes electromagnetic eigenstates in complex dielectric structures. It efficiently handles anisotropy and magnetic materials, offering accurate results for defect modes.

    Area of Science:

    • Computational physics
    • Electromagnetism
    • Materials science

    Background:

    • Maxwell's equations govern electromagnetic phenomena.
    • Accurate computation of eigenstates is crucial for designing photonic and metamaterials.
    • Existing methods face challenges with complex structures, anisotropy, and magnetic properties.

    Purpose of the Study:

    • To develop a fully-vectorial, three-dimensional algorithm for computing definite-frequency eigenstates.
    • To handle arbitrary periodic dielectric structures, including anisotropic and magnetic materials.
    • To improve computational efficiency and accuracy for complex electromagnetic systems.

    Main Methods:

    • Utilized preconditioned block-iterative eigensolvers in a planewave basis.
    • Developed a new effective dielectric tensor for anisotropic structures.

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  • Compared Davidson method with preconditioned conjugate-gradient Rayleigh-quotient minimization.

    • Main Results:

    • Demonstrated favorable scaling with system size and computed bands.
    • Achieved second-order convergence (O(Δx;2)) even with sharp material discontinuities.
    • Showcased the ability to solve for interior eigenvalues (defect modes) without computing all underlying eigenstates.
    • Characterized various iteration variants and preconditioners.

    Conclusions:

    • The developed algorithm provides an efficient and accurate method for solving Maxwell's equations in complex periodic dielectric systems.
    • The approach is versatile, applicable to anisotropic and magnetic materials, and capable of finding localized defect modes.
    • The freely available implementation facilitates further research in computational electromagnetism and materials design.