Differential Form of Maxwell's Equations

The differential form of Gauss's law is derived by applying the divergence theorem to the integral form of Gauss's law and rewriting the enclosed charge in terms of charge density. This mathematical transformation converts the integral equation, which applies to a region, into a differential equation that describes the electric field at individual points. The result relates the divergence of the electric field directly to the charge density at that point.

The integral form of Maxwell's equations applies to fields in a region containing charge or current, providing information about field behavior over an entire surface or volume. The differential form applies at a given point with charge and current densities, allowing study of how field vectors interact and relate to source densities at individual locations. Differential forms reveal spatial and temporal variations of fields at specific points.

Faraday's law in differential form is obtained by rearranging the integral form and applying Stoke's theorem. This mathematical operation converts the line integral around a closed loop into a surface integral, ultimately yielding a differential equation. The resulting equation relates the curl of the electric field to the rate of change of the magnetic field at a given point.

Stoke's theorem is applied to the Ampère-Maxwell equation after expressing enclosed current in terms of current density. The theorem converts the line integral of the magnetic field into a surface integral, allowing the equation to be rearranged into differential form. This yields a relationship between the curl of the magnetic field and both current density and the time-varying electric field at a point.

Maxwell's equations demonstrate that all electromagnetic fields are produced by charges and currents. By grouping electric and magnetic field terms on one side and source terms on the other, the equations show that charges and currents generate electromagnetic fields. Conversely, the Lorentz force law indicates that these fields exert forces on charges, establishing a complete interdependence between fields and sources.

Maxwell identified logical inconsistencies in earlier experimental results and discovered that Ampère's law was incomplete. He recognized that the law failed to account for time-varying electric fields, which are essential for a complete electromagnetic theory. Maxwell's modification, adding the displacement current term, resolved these inconsistencies and unified electricity and magnetism into a coherent theoretical framework.

The differential equations relate the spatial variations of electric and magnetic field vectors at a given point to their temporal variations. They also correlate spatial variations of both fields to charge and current densities at that specific location. This allows physicists to understand how fields change in space and time simultaneously, providing insight into local electromagnetic behavior.