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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.

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...

Potential Due to a Magnetized Object

Potential Due to a Magnetized Object

Magnetic dipoles in magnetic materials are aligned when placed under an external magnetic field. For paramagnets and ferromagnets, dipole alignment occurs in the direction of the magnetic field. However, the dipoles align opposite to the field in the case of diamagnets. This state of magnetic polarization due to the external field is called magnetization. Magnetization is defined as the dipole moment per unit volume. It plays a similar role to polarization in electrostatics.
The vector...

Magnetic Fields

Magnetic Fields

A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on any other moving charge or current that is present in the field. Like an electric field, the magnetic field is also a vector field. At any position, the direction of the magnetic field is defined as the direction in which the north pole of a compass needle points.
A magnetic field is defined by the force that a charged particle experiences...

Magnetic Vector Potential

Magnetic Vector Potential

In electrostatics, the electric field can be written as the negative gradient of the potential. In magnetostatics, the zero divergence of the magnetic field ensures that the magnetic field can be expressed as the curl of a vector potential. This potential is known as the magnetic vector potential.
Consider an ideal solenoid with n turns per unit length and radius R. If I is the current through the solenoid, the magnetic field inside the solenoid is expressed as the product of vacuum...

Diamagnetism

Diamagnetism

Materials consisting of paired electrons have zero net magnetic moments. However, when these materials are placed under an external magnetic field, the moments opposite to the field are induced. Such materials are called diamagnets. Diamagnetism is the response of the diamagnets when placed in an external magnetic field.
Diamagnetism was discovered by Anton Brugmans in 1778 when he observed that bismuth gets repelled by magnetic fields, thus theorizing that diamagnets get repelled by magnets.

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Related Experiment Video

Updated: Jul 11, 2026

Magnetometric Characterization of Intermediates in the Solid-State Electrochemistry of Redox-Active Metal-Organic Frameworks

Magnetometric Characterization of Intermediates in the Solid-State Electrochemistry of Redox-Active Metal-Organic Frameworks

Published on: June 9, 2023

Formulation of magnetically perturbed time-dependent density functional theory.

Michael Seth¹, Tom Ziegler

¹Department of Chemistry, University of Calgary, University Drive 2500, Calgary AB T2N-1N4, Canada.

The Journal of Chemical Physics

|October 9, 2007

Summary

This study introduces magnetically perturbed time-dependent density functional theory (TDDFT) to accurately calculate excitation energies and transition densities. The new formulation corrects conventional TDDFT for magnetic fields and spin-orbit coupling effects.

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Magnetometric Characterization of Intermediates in the Solid-State Electrochemistry of Redox-Active Metal-Organic Frameworks

Magnetometric Characterization of Intermediates in the Solid-State Electrochemistry of Redox-Active Metal-Organic Frameworks

Published on: June 9, 2023

Magnetically Induced Rotating Rayleigh-Taylor Instability

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Published on: March 3, 2017

Area of Science:

Quantum Chemistry
Computational Physics
Theoretical Chemistry

Background:

Time-dependent density functional theory (TDDFT) is a powerful method for calculating electronic excitation properties.
Incorporating external fields like magnetic fields and spin-orbit coupling into TDDFT calculations presents significant theoretical challenges.

Purpose of the Study:

To develop a theoretical framework for including static magnetic fields and spin-orbit coupling in TDDFT.
To derive and simplify equations for corrections to excitation energies and transition densities.

Main Methods:

Derivation of a time-dependent density functional theory (TDDFT) formulation with a static imaginary perturbation.
Application of a perturbational approach to obtain corrections to excitation energies and transition densities.
Simplification of the derived perturbed TDDFT equations.

Main Results:

Formulation of magnetically perturbed TDDFT.
Derivation of equations for first- and second-order corrections to excitation energy.
Derivation of equations for first-order corrections to transition density.

Conclusions:

The developed magnetically perturbed TDDFT allows for the inclusion of static magnetic fields and spin-orbit coupling in conventional TDDFT calculations.
The formulation provides a pathway to more accurate electronic structure calculations in the presence of external magnetic influences.