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.

Maxwell's four equations explain the fundamentals of electromagnetism.

Applying the divergence theorem to Gauss's law and rewriting the enclosed charge in terms of the total charge density gives the differential form of Gauss's law.

Similarly, applying divergence theorem to Gauss's law for the magnetic field gives the differential form of Gauss's law in magnetism.

Furthermore, rearranging Faraday's law and applying Stoke's theorem to it gives Faraday's law in the differential form.

Consider the Ampère-Maxwell equation, in which the enclosed current can be expressed in terms of the integral of current density.

Now, rearranging the terms and applying Stoke's theorem to it gives the differential form of the Ampère-Maxwell equation.

Grouping the terms of electric and magnetic fields on one side and the sources producing these fields on the other side suggests that all electromagnetic fields are produced by charges and currents.

The integral form of Maxwell's equation applies to fields in a region containing charge or current, whereas the differential form applies at a given point with charge and current densities.