Now work on the right side of (4) and show that The last of Maxwell's Laws is Ampere's Law with a crucial generalization first proposed by Maxwell. The French physicist Andre-Marie Ampere (1775-1836) discovered that an electrical current (moving charges) produces a magnetic field. If C is a closed curve that encloses a current I, then Ampere's Law states that the magnetic permeability of the medium. The law says that the magnetic field induced by the current, when integrated along the closed loop C is proportional to the current enclosed by the loop. Maxwell understood that the Ampere's Law does not apply to all situations and saw the need for another term - called the displacement current - that also contributes to the induced magnetic field. Maxwell's Law, which extends Ampere's Law, is where S is any surface with C as its boundary. Notice the important parallel between the changing flux of the electric field in this law. Suppose the enclosed current I can be expressed in terms of a current density J as Use stokes, Theorem to show that the differential form of this low is
