In this section, we derive the differential form of Amperes’ Circuital Law. In some applications, this differential equation, combined with boundary conditions associated with discontinuities in struc...In this section, we derive the differential form of Amperes’ Circuital Law. In some applications, this differential equation, combined with boundary conditions associated with discontinuities in structure and materials, can be used to solve for the magnetic field in arbitrarily complicated scenarios. A more direct reason for seeking out this differential equation is that we gain a little more insight into the relationship between current and the magnetic field.