Simply Lecture

The Maxwell fourth equation is called "the modified Ampere's circuital law". Statement: It states that the line integral of the magnetic field H around any closed part or circuit is equal to the current enclosed by the path. Differential form (without modification): That is, $$\int H.dL = I$$ Let the current be distributed through the current with current density J, then: $$I = \int J. ds$$ This implies that: $$\int H.dL = \int J.ds$$ .........(9) Applying Stokes theorem to the LHS of equ(9) to change line integral to surface integral we have: $$\int_{s} (\nabla X H).ds = \int_{s} J.ds$$ Since, two surface integrals are equal only if their integrands are equal. Thus, $$\nabla X H =J$$ .........(10) Equ(10) is the differential form of Maxwell fourth equation (without modification) Take divergence of equ(10) $$\nabla.(\nabla X H) = \nabla.J$$ Since, the divergence of the curl of a vector is zero.  Therefore, $$\nabla.(\nabla X H) = 0$$ It means that $...