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Differential Form of Maxwell's Equations

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.

Symmetry in Maxwell's Equations

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...

Maxwell-Boltzmann Distribution: Problem Solving

Maxwell-Boltzmann Distribution: Problem Solving

Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by

Magnetostatic Boundary Conditions

Magnetostatic Boundary Conditions

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...

Poisson's And Laplace's Equation

Poisson's And Laplace's Equation

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.

Electrostatic Boundary Conditions in Dielectrics

Electrostatic Boundary Conditions in Dielectrics

When an electric field passes from one homogeneous medium to another, crossing the boundary between the two mediums imparts a discontinuity in the electric field. This results in electrostatic boundary conditions that depend on the type of mediums the field propagates through.
Consider a case where both the mediums across a boundary are two different dielectric materials. Recall that the electric field and electric displacement are proportional and related through the material's permittivity.

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Related Experiment Video

Updated: Jun 26, 2026

Characterization of Thermal Transport in One-dimensional Solid Materials

Characterization of Thermal Transport in One-dimensional Solid Materials

Published on: January 26, 2014

Multipole surface solitons in thermal media.

Yaroslav V Kartashov¹, Victor A Vysloukh, Lluis Torner

¹ICFO-Institut de Ciencies Fotoniques, Universitat Politecnica de Catalunya, Mediterranean Technology Park, Barcelona, Spain. Yaroslav.Kartashov@icfo.es

|February 3, 2009

Summary

We studied multipole solitons at thermal interfaces. Higher-order multipoles are stable in layered media, unlike in uniform media where only lower-order ones are stable.

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Last Updated: Jun 26, 2026

Characterization of Thermal Transport in One-dimensional Solid Materials

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Area of Science:

Nonlinear optics
Thermal physics
Materials science

Background:

Solitons are self-reinforcing wave packets.
Thermal interfaces play a role in heat transfer and optical phenomena.
Multipole solitons are complex soliton structures with multiple poles.

Purpose of the Study:

To investigate the existence and stability of multipole solitons.
To explore the influence of uniform versus layered thermal media on soliton properties.
To identify conditions for stabilizing higher-order multipole solitons.

Main Methods:

Theoretical analysis of soliton solutions.
Investigation of stability properties at the interface.
Modeling of thermal and dielectric properties of media.

Main Results:

In uniform thermal media, only surface multipoles with fewer than three poles are stable.
Layered thermal media with alternating thermo-optic coefficients enable the stabilization of higher-order multipoles.
The interface between thermal media and a linear dielectric is crucial for soliton localization.

Conclusions:

The structure of thermal media significantly impacts multipole soliton stability.
Layered media offer a pathway to control and stabilize complex soliton formations.
This research advances understanding of nonlinear phenomena at interfaces.