### 30.15: 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. Maxwell discovered logical inconsistencies in these earlier results and identified the incompleteness of Ampère's law as their cause. Maxwell's equations and the Lorentz force law encompass all the laws of electricity and magnetism.

The integral forms of Maxwell's equations contain all the information about the interdependence of the field and source quantities over a given region in space. However, these equations do not permit one to study the interaction between the field vectors and their relationships with the source densities at individual points. Maxwell's equations in differential form can be derived by applying Maxwell's equations in the integral form to infinitesimal closed paths, surfaces, and volumes, such that the limit shrinks to points. The differential equations relate the spatial variations of the electric and magnetic field vectors at a given point to their temporal variations.

Furthermore, the differential form of Maxwell's equations also correlates the spatial variations of both fields to the charge and current densities at a given point. Grouping the terms of electric and magnetic fields on one side and the sources producing these fields on the other suggests that charges and currents produce all electromagnetic fields. Maxwell's equations show that charges produce electromagnetic fields, and the force laws state that fields affect the charges.

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